Theorem dfac12lem2 9416

Description: Lemma for dfac12 9421. (Contributed by Mario Carneiro, 29-May-2015.)

Hypotheses

Ref	Expression
dfac12.1	⊢ (𝜑 → 𝐴 ∈ On)
dfac12.3	⊢ (𝜑 → 𝐹:𝒫 (har‘(𝑅1‘𝐴))–1-1→On)
dfac12.4	⊢ 𝐺 = recs((𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = ∪ dom 𝑥, ((suc ∪ ran ∪ ran 𝑥 ·o (rank‘𝑦)) +o ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) “ 𝑦))))))
dfac12.5	⊢ (𝜑 → 𝐶 ∈ On)
dfac12.h	⊢ 𝐻 = (◡OrdIso( E , ran (𝐺‘∪ 𝐶)) ∘ (𝐺‘∪ 𝐶))
dfac12.6	⊢ (𝜑 → 𝐶 ⊆ 𝐴)
dfac12.8	⊢ (𝜑 → ∀𝑧 ∈ 𝐶 (𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On)

Assertion

Ref	Expression
dfac12lem2	⊢ (𝜑 → (𝐺‘𝐶):(𝑅1‘𝐶)–1-1→On)

Distinct variable groups:   𝑦,𝑧,𝐴   𝑥,𝑦,𝑧,𝐶   𝑥,𝐺,𝑦,𝑧   𝜑,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝑦,𝐻,𝑧

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐻(𝑥)

Proof of Theorem dfac12lem2

Step	Hyp	Ref	Expression
1		dfac12.4	. . . . . . . . . . . . . 14 ⊢ 𝐺 = recs((𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = ∪ dom 𝑥, ((suc ∪ ran ∪ ran 𝑥 ·o (rank‘𝑦)) +o ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) “ 𝑦))))))
2	1	tfr1 7885	. . . . . . . . . . . . 13 ⊢ 𝐺 Fn On
3		fnfun 6323	. . . . . . . . . . . . 13 ⊢ (𝐺 Fn On → Fun 𝐺)
4	2, 3	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun 𝐺
5		dfac12.5	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ∈ On)
6		funimaexg 6310	. . . . . . . . . . . 12 ⊢ ((Fun 𝐺 ∧ 𝐶 ∈ On) → (𝐺 “ 𝐶) ∈ V)
7	4, 5, 6	sylancr 587	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺 “ 𝐶) ∈ V)
8		uniexg 7325	. . . . . . . . . . 11 ⊢ ((𝐺 “ 𝐶) ∈ V → ∪ (𝐺 “ 𝐶) ∈ V)
9		rnexg 7470	. . . . . . . . . . 11 ⊢ (∪ (𝐺 “ 𝐶) ∈ V → ran ∪ (𝐺 “ 𝐶) ∈ V)
10	7, 8, 9	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ran ∪ (𝐺 “ 𝐶) ∈ V)
11		dfac12.8	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑧 ∈ 𝐶 (𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On)
12		f1f 6443	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On → (𝐺‘𝑧):(𝑅1‘𝑧)⟶On)
13		fssxp 6402	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺‘𝑧):(𝑅1‘𝑧)⟶On → (𝐺‘𝑧) ⊆ ((𝑅1‘𝑧) × On))
14		ssv 3912	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑅1‘𝑧) ⊆ V
15		xpss1 5462	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅1‘𝑧) ⊆ V → ((𝑅1‘𝑧) × On) ⊆ (V × On))
16	14, 15	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅1‘𝑧) × On) ⊆ (V × On)
17		sstr 3897	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐺‘𝑧) ⊆ ((𝑅1‘𝑧) × On) ∧ ((𝑅1‘𝑧) × On) ⊆ (V × On)) → (𝐺‘𝑧) ⊆ (V × On))
18	16, 17	mpan2 687	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺‘𝑧) ⊆ ((𝑅1‘𝑧) × On) → (𝐺‘𝑧) ⊆ (V × On))
19		fvex 6551	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐺‘𝑧) ∈ V
20	19	elpw 4459	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺‘𝑧) ∈ 𝒫 (V × On) ↔ (𝐺‘𝑧) ⊆ (V × On))
21	18, 20	sylibr 235	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺‘𝑧) ⊆ ((𝑅1‘𝑧) × On) → (𝐺‘𝑧) ∈ 𝒫 (V × On))
22	12, 13, 21	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On → (𝐺‘𝑧) ∈ 𝒫 (V × On))
23	22	ralimi 3127	. . . . . . . . . . . . . . 15 ⊢ (∀𝑧 ∈ 𝐶 (𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On → ∀𝑧 ∈ 𝐶 (𝐺‘𝑧) ∈ 𝒫 (V × On))
24	11, 23	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑧 ∈ 𝐶 (𝐺‘𝑧) ∈ 𝒫 (V × On))
25		onss 7361	. . . . . . . . . . . . . . . . 17 ⊢ (𝐶 ∈ On → 𝐶 ⊆ On)
26	5, 25	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐶 ⊆ On)
27		fndm 6325	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 Fn On → dom 𝐺 = On)
28	2, 27	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ dom 𝐺 = On
29	26, 28	syl6sseqr 3939	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐶 ⊆ dom 𝐺)
30		funimass4 6598	. . . . . . . . . . . . . . 15 ⊢ ((Fun 𝐺 ∧ 𝐶 ⊆ dom 𝐺) → ((𝐺 “ 𝐶) ⊆ 𝒫 (V × On) ↔ ∀𝑧 ∈ 𝐶 (𝐺‘𝑧) ∈ 𝒫 (V × On)))
31	4, 29, 30	sylancr 587	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐺 “ 𝐶) ⊆ 𝒫 (V × On) ↔ ∀𝑧 ∈ 𝐶 (𝐺‘𝑧) ∈ 𝒫 (V × On)))
32	24, 31	mpbird 258	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺 “ 𝐶) ⊆ 𝒫 (V × On))
33		sspwuni 4921	. . . . . . . . . . . . 13 ⊢ ((𝐺 “ 𝐶) ⊆ 𝒫 (V × On) ↔ ∪ (𝐺 “ 𝐶) ⊆ (V × On))
34	32, 33	sylib 219	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ (𝐺 “ 𝐶) ⊆ (V × On))
35		rnss 5691	. . . . . . . . . . . 12 ⊢ (∪ (𝐺 “ 𝐶) ⊆ (V × On) → ran ∪ (𝐺 “ 𝐶) ⊆ ran (V × On))
36	34, 35	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ran ∪ (𝐺 “ 𝐶) ⊆ ran (V × On))
37		rnxpss 5905	. . . . . . . . . . 11 ⊢ ran (V × On) ⊆ On
38	36, 37	syl6ss 3901	. . . . . . . . . 10 ⊢ (𝜑 → ran ∪ (𝐺 “ 𝐶) ⊆ On)
39		ssonuni 7357	. . . . . . . . . 10 ⊢ (ran ∪ (𝐺 “ 𝐶) ∈ V → (ran ∪ (𝐺 “ 𝐶) ⊆ On → ∪ ran ∪ (𝐺 “ 𝐶) ∈ On))
40	10, 38, 39	sylc 65	. . . . . . . . 9 ⊢ (𝜑 → ∪ ran ∪ (𝐺 “ 𝐶) ∈ On)
41		suceloni 7384	. . . . . . . . 9 ⊢ (∪ ran ∪ (𝐺 “ 𝐶) ∈ On → suc ∪ ran ∪ (𝐺 “ 𝐶) ∈ On)
42	40, 41	syl 17	. . . . . . . 8 ⊢ (𝜑 → suc ∪ ran ∪ (𝐺 “ 𝐶) ∈ On)
43	42	ad2antrr 722	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → suc ∪ ran ∪ (𝐺 “ 𝐶) ∈ On)
44		rankon 9070	. . . . . . 7 ⊢ (rank‘𝑦) ∈ On
45		omcl 8012	. . . . . . 7 ⊢ ((suc ∪ ran ∪ (𝐺 “ 𝐶) ∈ On ∧ (rank‘𝑦) ∈ On) → (suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) ∈ On)
46	43, 44, 45	sylancl 586	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → (suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) ∈ On)
47		fveq2 6538	. . . . . . . . . . 11 ⊢ (𝑧 = suc (rank‘𝑦) → (𝐺‘𝑧) = (𝐺‘suc (rank‘𝑦)))
48		f1eq1 6438	. . . . . . . . . . 11 ⊢ ((𝐺‘𝑧) = (𝐺‘suc (rank‘𝑦)) → ((𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On ↔ (𝐺‘suc (rank‘𝑦)):(𝑅1‘𝑧)–1-1→On))
49	47, 48	syl 17	. . . . . . . . . 10 ⊢ (𝑧 = suc (rank‘𝑦) → ((𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On ↔ (𝐺‘suc (rank‘𝑦)):(𝑅1‘𝑧)–1-1→On))
50		fveq2 6538	. . . . . . . . . . 11 ⊢ (𝑧 = suc (rank‘𝑦) → (𝑅1‘𝑧) = (𝑅1‘suc (rank‘𝑦)))
51		f1eq2 6439	. . . . . . . . . . 11 ⊢ ((𝑅1‘𝑧) = (𝑅1‘suc (rank‘𝑦)) → ((𝐺‘suc (rank‘𝑦)):(𝑅1‘𝑧)–1-1→On ↔ (𝐺‘suc (rank‘𝑦)):(𝑅1‘suc (rank‘𝑦))–1-1→On))
52	50, 51	syl 17	. . . . . . . . . 10 ⊢ (𝑧 = suc (rank‘𝑦) → ((𝐺‘suc (rank‘𝑦)):(𝑅1‘𝑧)–1-1→On ↔ (𝐺‘suc (rank‘𝑦)):(𝑅1‘suc (rank‘𝑦))–1-1→On))
53	49, 52	bitrd 280	. . . . . . . . 9 ⊢ (𝑧 = suc (rank‘𝑦) → ((𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On ↔ (𝐺‘suc (rank‘𝑦)):(𝑅1‘suc (rank‘𝑦))–1-1→On))
54	11	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → ∀𝑧 ∈ 𝐶 (𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On)
55		rankr1ai 9073	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝑅1‘𝐶) → (rank‘𝑦) ∈ 𝐶)
56	55	ad2antlr 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → (rank‘𝑦) ∈ 𝐶)
57		simpr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → 𝐶 = ∪ 𝐶)
58	56, 57	eleqtrd 2885	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → (rank‘𝑦) ∈ ∪ 𝐶)
59		eloni 6076	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ On → Ord 𝐶)
60	5, 59	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → Ord 𝐶)
61	60	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → Ord 𝐶)
62		ordsucuniel 7395	. . . . . . . . . . 11 ⊢ (Ord 𝐶 → ((rank‘𝑦) ∈ ∪ 𝐶 ↔ suc (rank‘𝑦) ∈ 𝐶))
63	61, 62	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → ((rank‘𝑦) ∈ ∪ 𝐶 ↔ suc (rank‘𝑦) ∈ 𝐶))
64	58, 63	mpbid 233	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → suc (rank‘𝑦) ∈ 𝐶)
65	53, 54, 64	rspcdva 3565	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → (𝐺‘suc (rank‘𝑦)):(𝑅1‘suc (rank‘𝑦))–1-1→On)
66		f1f 6443	. . . . . . . 8 ⊢ ((𝐺‘suc (rank‘𝑦)):(𝑅1‘suc (rank‘𝑦))–1-1→On → (𝐺‘suc (rank‘𝑦)):(𝑅1‘suc (rank‘𝑦))⟶On)
67	65, 66	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → (𝐺‘suc (rank‘𝑦)):(𝑅1‘suc (rank‘𝑦))⟶On)
68		r1elwf 9071	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑅1‘𝐶) → 𝑦 ∈ ∪ (𝑅1 “ On))
69	68	ad2antlr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → 𝑦 ∈ ∪ (𝑅1 “ On))
70		rankidb 9075	. . . . . . . 8 ⊢ (𝑦 ∈ ∪ (𝑅1 “ On) → 𝑦 ∈ (𝑅1‘suc (rank‘𝑦)))
71	69, 70	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → 𝑦 ∈ (𝑅1‘suc (rank‘𝑦)))
72	67, 71	ffvelrnd 6717	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → ((𝐺‘suc (rank‘𝑦))‘𝑦) ∈ On)
73		oacl 8011	. . . . . 6 ⊢ (((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) ∈ On ∧ ((𝐺‘suc (rank‘𝑦))‘𝑦) ∈ On) → ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o ((𝐺‘suc (rank‘𝑦))‘𝑦)) ∈ On)
74	46, 72, 73	syl2anc 584	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o ((𝐺‘suc (rank‘𝑦))‘𝑦)) ∈ On)
75		dfac12.3	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝒫 (har‘(𝑅1‘𝐴))–1-1→On)
76		f1f 6443	. . . . . . . 8 ⊢ (𝐹:𝒫 (har‘(𝑅1‘𝐴))–1-1→On → 𝐹:𝒫 (har‘(𝑅1‘𝐴))⟶On)
77	75, 76	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝒫 (har‘(𝑅1‘𝐴))⟶On)
78	77	ad2antrr 722	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝐹:𝒫 (har‘(𝑅1‘𝐴))⟶On)
79		imassrn 5817	. . . . . . . 8 ⊢ (𝐻 “ 𝑦) ⊆ ran 𝐻
80		fvex 6551	. . . . . . . . . . . . . . 15 ⊢ (𝐺‘∪ 𝐶) ∈ V
81	80	rnex 7473	. . . . . . . . . . . . . 14 ⊢ ran (𝐺‘∪ 𝐶) ∈ V
82		fveq2 6538	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = ∪ 𝐶 → (𝐺‘𝑧) = (𝐺‘∪ 𝐶))
83		f1eq1 6438	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺‘𝑧) = (𝐺‘∪ 𝐶) → ((𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On ↔ (𝐺‘∪ 𝐶):(𝑅1‘𝑧)–1-1→On))
84	82, 83	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = ∪ 𝐶 → ((𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On ↔ (𝐺‘∪ 𝐶):(𝑅1‘𝑧)–1-1→On))
85		fveq2 6538	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = ∪ 𝐶 → (𝑅1‘𝑧) = (𝑅1‘∪ 𝐶))
86		f1eq2 6439	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅1‘𝑧) = (𝑅1‘∪ 𝐶) → ((𝐺‘∪ 𝐶):(𝑅1‘𝑧)–1-1→On ↔ (𝐺‘∪ 𝐶):(𝑅1‘∪ 𝐶)–1-1→On))
87	85, 86	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = ∪ 𝐶 → ((𝐺‘∪ 𝐶):(𝑅1‘𝑧)–1-1→On ↔ (𝐺‘∪ 𝐶):(𝑅1‘∪ 𝐶)–1-1→On))
88	84, 87	bitrd 280	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = ∪ 𝐶 → ((𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On ↔ (𝐺‘∪ 𝐶):(𝑅1‘∪ 𝐶)–1-1→On))
89	11	ad2antrr 722	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ∀𝑧 ∈ 𝐶 (𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On)
90	5	ad2antrr 722	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝐶 ∈ On)
91		onuni 7364	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐶 ∈ On → ∪ 𝐶 ∈ On)
92		sucidg 6144	. . . . . . . . . . . . . . . . . . 19 ⊢ (∪ 𝐶 ∈ On → ∪ 𝐶 ∈ suc ∪ 𝐶)
93	90, 91, 92	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ∪ 𝐶 ∈ suc ∪ 𝐶)
94	60	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) → Ord 𝐶)
95		orduniorsuc 7401	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Ord 𝐶 → (𝐶 = ∪ 𝐶 ∨ 𝐶 = suc ∪ 𝐶))
96	94, 95	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) → (𝐶 = ∪ 𝐶 ∨ 𝐶 = suc ∪ 𝐶))
97	96	orcanai 997	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝐶 = suc ∪ 𝐶)
98	93, 97	eleqtrrd 2886	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ∪ 𝐶 ∈ 𝐶)
99	88, 89, 98	rspcdva 3565	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (𝐺‘∪ 𝐶):(𝑅1‘∪ 𝐶)–1-1→On)
100		f1f 6443	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺‘∪ 𝐶):(𝑅1‘∪ 𝐶)–1-1→On → (𝐺‘∪ 𝐶):(𝑅1‘∪ 𝐶)⟶On)
101		frn 6388	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺‘∪ 𝐶):(𝑅1‘∪ 𝐶)⟶On → ran (𝐺‘∪ 𝐶) ⊆ On)
102	99, 100, 101	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ran (𝐺‘∪ 𝐶) ⊆ On)
103		epweon 7353	. . . . . . . . . . . . . . 15 ⊢ E We On
104		wess 5430	. . . . . . . . . . . . . . 15 ⊢ (ran (𝐺‘∪ 𝐶) ⊆ On → ( E We On → E We ran (𝐺‘∪ 𝐶)))
105	102, 103, 104	mpisyl 21	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → E We ran (𝐺‘∪ 𝐶))
106		eqid 2795	. . . . . . . . . . . . . . 15 ⊢ OrdIso( E , ran (𝐺‘∪ 𝐶)) = OrdIso( E , ran (𝐺‘∪ 𝐶))
107	106	oiiso 8847	. . . . . . . . . . . . . 14 ⊢ ((ran (𝐺‘∪ 𝐶) ∈ V ∧ E We ran (𝐺‘∪ 𝐶)) → OrdIso( E , ran (𝐺‘∪ 𝐶)) Isom E , E (dom OrdIso( E , ran (𝐺‘∪ 𝐶)), ran (𝐺‘∪ 𝐶)))
108	81, 105, 107	sylancr 587	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → OrdIso( E , ran (𝐺‘∪ 𝐶)) Isom E , E (dom OrdIso( E , ran (𝐺‘∪ 𝐶)), ran (𝐺‘∪ 𝐶)))
109		isof1o 6939	. . . . . . . . . . . . 13 ⊢ (OrdIso( E , ran (𝐺‘∪ 𝐶)) Isom E , E (dom OrdIso( E , ran (𝐺‘∪ 𝐶)), ran (𝐺‘∪ 𝐶)) → OrdIso( E , ran (𝐺‘∪ 𝐶)):dom OrdIso( E , ran (𝐺‘∪ 𝐶))–1-1-onto→ran (𝐺‘∪ 𝐶))
110		f1ocnv 6495	. . . . . . . . . . . . 13 ⊢ (OrdIso( E , ran (𝐺‘∪ 𝐶)):dom OrdIso( E , ran (𝐺‘∪ 𝐶))–1-1-onto→ran (𝐺‘∪ 𝐶) → ◡OrdIso( E , ran (𝐺‘∪ 𝐶)):ran (𝐺‘∪ 𝐶)–1-1-onto→dom OrdIso( E , ran (𝐺‘∪ 𝐶)))
111		f1of1 6482	. . . . . . . . . . . . 13 ⊢ (◡OrdIso( E , ran (𝐺‘∪ 𝐶)):ran (𝐺‘∪ 𝐶)–1-1-onto→dom OrdIso( E , ran (𝐺‘∪ 𝐶)) → ◡OrdIso( E , ran (𝐺‘∪ 𝐶)):ran (𝐺‘∪ 𝐶)–1-1→dom OrdIso( E , ran (𝐺‘∪ 𝐶)))
112	108, 109, 110, 111	4syl 19	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ◡OrdIso( E , ran (𝐺‘∪ 𝐶)):ran (𝐺‘∪ 𝐶)–1-1→dom OrdIso( E , ran (𝐺‘∪ 𝐶)))
113		f1f1orn 6494	. . . . . . . . . . . . 13 ⊢ ((𝐺‘∪ 𝐶):(𝑅1‘∪ 𝐶)–1-1→On → (𝐺‘∪ 𝐶):(𝑅1‘∪ 𝐶)–1-1-onto→ran (𝐺‘∪ 𝐶))
114		f1of1 6482	. . . . . . . . . . . . 13 ⊢ ((𝐺‘∪ 𝐶):(𝑅1‘∪ 𝐶)–1-1-onto→ran (𝐺‘∪ 𝐶) → (𝐺‘∪ 𝐶):(𝑅1‘∪ 𝐶)–1-1→ran (𝐺‘∪ 𝐶))
115	99, 113, 114	3syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (𝐺‘∪ 𝐶):(𝑅1‘∪ 𝐶)–1-1→ran (𝐺‘∪ 𝐶))
116		f1co 6453	. . . . . . . . . . . 12 ⊢ ((◡OrdIso( E , ran (𝐺‘∪ 𝐶)):ran (𝐺‘∪ 𝐶)–1-1→dom OrdIso( E , ran (𝐺‘∪ 𝐶)) ∧ (𝐺‘∪ 𝐶):(𝑅1‘∪ 𝐶)–1-1→ran (𝐺‘∪ 𝐶)) → (◡OrdIso( E , ran (𝐺‘∪ 𝐶)) ∘ (𝐺‘∪ 𝐶)):(𝑅1‘∪ 𝐶)–1-1→dom OrdIso( E , ran (𝐺‘∪ 𝐶)))
117	112, 115, 116	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (◡OrdIso( E , ran (𝐺‘∪ 𝐶)) ∘ (𝐺‘∪ 𝐶)):(𝑅1‘∪ 𝐶)–1-1→dom OrdIso( E , ran (𝐺‘∪ 𝐶)))
118		dfac12.h	. . . . . . . . . . . 12 ⊢ 𝐻 = (◡OrdIso( E , ran (𝐺‘∪ 𝐶)) ∘ (𝐺‘∪ 𝐶))
119		f1eq1 6438	. . . . . . . . . . . 12 ⊢ (𝐻 = (◡OrdIso( E , ran (𝐺‘∪ 𝐶)) ∘ (𝐺‘∪ 𝐶)) → (𝐻:(𝑅1‘∪ 𝐶)–1-1→dom OrdIso( E , ran (𝐺‘∪ 𝐶)) ↔ (◡OrdIso( E , ran (𝐺‘∪ 𝐶)) ∘ (𝐺‘∪ 𝐶)):(𝑅1‘∪ 𝐶)–1-1→dom OrdIso( E , ran (𝐺‘∪ 𝐶))))
120	118, 119	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐻:(𝑅1‘∪ 𝐶)–1-1→dom OrdIso( E , ran (𝐺‘∪ 𝐶)) ↔ (◡OrdIso( E , ran (𝐺‘∪ 𝐶)) ∘ (𝐺‘∪ 𝐶)):(𝑅1‘∪ 𝐶)–1-1→dom OrdIso( E , ran (𝐺‘∪ 𝐶)))
121	117, 120	sylibr 235	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝐻:(𝑅1‘∪ 𝐶)–1-1→dom OrdIso( E , ran (𝐺‘∪ 𝐶)))
122		f1f 6443	. . . . . . . . . 10 ⊢ (𝐻:(𝑅1‘∪ 𝐶)–1-1→dom OrdIso( E , ran (𝐺‘∪ 𝐶)) → 𝐻:(𝑅1‘∪ 𝐶)⟶dom OrdIso( E , ran (𝐺‘∪ 𝐶)))
123		frn 6388	. . . . . . . . . 10 ⊢ (𝐻:(𝑅1‘∪ 𝐶)⟶dom OrdIso( E , ran (𝐺‘∪ 𝐶)) → ran 𝐻 ⊆ dom OrdIso( E , ran (𝐺‘∪ 𝐶)))
124	121, 122, 123	3syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ran 𝐻 ⊆ dom OrdIso( E , ran (𝐺‘∪ 𝐶)))
125		harcl 8871	. . . . . . . . . . 11 ⊢ (har‘(𝑅1‘𝐴)) ∈ On
126	125	onordi 6170	. . . . . . . . . 10 ⊢ Ord (har‘(𝑅1‘𝐴))
127	106	oion 8846	. . . . . . . . . . . 12 ⊢ (ran (𝐺‘∪ 𝐶) ∈ V → dom OrdIso( E , ran (𝐺‘∪ 𝐶)) ∈ On)
128	81, 127	mp1i 13	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → dom OrdIso( E , ran (𝐺‘∪ 𝐶)) ∈ On)
129	106	oien 8848	. . . . . . . . . . . . 13 ⊢ ((ran (𝐺‘∪ 𝐶) ∈ V ∧ E We ran (𝐺‘∪ 𝐶)) → dom OrdIso( E , ran (𝐺‘∪ 𝐶)) ≈ ran (𝐺‘∪ 𝐶))
130	81, 105, 129	sylancr 587	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → dom OrdIso( E , ran (𝐺‘∪ 𝐶)) ≈ ran (𝐺‘∪ 𝐶))
131		fvex 6551	. . . . . . . . . . . . . . 15 ⊢ (𝑅1‘∪ 𝐶) ∈ V
132	131	f1oen 8378	. . . . . . . . . . . . . 14 ⊢ ((𝐺‘∪ 𝐶):(𝑅1‘∪ 𝐶)–1-1-onto→ran (𝐺‘∪ 𝐶) → (𝑅1‘∪ 𝐶) ≈ ran (𝐺‘∪ 𝐶))
133		ensym 8406	. . . . . . . . . . . . . 14 ⊢ ((𝑅1‘∪ 𝐶) ≈ ran (𝐺‘∪ 𝐶) → ran (𝐺‘∪ 𝐶) ≈ (𝑅1‘∪ 𝐶))
134	99, 113, 132, 133	4syl 19	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ran (𝐺‘∪ 𝐶) ≈ (𝑅1‘∪ 𝐶))
135		fvex 6551	. . . . . . . . . . . . . 14 ⊢ (𝑅1‘𝐴) ∈ V
136		dfac12.1	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ∈ On)
137	136	ad2antrr 722	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝐴 ∈ On)
138		dfac12.6	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
139	138	ad2antrr 722	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝐶 ⊆ 𝐴)
140	139, 98	sseldd 3890	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ∪ 𝐶 ∈ 𝐴)
141		r1ord2 9056	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ On → (∪ 𝐶 ∈ 𝐴 → (𝑅1‘∪ 𝐶) ⊆ (𝑅1‘𝐴)))
142	137, 140, 141	sylc 65	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (𝑅1‘∪ 𝐶) ⊆ (𝑅1‘𝐴))
143		ssdomg 8403	. . . . . . . . . . . . . 14 ⊢ ((𝑅1‘𝐴) ∈ V → ((𝑅1‘∪ 𝐶) ⊆ (𝑅1‘𝐴) → (𝑅1‘∪ 𝐶) ≼ (𝑅1‘𝐴)))
144	135, 142, 143	mpsyl 68	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (𝑅1‘∪ 𝐶) ≼ (𝑅1‘𝐴))
145		endomtr 8415	. . . . . . . . . . . . 13 ⊢ ((ran (𝐺‘∪ 𝐶) ≈ (𝑅1‘∪ 𝐶) ∧ (𝑅1‘∪ 𝐶) ≼ (𝑅1‘𝐴)) → ran (𝐺‘∪ 𝐶) ≼ (𝑅1‘𝐴))
146	134, 144, 145	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ran (𝐺‘∪ 𝐶) ≼ (𝑅1‘𝐴))
147		endomtr 8415	. . . . . . . . . . . 12 ⊢ ((dom OrdIso( E , ran (𝐺‘∪ 𝐶)) ≈ ran (𝐺‘∪ 𝐶) ∧ ran (𝐺‘∪ 𝐶) ≼ (𝑅1‘𝐴)) → dom OrdIso( E , ran (𝐺‘∪ 𝐶)) ≼ (𝑅1‘𝐴))
148	130, 146, 147	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → dom OrdIso( E , ran (𝐺‘∪ 𝐶)) ≼ (𝑅1‘𝐴))
149		elharval 8873	. . . . . . . . . . 11 ⊢ (dom OrdIso( E , ran (𝐺‘∪ 𝐶)) ∈ (har‘(𝑅1‘𝐴)) ↔ (dom OrdIso( E , ran (𝐺‘∪ 𝐶)) ∈ On ∧ dom OrdIso( E , ran (𝐺‘∪ 𝐶)) ≼ (𝑅1‘𝐴)))
150	128, 148, 149	sylanbrc 583	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → dom OrdIso( E , ran (𝐺‘∪ 𝐶)) ∈ (har‘(𝑅1‘𝐴)))
151		ordelss 6082	. . . . . . . . . 10 ⊢ ((Ord (har‘(𝑅1‘𝐴)) ∧ dom OrdIso( E , ran (𝐺‘∪ 𝐶)) ∈ (har‘(𝑅1‘𝐴))) → dom OrdIso( E , ran (𝐺‘∪ 𝐶)) ⊆ (har‘(𝑅1‘𝐴)))
152	126, 150, 151	sylancr 587	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → dom OrdIso( E , ran (𝐺‘∪ 𝐶)) ⊆ (har‘(𝑅1‘𝐴)))
153	124, 152	sstrd 3899	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ran 𝐻 ⊆ (har‘(𝑅1‘𝐴)))
154	79, 153	syl5ss 3900	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (𝐻 “ 𝑦) ⊆ (har‘(𝑅1‘𝐴)))
155		fvex 6551	. . . . . . . 8 ⊢ (har‘(𝑅1‘𝐴)) ∈ V
156	155	elpw2 5139	. . . . . . 7 ⊢ ((𝐻 “ 𝑦) ∈ 𝒫 (har‘(𝑅1‘𝐴)) ↔ (𝐻 “ 𝑦) ⊆ (har‘(𝑅1‘𝐴)))
157	154, 156	sylibr 235	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (𝐻 “ 𝑦) ∈ 𝒫 (har‘(𝑅1‘𝐴)))
158	78, 157	ffvelrnd 6717	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (𝐹‘(𝐻 “ 𝑦)) ∈ On)
159	74, 158	ifclda 4415	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) → if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦))) ∈ On)
160	159	ex 413	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝑅1‘𝐶) → if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦))) ∈ On))
161		iftrue 4387	. . . . . . . 8 ⊢ (𝐶 = ∪ 𝐶 → if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦))) = ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o ((𝐺‘suc (rank‘𝑦))‘𝑦)))
162		iftrue 4387	. . . . . . . 8 ⊢ (𝐶 = ∪ 𝐶 → if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑧)) +o ((𝐺‘suc (rank‘𝑧))‘𝑧)), (𝐹‘(𝐻 “ 𝑧))) = ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑧)) +o ((𝐺‘suc (rank‘𝑧))‘𝑧)))
163	161, 162	eqeq12d 2810	. . . . . . 7 ⊢ (𝐶 = ∪ 𝐶 → (if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦))) = if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑧)) +o ((𝐺‘suc (rank‘𝑧))‘𝑧)), (𝐹‘(𝐻 “ 𝑧))) ↔ ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o ((𝐺‘suc (rank‘𝑦))‘𝑦)) = ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑧)) +o ((𝐺‘suc (rank‘𝑧))‘𝑧))))
164	163	adantl 482	. . . . . 6 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) → (if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦))) = if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑧)) +o ((𝐺‘suc (rank‘𝑧))‘𝑧)), (𝐹‘(𝐻 “ 𝑧))) ↔ ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o ((𝐺‘suc (rank‘𝑦))‘𝑦)) = ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑧)) +o ((𝐺‘suc (rank‘𝑧))‘𝑧))))
165	42	ad2antrr 722	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) → suc ∪ ran ∪ (𝐺 “ 𝐶) ∈ On)
166		nsuceq0 6146	. . . . . . . 8 ⊢ suc ∪ ran ∪ (𝐺 “ 𝐶) ≠ ∅
167	166	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) → suc ∪ ran ∪ (𝐺 “ 𝐶) ≠ ∅)
168	44	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) → (rank‘𝑦) ∈ On)
169		onsucuni 7399	. . . . . . . . . . 11 ⊢ (ran ∪ (𝐺 “ 𝐶) ⊆ On → ran ∪ (𝐺 “ 𝐶) ⊆ suc ∪ ran ∪ (𝐺 “ 𝐶))
170	38, 169	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ran ∪ (𝐺 “ 𝐶) ⊆ suc ∪ ran ∪ (𝐺 “ 𝐶))
171	170	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → ran ∪ (𝐺 “ 𝐶) ⊆ suc ∪ ran ∪ (𝐺 “ 𝐶))
172	26	ad2antrr 722	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → 𝐶 ⊆ On)
173		fnfvima 6860	. . . . . . . . . . . 12 ⊢ ((𝐺 Fn On ∧ 𝐶 ⊆ On ∧ suc (rank‘𝑦) ∈ 𝐶) → (𝐺‘suc (rank‘𝑦)) ∈ (𝐺 “ 𝐶))
174	2, 172, 64, 173	mp3an2i 1458	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → (𝐺‘suc (rank‘𝑦)) ∈ (𝐺 “ 𝐶))
175		elssuni 4774	. . . . . . . . . . 11 ⊢ ((𝐺‘suc (rank‘𝑦)) ∈ (𝐺 “ 𝐶) → (𝐺‘suc (rank‘𝑦)) ⊆ ∪ (𝐺 “ 𝐶))
176		rnss 5691	. . . . . . . . . . 11 ⊢ ((𝐺‘suc (rank‘𝑦)) ⊆ ∪ (𝐺 “ 𝐶) → ran (𝐺‘suc (rank‘𝑦)) ⊆ ran ∪ (𝐺 “ 𝐶))
177	174, 175, 176	3syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → ran (𝐺‘suc (rank‘𝑦)) ⊆ ran ∪ (𝐺 “ 𝐶))
178		f1fn 6444	. . . . . . . . . . . 12 ⊢ ((𝐺‘suc (rank‘𝑦)):(𝑅1‘suc (rank‘𝑦))–1-1→On → (𝐺‘suc (rank‘𝑦)) Fn (𝑅1‘suc (rank‘𝑦)))
179	65, 178	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → (𝐺‘suc (rank‘𝑦)) Fn (𝑅1‘suc (rank‘𝑦)))
180		fnfvelrn 6713	. . . . . . . . . . 11 ⊢ (((𝐺‘suc (rank‘𝑦)) Fn (𝑅1‘suc (rank‘𝑦)) ∧ 𝑦 ∈ (𝑅1‘suc (rank‘𝑦))) → ((𝐺‘suc (rank‘𝑦))‘𝑦) ∈ ran (𝐺‘suc (rank‘𝑦)))
181	179, 71, 180	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → ((𝐺‘suc (rank‘𝑦))‘𝑦) ∈ ran (𝐺‘suc (rank‘𝑦)))
182	177, 181	sseldd 3890	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → ((𝐺‘suc (rank‘𝑦))‘𝑦) ∈ ran ∪ (𝐺 “ 𝐶))
183	171, 182	sseldd 3890	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → ((𝐺‘suc (rank‘𝑦))‘𝑦) ∈ suc ∪ ran ∪ (𝐺 “ 𝐶))
184	183	adantlrr 717	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) → ((𝐺‘suc (rank‘𝑦))‘𝑦) ∈ suc ∪ ran ∪ (𝐺 “ 𝐶))
185		rankon 9070	. . . . . . . 8 ⊢ (rank‘𝑧) ∈ On
186	185	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) → (rank‘𝑧) ∈ On)
187		eleq1w 2865	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ (𝑅1‘𝐶) ↔ 𝑧 ∈ (𝑅1‘𝐶)))
188	187	anbi2d 628	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ↔ (𝜑 ∧ 𝑧 ∈ (𝑅1‘𝐶))))
189	188	anbi1d 629	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) ↔ ((𝜑 ∧ 𝑧 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶)))
190		fveq2 6538	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → (rank‘𝑦) = (rank‘𝑧))
191		suceq 6131	. . . . . . . . . . . . . 14 ⊢ ((rank‘𝑦) = (rank‘𝑧) → suc (rank‘𝑦) = suc (rank‘𝑧))
192	190, 191	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → suc (rank‘𝑦) = suc (rank‘𝑧))
193	192	fveq2d 6542	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → (𝐺‘suc (rank‘𝑦)) = (𝐺‘suc (rank‘𝑧)))
194		id 22	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → 𝑦 = 𝑧)
195	193, 194	fveq12d 6545	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → ((𝐺‘suc (rank‘𝑦))‘𝑦) = ((𝐺‘suc (rank‘𝑧))‘𝑧))
196	195	eleq1d 2867	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (((𝐺‘suc (rank‘𝑦))‘𝑦) ∈ suc ∪ ran ∪ (𝐺 “ 𝐶) ↔ ((𝐺‘suc (rank‘𝑧))‘𝑧) ∈ suc ∪ ran ∪ (𝐺 “ 𝐶)))
197	189, 196	imbi12d 346	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → ((((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → ((𝐺‘suc (rank‘𝑦))‘𝑦) ∈ suc ∪ ran ∪ (𝐺 “ 𝐶)) ↔ (((𝜑 ∧ 𝑧 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → ((𝐺‘suc (rank‘𝑧))‘𝑧) ∈ suc ∪ ran ∪ (𝐺 “ 𝐶))))
198	197, 183	chvarv 2370	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → ((𝐺‘suc (rank‘𝑧))‘𝑧) ∈ suc ∪ ran ∪ (𝐺 “ 𝐶))
199	198	adantlrl 716	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) → ((𝐺‘suc (rank‘𝑧))‘𝑧) ∈ suc ∪ ran ∪ (𝐺 “ 𝐶))
200		omopth2 8060	. . . . . . 7 ⊢ (((suc ∪ ran ∪ (𝐺 “ 𝐶) ∈ On ∧ suc ∪ ran ∪ (𝐺 “ 𝐶) ≠ ∅) ∧ ((rank‘𝑦) ∈ On ∧ ((𝐺‘suc (rank‘𝑦))‘𝑦) ∈ suc ∪ ran ∪ (𝐺 “ 𝐶)) ∧ ((rank‘𝑧) ∈ On ∧ ((𝐺‘suc (rank‘𝑧))‘𝑧) ∈ suc ∪ ran ∪ (𝐺 “ 𝐶))) → (((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o ((𝐺‘suc (rank‘𝑦))‘𝑦)) = ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑧)) +o ((𝐺‘suc (rank‘𝑧))‘𝑧)) ↔ ((rank‘𝑦) = (rank‘𝑧) ∧ ((𝐺‘suc (rank‘𝑦))‘𝑦) = ((𝐺‘suc (rank‘𝑧))‘𝑧))))
201	165, 167, 168, 184, 186, 199, 200	syl222anc 1379	. . . . . 6 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) → (((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o ((𝐺‘suc (rank‘𝑦))‘𝑦)) = ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑧)) +o ((𝐺‘suc (rank‘𝑧))‘𝑧)) ↔ ((rank‘𝑦) = (rank‘𝑧) ∧ ((𝐺‘suc (rank‘𝑦))‘𝑦) = ((𝐺‘suc (rank‘𝑧))‘𝑧))))
202	191	adantl 482	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) ∧ (rank‘𝑦) = (rank‘𝑧)) → suc (rank‘𝑦) = suc (rank‘𝑧))
203	202	fveq2d 6542	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) ∧ (rank‘𝑦) = (rank‘𝑧)) → (𝐺‘suc (rank‘𝑦)) = (𝐺‘suc (rank‘𝑧)))
204	203	fveq1d 6540	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) ∧ (rank‘𝑦) = (rank‘𝑧)) → ((𝐺‘suc (rank‘𝑦))‘𝑧) = ((𝐺‘suc (rank‘𝑧))‘𝑧))
205	204	eqeq2d 2805	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) ∧ (rank‘𝑦) = (rank‘𝑧)) → (((𝐺‘suc (rank‘𝑦))‘𝑦) = ((𝐺‘suc (rank‘𝑦))‘𝑧) ↔ ((𝐺‘suc (rank‘𝑦))‘𝑦) = ((𝐺‘suc (rank‘𝑧))‘𝑧)))
206	65	adantlrr 717	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) → (𝐺‘suc (rank‘𝑦)):(𝑅1‘suc (rank‘𝑦))–1-1→On)
207	206	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) ∧ (rank‘𝑦) = (rank‘𝑧)) → (𝐺‘suc (rank‘𝑦)):(𝑅1‘suc (rank‘𝑦))–1-1→On)
208	71	adantlrr 717	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) → 𝑦 ∈ (𝑅1‘suc (rank‘𝑦)))
209	208	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) ∧ (rank‘𝑦) = (rank‘𝑧)) → 𝑦 ∈ (𝑅1‘suc (rank‘𝑦)))
210		r1elwf 9071	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ (𝑅1‘𝐶) → 𝑧 ∈ ∪ (𝑅1 “ On))
211		rankidb 9075	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ∪ (𝑅1 “ On) → 𝑧 ∈ (𝑅1‘suc (rank‘𝑧)))
212	210, 211	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝑅1‘𝐶) → 𝑧 ∈ (𝑅1‘suc (rank‘𝑧)))
213	212	ad2antll 725	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) → 𝑧 ∈ (𝑅1‘suc (rank‘𝑧)))
214	213	ad2antrr 722	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) ∧ (rank‘𝑦) = (rank‘𝑧)) → 𝑧 ∈ (𝑅1‘suc (rank‘𝑧)))
215	202	fveq2d 6542	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) ∧ (rank‘𝑦) = (rank‘𝑧)) → (𝑅1‘suc (rank‘𝑦)) = (𝑅1‘suc (rank‘𝑧)))
216	214, 215	eleqtrrd 2886	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) ∧ (rank‘𝑦) = (rank‘𝑧)) → 𝑧 ∈ (𝑅1‘suc (rank‘𝑦)))
217		f1fveq 6885	. . . . . . . . . . 11 ⊢ (((𝐺‘suc (rank‘𝑦)):(𝑅1‘suc (rank‘𝑦))–1-1→On ∧ (𝑦 ∈ (𝑅1‘suc (rank‘𝑦)) ∧ 𝑧 ∈ (𝑅1‘suc (rank‘𝑦)))) → (((𝐺‘suc (rank‘𝑦))‘𝑦) = ((𝐺‘suc (rank‘𝑦))‘𝑧) ↔ 𝑦 = 𝑧))
218	207, 209, 216, 217	syl12anc 833	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) ∧ (rank‘𝑦) = (rank‘𝑧)) → (((𝐺‘suc (rank‘𝑦))‘𝑦) = ((𝐺‘suc (rank‘𝑦))‘𝑧) ↔ 𝑦 = 𝑧))
219	205, 218	bitr3d 282	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) ∧ (rank‘𝑦) = (rank‘𝑧)) → (((𝐺‘suc (rank‘𝑦))‘𝑦) = ((𝐺‘suc (rank‘𝑧))‘𝑧) ↔ 𝑦 = 𝑧))
220	219	biimpd 230	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) ∧ (rank‘𝑦) = (rank‘𝑧)) → (((𝐺‘suc (rank‘𝑦))‘𝑦) = ((𝐺‘suc (rank‘𝑧))‘𝑧) → 𝑦 = 𝑧))
221	220	expimpd 454	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) → (((rank‘𝑦) = (rank‘𝑧) ∧ ((𝐺‘suc (rank‘𝑦))‘𝑦) = ((𝐺‘suc (rank‘𝑧))‘𝑧)) → 𝑦 = 𝑧))
222	190, 195	jca 512	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ((rank‘𝑦) = (rank‘𝑧) ∧ ((𝐺‘suc (rank‘𝑦))‘𝑦) = ((𝐺‘suc (rank‘𝑧))‘𝑧)))
223	221, 222	impbid1 226	. . . . . 6 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) → (((rank‘𝑦) = (rank‘𝑧) ∧ ((𝐺‘suc (rank‘𝑦))‘𝑦) = ((𝐺‘suc (rank‘𝑧))‘𝑧)) ↔ 𝑦 = 𝑧))
224	164, 201, 223	3bitrd 306	. . . . 5 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) → (if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦))) = if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑧)) +o ((𝐺‘suc (rank‘𝑧))‘𝑧)), (𝐹‘(𝐻 “ 𝑧))) ↔ 𝑦 = 𝑧))
225		iffalse 4390	. . . . . . . 8 ⊢ (¬ 𝐶 = ∪ 𝐶 → if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦))) = (𝐹‘(𝐻 “ 𝑦)))
226		iffalse 4390	. . . . . . . 8 ⊢ (¬ 𝐶 = ∪ 𝐶 → if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑧)) +o ((𝐺‘suc (rank‘𝑧))‘𝑧)), (𝐹‘(𝐻 “ 𝑧))) = (𝐹‘(𝐻 “ 𝑧)))
227	225, 226	eqeq12d 2810	. . . . . . 7 ⊢ (¬ 𝐶 = ∪ 𝐶 → (if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦))) = if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑧)) +o ((𝐺‘suc (rank‘𝑧))‘𝑧)), (𝐹‘(𝐻 “ 𝑧))) ↔ (𝐹‘(𝐻 “ 𝑦)) = (𝐹‘(𝐻 “ 𝑧))))
228	227	adantl 482	. . . . . 6 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ ¬ 𝐶 = ∪ 𝐶) → (if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦))) = if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑧)) +o ((𝐺‘suc (rank‘𝑧))‘𝑧)), (𝐹‘(𝐻 “ 𝑧))) ↔ (𝐹‘(𝐻 “ 𝑦)) = (𝐹‘(𝐻 “ 𝑧))))
229	75	ad2antrr 722	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝐹:𝒫 (har‘(𝑅1‘𝐴))–1-1→On)
230	157	adantlrr 717	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ ¬ 𝐶 = ∪ 𝐶) → (𝐻 “ 𝑦) ∈ 𝒫 (har‘(𝑅1‘𝐴)))
231	188	anbi1d 629	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) ↔ ((𝜑 ∧ 𝑧 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶)))
232		imaeq2 5802	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (𝐻 “ 𝑦) = (𝐻 “ 𝑧))
233	232	eleq1d 2867	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → ((𝐻 “ 𝑦) ∈ 𝒫 (har‘(𝑅1‘𝐴)) ↔ (𝐻 “ 𝑧) ∈ 𝒫 (har‘(𝑅1‘𝐴))))
234	231, 233	imbi12d 346	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → ((((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (𝐻 “ 𝑦) ∈ 𝒫 (har‘(𝑅1‘𝐴))) ↔ (((𝜑 ∧ 𝑧 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (𝐻 “ 𝑧) ∈ 𝒫 (har‘(𝑅1‘𝐴)))))
235	234, 157	chvarv 2370	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (𝐻 “ 𝑧) ∈ 𝒫 (har‘(𝑅1‘𝐴)))
236	235	adantlrl 716	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ ¬ 𝐶 = ∪ 𝐶) → (𝐻 “ 𝑧) ∈ 𝒫 (har‘(𝑅1‘𝐴)))
237		f1fveq 6885	. . . . . . 7 ⊢ ((𝐹:𝒫 (har‘(𝑅1‘𝐴))–1-1→On ∧ ((𝐻 “ 𝑦) ∈ 𝒫 (har‘(𝑅1‘𝐴)) ∧ (𝐻 “ 𝑧) ∈ 𝒫 (har‘(𝑅1‘𝐴)))) → ((𝐹‘(𝐻 “ 𝑦)) = (𝐹‘(𝐻 “ 𝑧)) ↔ (𝐻 “ 𝑦) = (𝐻 “ 𝑧)))
238	229, 230, 236, 237	syl12anc 833	. . . . . 6 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ ¬ 𝐶 = ∪ 𝐶) → ((𝐹‘(𝐻 “ 𝑦)) = (𝐹‘(𝐻 “ 𝑧)) ↔ (𝐻 “ 𝑦) = (𝐻 “ 𝑧)))
239	121	adantlrr 717	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝐻:(𝑅1‘∪ 𝐶)–1-1→dom OrdIso( E , ran (𝐺‘∪ 𝐶)))
240		simplrl 773	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝑦 ∈ (𝑅1‘𝐶))
241	97	fveq2d 6542	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (𝑅1‘𝐶) = (𝑅1‘suc ∪ 𝐶))
242		r1suc 9045	. . . . . . . . . . . 12 ⊢ (∪ 𝐶 ∈ On → (𝑅1‘suc ∪ 𝐶) = 𝒫 (𝑅1‘∪ 𝐶))
243	90, 91, 242	3syl 18	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (𝑅1‘suc ∪ 𝐶) = 𝒫 (𝑅1‘∪ 𝐶))
244	241, 243	eqtrd 2831	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (𝑅1‘𝐶) = 𝒫 (𝑅1‘∪ 𝐶))
245	244	adantlrr 717	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ ¬ 𝐶 = ∪ 𝐶) → (𝑅1‘𝐶) = 𝒫 (𝑅1‘∪ 𝐶))
246	240, 245	eleqtrd 2885	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝑦 ∈ 𝒫 (𝑅1‘∪ 𝐶))
247	246	elpwid 4465	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝑦 ⊆ (𝑅1‘∪ 𝐶))
248		simplrr 774	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝑧 ∈ (𝑅1‘𝐶))
249	248, 245	eleqtrd 2885	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝑧 ∈ 𝒫 (𝑅1‘∪ 𝐶))
250	249	elpwid 4465	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝑧 ⊆ (𝑅1‘∪ 𝐶))
251		f1imaeq 6888	. . . . . . 7 ⊢ ((𝐻:(𝑅1‘∪ 𝐶)–1-1→dom OrdIso( E , ran (𝐺‘∪ 𝐶)) ∧ (𝑦 ⊆ (𝑅1‘∪ 𝐶) ∧ 𝑧 ⊆ (𝑅1‘∪ 𝐶))) → ((𝐻 “ 𝑦) = (𝐻 “ 𝑧) ↔ 𝑦 = 𝑧))
252	239, 247, 250, 251	syl12anc 833	. . . . . 6 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ ¬ 𝐶 = ∪ 𝐶) → ((𝐻 “ 𝑦) = (𝐻 “ 𝑧) ↔ 𝑦 = 𝑧))
253	228, 238, 252	3bitrd 306	. . . . 5 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ ¬ 𝐶 = ∪ 𝐶) → (if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦))) = if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑧)) +o ((𝐺‘suc (rank‘𝑧))‘𝑧)), (𝐹‘(𝐻 “ 𝑧))) ↔ 𝑦 = 𝑧))
254	224, 253	pm2.61dan 809	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) → (if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦))) = if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑧)) +o ((𝐺‘suc (rank‘𝑧))‘𝑧)), (𝐹‘(𝐻 “ 𝑧))) ↔ 𝑦 = 𝑧))
255	254	ex 413	. . 3 ⊢ (𝜑 → ((𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶)) → (if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦))) = if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑧)) +o ((𝐺‘suc (rank‘𝑧))‘𝑧)), (𝐹‘(𝐻 “ 𝑧))) ↔ 𝑦 = 𝑧)))
256	160, 255	dom2lem 8397	. 2 ⊢ (𝜑 → (𝑦 ∈ (𝑅1‘𝐶) ↦ if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦)))):(𝑅1‘𝐶)–1-1→On)
257	136, 75, 1, 5, 118	dfac12lem1 9415	. . 3 ⊢ (𝜑 → (𝐺‘𝐶) = (𝑦 ∈ (𝑅1‘𝐶) ↦ if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦)))))
258		f1eq1 6438	. . 3 ⊢ ((𝐺‘𝐶) = (𝑦 ∈ (𝑅1‘𝐶) ↦ if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦)))) → ((𝐺‘𝐶):(𝑅1‘𝐶)–1-1→On ↔ (𝑦 ∈ (𝑅1‘𝐶) ↦ if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦)))):(𝑅1‘𝐶)–1-1→On))
259	257, 258	syl 17	. 2 ⊢ (𝜑 → ((𝐺‘𝐶):(𝑅1‘𝐶)–1-1→On ↔ (𝑦 ∈ (𝑅1‘𝐶) ↦ if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦)))):(𝑅1‘𝐶)–1-1→On))
260	256, 259	mpbird 258	1 ⊢ (𝜑 → (𝐺‘𝐶):(𝑅1‘𝐶)–1-1→On)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 842   = wceq 1522   ∈ wcel 2081   ≠ wne 2984  ∀wral 3105  Vcvv 3437   ⊆ wss 3859  ∅c0 4211  ifcif 4381  𝒫 cpw 4453  ∪ cuni 4745   class class class wbr 4962   ↦ cmpt 5041   E cep 5352   We wwe 5401   × cxp 5441  ^◡ccnv 5442  dom cdm 5443  ran crn 5444   “ cima 5446   ∘ ccom 5447  Ord word 6065  Oncon0 6066  suc csuc 6068  Fun wfun 6219   Fn wfn 6220  ⟶wf 6221  –1-1→wf1 6222  –1-1-onto→wf1o 6224  ‘cfv 6225   Isom wiso 6226  (class class class)co 7016  recscrecs 7859   +_o coa 7950   ·_o comu 7951   ≈ cen 8354   ≼ cdom 8355  OrdIsocoi 8819  harchar 8866  𝑅₁cr1 9037  rankcrnk 9038

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-se 5403  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-isom 6234  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-oadd 7957  df-omul 7958  df-er 8139  df-en 8358  df-dom 8359  df-oi 8820  df-har 8868  df-r1 9039  df-rank 9040

This theorem is referenced by:  dfac12lem3  9417