Theorem hartogslem1 9520

Description: Lemma for hartogs 9522. (Contributed by Mario Carneiro, 14-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.)

Hypotheses

Ref	Expression
hartogslem.2	⊢ 𝐹 = {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}
hartogslem.3	⊢ 𝑅 = {⟨𝑠, 𝑡⟩ ∣ ∃𝑤 ∈ 𝑦 ∃𝑧 ∈ 𝑦 ((𝑠 = (𝑓‘𝑤) ∧ 𝑡 = (𝑓‘𝑧)) ∧ 𝑤 E 𝑧)}

Assertion

Ref	Expression
hartogslem1	⊢ (dom 𝐹 ⊆ 𝒫 (𝐴 × 𝐴) ∧ Fun 𝐹 ∧ (𝐴 ∈ 𝑉 → ran 𝐹 = {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}))

Distinct variable groups:   𝑓,𝑠,𝑡,𝑤,𝑦,𝑧   𝑓,𝑟,𝑥,𝐴,𝑦   𝑅,𝑟,𝑥   𝑉,𝑟,𝑦

Allowed substitution hints:   𝐴(𝑧, 𝑤, 𝑡, 𝑠)   𝑅(𝑦, 𝑧, 𝑤, 𝑡, 𝑓, 𝑠)   𝐹(𝑥, 𝑦, 𝑧, 𝑤, 𝑡, 𝑓, 𝑠, 𝑟)   𝑉(𝑥, 𝑧, 𝑤, 𝑡, 𝑓, 𝑠)

Proof of Theorem hartogslem1

Step	Hyp	Ref	Expression
1		hartogslem.2	. . . . 5 ⊢ 𝐹 = {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}
2	1	dmeqi 5886	. . . 4 ⊢ dom 𝐹 = dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}
3		dmopab 5897	. . . 4 ⊢ dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} = {𝑟 ∣ ∃𝑦(((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}
4	2, 3	eqtri 2784	. . 3 ⊢ dom 𝐹 = {𝑟 ∣ ∃𝑦(((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}
5		simp3 1156	. . . . . . . 8 ⊢ ((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) → 𝑟 ⊆ (dom 𝑟 × dom 𝑟))
6		simp1 1154	. . . . . . . . 9 ⊢ ((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) → dom 𝑟 ⊆ 𝐴)
7		xpss12 5666	. . . . . . . . 9 ⊢ ((dom 𝑟 ⊆ 𝐴 ∧ dom 𝑟 ⊆ 𝐴) → (dom 𝑟 × dom 𝑟) ⊆ (𝐴 × 𝐴))
8	6, 6, 7	syl2anc 596	. . . . . . . 8 ⊢ ((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) → (dom 𝑟 × dom 𝑟) ⊆ (𝐴 × 𝐴))
9	5, 8	sstrd 3941	. . . . . . 7 ⊢ ((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) → 𝑟 ⊆ (𝐴 × 𝐴))
10		velpw 4562	. . . . . . 7 ⊢ (𝑟 ∈ 𝒫 (𝐴 × 𝐴) ↔ 𝑟 ⊆ (𝐴 × 𝐴))
11	9, 10	sylibr 237	. . . . . 6 ⊢ ((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) → 𝑟 ∈ 𝒫 (𝐴 × 𝐴))
12	11	ad2antrr 739	. . . . 5 ⊢ ((((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)) → 𝑟 ∈ 𝒫 (𝐴 × 𝐴))
13	12	exlimiv 1963	. . . 4 ⊢ (∃𝑦(((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)) → 𝑟 ∈ 𝒫 (𝐴 × 𝐴))
14	13	abssi 4016	. . 3 ⊢ {𝑟 ∣ ∃𝑦(((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ⊆ 𝒫 (𝐴 × 𝐴)
15	4, 14	eqsstri 3977	. 2 ⊢ dom 𝐹 ⊆ 𝒫 (𝐴 × 𝐴)
16		funopab4 6569	. . 3 ⊢ Fun {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}
17	1	funeqi 6552	. . 3 ⊢ (Fun 𝐹 ↔ Fun {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))})
18	16, 17	mpbir 234	. 2 ⊢ Fun 𝐹
19	1	rneqi 5919	. . . 4 ⊢ ran 𝐹 = ran {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}
20		rnopab 5936	. . . 4 ⊢ ran {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} = {𝑦 ∣ ∃𝑟(((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}
21	19, 20	eqtri 2784	. . 3 ⊢ ran 𝐹 = {𝑦 ∣ ∃𝑟(((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}
22		breq1 5106	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ≼ 𝐴 ↔ 𝑦 ≼ 𝐴))
23	22	elrab 3645	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ↔ (𝑦 ∈ On ∧ 𝑦 ≼ 𝐴))
24		f1f 6770	. . . . . . . . . . . . 13 ⊢ (𝑓:𝑦–1-1→𝐴 → 𝑓:𝑦⟶𝐴)
25	24	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → 𝑓:𝑦⟶𝐴)
26	25	frnd 6710	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → ran 𝑓 ⊆ 𝐴)
27		resss 5992	. . . . . . . . . . . . . 14 ⊢ ( I ↾ ran 𝑓) ⊆ I
28		ssun2 4125	. . . . . . . . . . . . . 14 ⊢ I ⊆ (𝑅 ∪ I )
29	27, 28	sstri 3940	. . . . . . . . . . . . 13 ⊢ ( I ↾ ran 𝑓) ⊆ (𝑅 ∪ I )
30		idssxp 6043	. . . . . . . . . . . . 13 ⊢ ( I ↾ ran 𝑓) ⊆ (ran 𝑓 × ran 𝑓)
31	29, 30	ssini 4185	. . . . . . . . . . . 12 ⊢ ( I ↾ ran 𝑓) ⊆ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓))
32	31	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → ( I ↾ ran 𝑓) ⊆ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)))
33		inss2 4183	. . . . . . . . . . . 12 ⊢ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ⊆ (ran 𝑓 × ran 𝑓)
34	33	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ⊆ (ran 𝑓 × ran 𝑓))
35	26, 32, 34	3jca 1146	. . . . . . . . . 10 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → (ran 𝑓 ⊆ 𝐴 ∧ ( I ↾ ran 𝑓) ⊆ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ∧ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ⊆ (ran 𝑓 × ran 𝑓)))
36		eloni 6365	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ On → Ord 𝑦)
37		ordwe 6368	. . . . . . . . . . . . . 14 ⊢ (Ord 𝑦 → E We 𝑦)
38	36, 37	syl 18	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ On → E We 𝑦)
39	38	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → E We 𝑦)
40		f1f1orn 6828	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:𝑦–1-1→𝐴 → 𝑓:𝑦–1-1-onto→ran 𝑓)
41	40	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → 𝑓:𝑦–1-1-onto→ran 𝑓)
42		hartogslem.3	. . . . . . . . . . . . . . 15 ⊢ 𝑅 = {⟨𝑠, 𝑡⟩ ∣ ∃𝑤 ∈ 𝑦 ∃𝑧 ∈ 𝑦 ((𝑠 = (𝑓‘𝑤) ∧ 𝑡 = (𝑓‘𝑧)) ∧ 𝑤 E 𝑧)}
43		f1oiso 7351	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:𝑦–1-1-onto→ran 𝑓 ∧ 𝑅 = {⟨𝑠, 𝑡⟩ ∣ ∃𝑤 ∈ 𝑦 ∃𝑧 ∈ 𝑦 ((𝑠 = (𝑓‘𝑤) ∧ 𝑡 = (𝑓‘𝑧)) ∧ 𝑤 E 𝑧)}) → 𝑓 Isom E , 𝑅 (𝑦, ran 𝑓))
44	41, 42, 43	sylancl 598	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → 𝑓 Isom E , 𝑅 (𝑦, ran 𝑓))
45		isores2 7333	. . . . . . . . . . . . . 14 ⊢ (𝑓 Isom E , 𝑅 (𝑦, ran 𝑓) ↔ 𝑓 Isom E , (𝑅 ∩ (ran 𝑓 × ran 𝑓))(𝑦, ran 𝑓))
46	44, 45	sylib 221	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → 𝑓 Isom E , (𝑅 ∩ (ran 𝑓 × ran 𝑓))(𝑦, ran 𝑓))
47		isowe 7349	. . . . . . . . . . . . 13 ⊢ (𝑓 Isom E , (𝑅 ∩ (ran 𝑓 × ran 𝑓))(𝑦, ran 𝑓) → ( E We 𝑦 ↔ (𝑅 ∩ (ran 𝑓 × ran 𝑓)) We ran 𝑓))
48	46, 47	syl 18	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → ( E We 𝑦 ↔ (𝑅 ∩ (ran 𝑓 × ran 𝑓)) We ran 𝑓))
49	39, 48	mpbid 235	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → (𝑅 ∩ (ran 𝑓 × ran 𝑓)) We ran 𝑓)
50		weso 5642	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) We ran 𝑓 → (𝑅 ∩ (ran 𝑓 × ran 𝑓)) Or ran 𝑓)
51	49, 50	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → (𝑅 ∩ (ran 𝑓 × ran 𝑓)) Or ran 𝑓)
52		inss2 4183	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑅 ∩ (ran 𝑓 × ran 𝑓)) ⊆ (ran 𝑓 × ran 𝑓)
53	52	brel 5716	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥(𝑅 ∩ (ran 𝑓 × ran 𝑓))𝑥 → (𝑥 ∈ ran 𝑓 ∧ 𝑥 ∈ ran 𝑓))
54	53	simpld 500	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥(𝑅 ∩ (ran 𝑓 × ran 𝑓))𝑥 → 𝑥 ∈ ran 𝑓)
55		sonr 5583	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∩ (ran 𝑓 × ran 𝑓)) Or ran 𝑓 ∧ 𝑥 ∈ ran 𝑓) → ¬ 𝑥(𝑅 ∩ (ran 𝑓 × ran 𝑓))𝑥)
56	51, 54, 55	syl2an 608	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) ∧ 𝑥(𝑅 ∩ (ran 𝑓 × ran 𝑓))𝑥) → ¬ 𝑥(𝑅 ∩ (ran 𝑓 × ran 𝑓))𝑥)
57	56	pm2.01da 811	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → ¬ 𝑥(𝑅 ∩ (ran 𝑓 × ran 𝑓))𝑥)
58	57	alrimiv 1960	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → ∀𝑥 ¬ 𝑥(𝑅 ∩ (ran 𝑓 × ran 𝑓))𝑥)
59		intirr 6110	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∩ I ) = ∅ ↔ ∀𝑥 ¬ 𝑥(𝑅 ∩ (ran 𝑓 × ran 𝑓))𝑥)
60	58, 59	sylibr 237	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∩ I ) = ∅)
61		disj3 4407	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∩ I ) = ∅ ↔ (𝑅 ∩ (ran 𝑓 × ran 𝑓)) = ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ))
62	60, 61	sylib 221	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → (𝑅 ∩ (ran 𝑓 × ran 𝑓)) = ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ))
63		weeq1 5638	. . . . . . . . . . . 12 ⊢ ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) = ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) → ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) We ran 𝑓 ↔ ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) We ran 𝑓))
64	62, 63	syl 18	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) We ran 𝑓 ↔ ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) We ran 𝑓))
65	49, 64	mpbid 235	. . . . . . . . . 10 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) We ran 𝑓)
66	36	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → Ord 𝑦)
67		isoeq3 7319	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) = ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) → (𝑓 Isom E , (𝑅 ∩ (ran 𝑓 × ran 𝑓))(𝑦, ran 𝑓) ↔ 𝑓 Isom E , ( (𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I )(𝑦, ran 𝑓)))
68	62, 67	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → (𝑓 Isom E , (𝑅 ∩ (ran 𝑓 × ran 𝑓))(𝑦, ran 𝑓) ↔ 𝑓 Isom E , ( (𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I )(𝑦, ran 𝑓)))
69	46, 68	mpbid 235	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → 𝑓 Isom E , ( (𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I )(𝑦, ran 𝑓))
70		vex 3455	. . . . . . . . . . . . . . 15 ⊢ 𝑓 ∈ V
71	70	rnex 7911	. . . . . . . . . . . . . 14 ⊢ ran 𝑓 ∈ V
72		exse 5611	. . . . . . . . . . . . . 14 ⊢ (ran 𝑓 ∈ V → ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) Se ran 𝑓)
73	71, 72	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) Se ran 𝑓
74		eqid 2761	. . . . . . . . . . . . . 14 ⊢ OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓) = OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓)
75	74	oieu 9517	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) We ran 𝑓 ∧ ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) Se ran 𝑓) → ((Ord 𝑦 ∧ 𝑓 Isom E , ( (𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I )(𝑦, ran 𝑓)) ↔ (𝑦 = dom OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓) ∧ 𝑓 = OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓))))
76	65, 73, 75	sylancl 598	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → ((Ord 𝑦 ∧ 𝑓 Isom E , ( (𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I )(𝑦, ran 𝑓)) ↔ (𝑦 = dom OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓) ∧ 𝑓 = OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓))))
77	66, 69, 76	mpbi2and 725	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → (𝑦 = dom OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓) ∧ 𝑓 = OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓)))
78	77	simpld 500	. . . . . . . . . 10 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → 𝑦 = dom OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓))
79	71, 71	xpex 7756	. . . . . . . . . . . 12 ⊢ (ran 𝑓 × ran 𝑓) ∈ V
80	79	inex2 5278	. . . . . . . . . . 11 ⊢ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ∈ V
81		sseq1 3956	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → (𝑟 ⊆ (ran 𝑓 × ran 𝑓) ↔ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ⊆ (ran 𝑓 × ran 𝑓)))
82	33, 81	mpbiri 261	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → 𝑟 ⊆ (ran 𝑓 × ran 𝑓))
83		dmss 5884	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 ⊆ (ran 𝑓 × ran 𝑓) → dom 𝑟 ⊆ dom (ran 𝑓 × ran 𝑓))
84	82, 83	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → dom 𝑟 ⊆ dom (ran 𝑓 × ran 𝑓))
85		dmxpid 5912	. . . . . . . . . . . . . . . . 17 ⊢ dom (ran 𝑓 × ran 𝑓) = ran 𝑓
86	84, 85	sseqtrdi 3971	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → dom 𝑟 ⊆ ran 𝑓)
87		dmresi 6046	. . . . . . . . . . . . . . . . 17 ⊢ dom ( I ↾ ran 𝑓) = ran 𝑓
88		sseq2 3957	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → (( I ↾ ran 𝑓) ⊆ 𝑟 ↔ ( I ↾ ran 𝑓) ⊆ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓))))
89	31, 88	mpbiri 261	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → ( I ↾ ran 𝑓) ⊆ 𝑟)
90		dmss 5884	. . . . . . . . . . . . . . . . . 18 ⊢ (( I ↾ ran 𝑓) ⊆ 𝑟 → dom ( I ↾ ran 𝑓) ⊆ dom 𝑟)
91	89, 90	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → dom ( I ↾ ran 𝑓) ⊆ dom 𝑟)
92	87, 91	eqsstrrid 3970	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → ran 𝑓 ⊆ dom 𝑟)
93	86, 92	eqssd 3948	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → dom 𝑟 = ran 𝑓)
94	93	sseq1d 3962	. . . . . . . . . . . . . 14 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → (dom 𝑟 ⊆ 𝐴 ↔ ran 𝑓 ⊆ 𝐴))
95	93	reseq2d 5970	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → ( I ↾ dom 𝑟) = ( I ↾ ran 𝑓))
96		id 23	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → 𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)))
97	95, 96	sseq12d 3964	. . . . . . . . . . . . . 14 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → (( I ↾ dom 𝑟) ⊆ 𝑟 ↔ ( I ↾ ran 𝑓) ⊆ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓))))
98	93	sqxpeqd 5683	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → (dom 𝑟 × dom 𝑟) = (ran 𝑓 × ran 𝑓))
99	96, 98	sseq12d 3964	. . . . . . . . . . . . . 14 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → (𝑟 ⊆ (dom 𝑟 × dom 𝑟) ↔ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ⊆ (ran 𝑓 × ran 𝑓)))
100	94, 97, 99	3anbi123d 1464	. . . . . . . . . . . . 13 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → ((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ↔ (ran 𝑓 ⊆ 𝐴 ∧ ( I ↾ ran 𝑓) ⊆ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ∧ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ⊆ (ran 𝑓 × ran 𝑓))))
101		difeq1 4067	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → (𝑟 ∖ I ) = (((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ∖ I ))
102		difun2 4437	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∪ I ) ∖ I ) = (𝑅 ∖ I )
103	102	ineq1i 4162	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∪ I ) ∖ I ) ∩ (ran 𝑓 × ran 𝑓)) = ((𝑅 ∖ I ) ∩ (ran 𝑓 × ran 𝑓))
104		indif1 4228	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∪ I ) ∖ I ) ∩ (ran 𝑓 × ran 𝑓)) = (((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ∖ I )
105		indif1 4228	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∖ I ) ∩ (ran 𝑓 × ran 𝑓)) = ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I )
106	103, 104, 105	3eqtr3i 2792	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) = ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I )
107	101, 106	eqtrdi 2812	. . . . . . . . . . . . . 14 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → (𝑟 ∖ I ) = ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ))
108	107, 93	weeq12d 5640	. . . . . . . . . . . . 13 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → ((𝑟 ∖ I ) We dom 𝑟 ↔ ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) We ran 𝑓))
109	100, 108	anbi12d 644	. . . . . . . . . . . 12 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → (((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ↔ ((ran 𝑓 ⊆ 𝐴 ∧ ( I ↾ ran 𝑓) ⊆ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ∧ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ⊆ (ran 𝑓 × ran 𝑓)) ∧ ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) We ran 𝑓)))
110		oieq1 9490	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 ∖ I ) = ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) → OrdIso((𝑟 ∖ I ), dom 𝑟) = OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), dom 𝑟))
111	107, 110	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → OrdIso((𝑟 ∖ I ), dom 𝑟) = OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), dom 𝑟))
112		oieq2 9491	. . . . . . . . . . . . . . . 16 ⊢ (dom 𝑟 = ran 𝑓 → OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), dom 𝑟) = OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓))
113	93, 112	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), dom 𝑟) = OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓))
114	111, 113	eqtrd 2796	. . . . . . . . . . . . . 14 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → OrdIso((𝑟 ∖ I ), dom 𝑟) = OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓))
115	114	dmeqd 5887	. . . . . . . . . . . . 13 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → dom OrdIso((𝑟 ∖ I ), dom 𝑟) = dom OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓))
116	115	eqeq2d 2772	. . . . . . . . . . . 12 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → (𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟) ↔ 𝑦 = dom OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓)))
117	109, 116	anbi12d 644	. . . . . . . . . . 11 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → ((((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)) ↔ (((ran 𝑓 ⊆ 𝐴 ∧ ( I ↾ ran 𝑓) ⊆ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ∧ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ⊆ (ran 𝑓 × ran 𝑓)) ∧ ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) We ran 𝑓) ∧ 𝑦 = dom OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓))))
118	80, 117	spcev 3561	. . . . . . . . . 10 ⊢ ((((ran 𝑓 ⊆ 𝐴 ∧ ( I ↾ ran 𝑓) ⊆ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ∧ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ⊆ (ran 𝑓 × ran 𝑓)) ∧ ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) We ran 𝑓) ∧ 𝑦 = dom OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓)) → ∃𝑟(((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)))
119	35, 65, 78, 118	syl21anc 851	. . . . . . . . 9 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → ∃𝑟(((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)))
120	119	ex 418	. . . . . . . 8 ⊢ (𝑦 ∈ On → (𝑓:𝑦–1-1→𝐴 → ∃𝑟(((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))))
121	120	exlimdv 1966	. . . . . . 7 ⊢ (𝑦 ∈ On → (∃𝑓 𝑓:𝑦–1-1→𝐴 → ∃𝑟(((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))))
122		brdomi 8970	. . . . . . 7 ⊢ (𝑦 ≼ 𝐴 → ∃𝑓 𝑓:𝑦–1-1→𝐴)
123	121, 122	impel 515	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ 𝑦 ≼ 𝐴) → ∃𝑟(((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)))
124		simpr 490	. . . . . . . . . . 11 ⊢ ((((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)) → 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))
125		vex 3455	. . . . . . . . . . . . 13 ⊢ 𝑟 ∈ V
126	125	dmex 7910	. . . . . . . . . . . 12 ⊢ dom 𝑟 ∈ V
127		eqid 2761	. . . . . . . . . . . . 13 ⊢ OrdIso((𝑟 ∖ I ), dom 𝑟) = OrdIso((𝑟 ∖ I ), dom 𝑟)
128	127	oion 9514	. . . . . . . . . . . 12 ⊢ (dom 𝑟 ∈ V → dom OrdIso((𝑟 ∖ I ), dom 𝑟) ∈ On)
129	126, 128	ax-mp 5	. . . . . . . . . . 11 ⊢ dom OrdIso((𝑟 ∖ I ), dom 𝑟) ∈ On
130	124, 129	eqeltrdi 2869	. . . . . . . . . 10 ⊢ ((((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)) → 𝑦 ∈ On)
131	130	adantl 487	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ (((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))) → 𝑦 ∈ On)
132		simplr 781	. . . . . . . . . . . 12 ⊢ ((((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)) → (𝑟 ∖ I ) We dom 𝑟)
133	127	oien 9516	. . . . . . . . . . . 12 ⊢ ((dom 𝑟 ∈ V ∧ (𝑟 ∖ I ) We dom 𝑟) → dom OrdIso((𝑟 ∖ I ), dom 𝑟) ≈ dom 𝑟)
134	126, 132, 133	sylancr 599	. . . . . . . . . . 11 ⊢ ((((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)) → dom OrdIso((𝑟 ∖ I ), dom 𝑟) ≈ dom 𝑟)
135	124, 134	eqbrtrd 5127	. . . . . . . . . 10 ⊢ ((((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)) → 𝑦 ≈ dom 𝑟)
136		ssdomg 9011	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → (dom 𝑟 ⊆ 𝐴 → dom 𝑟 ≼ 𝐴))
137		simpll1 1231	. . . . . . . . . . 11 ⊢ ((((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)) → dom 𝑟 ⊆ 𝐴)
138	136, 137	impel 515	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ (((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))) → dom 𝑟 ≼ 𝐴)
139		endomtr 9023	. . . . . . . . . 10 ⊢ ((𝑦 ≈ dom 𝑟 ∧ dom 𝑟 ≼ 𝐴) → 𝑦 ≼ 𝐴)
140	135, 138, 139	syl2an2 699	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ (((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))) → 𝑦 ≼ 𝐴)
141	131, 140	jca 521	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ (((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))) → (𝑦 ∈ On ∧ 𝑦 ≼ 𝐴))
142	141	ex 418	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ((((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)) → (𝑦 ∈ On ∧ 𝑦 ≼ 𝐴)))
143	142	exlimdv 1966	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (∃𝑟(((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)) → (𝑦 ∈ On ∧ 𝑦 ≼ 𝐴)))
144	123, 143	impbid2 229	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ((𝑦 ∈ On ∧ 𝑦 ≼ 𝐴) ↔ ∃𝑟(((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))))
145	23, 144	bitrid 286	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ↔ ∃𝑟(((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))))
146	145	eqabdv 2894	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} = {𝑦 ∣ ∃𝑟(((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))})
147	21, 146	eqtr4id 2815	. 2 ⊢ (𝐴 ∈ 𝑉 → ran 𝐹 = {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴})
148	15, 18, 147	3pm3.2i 1358	1 ⊢ (dom 𝐹 ⊆ 𝒫 (𝐴 × 𝐴) ∧ Fun 𝐹 ∧ (𝐴 ∈ 𝑉 → ran 𝐹 = {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2739  ∃wrex 3087  {crab 3413  Vcvv 3451   ∖ cdif 3896   ∪ cun 3897   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557   class class class wbr 5103  {copab 5167   I cid 5545   E cep 5550   Or wor 5558   Se wse 5602   We wwe 5603   × cxp 5649  dom cdm 5651  ran crn 5652   ↾ cres 5653  Ord word 6354  Oncon0 6355  Fun wfun 6525  ⟶wf 6527  –1-1→wf1 6528  –1-1-onto→wf1o 6530  ‘cfv 6531   Isom wiso 6532   ≈ cen 8954   ≼ cdom 8955  OrdIsocoi 9487

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7369  df-ov 7415  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-en 8958  df-dom 8959  df-oi 9488

This theorem is used by:  hartogslem2  9521  harwdom  9569