Theorem hartogslem1 9459

Description: Lemma for hartogs 9461. (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 5861	. . . 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 5872	. . . 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 2760	. . 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 1139	. . . . . . . 8 ⊢ ((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) → 𝑟 ⊆ (dom 𝑟 × dom 𝑟))
6		simp1 1137	. . . . . . . . 9 ⊢ ((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) → dom 𝑟 ⊆ 𝐴)
7		xpss12 5647	. . . . . . . . 9 ⊢ ((dom 𝑟 ⊆ 𝐴 ∧ dom 𝑟 ⊆ 𝐴) → (dom 𝑟 × dom 𝑟) ⊆ (𝐴 × 𝐴))
8	6, 6, 7	syl2anc 585	. . . . . . . 8 ⊢ ((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) → (dom 𝑟 × dom 𝑟) ⊆ (𝐴 × 𝐴))
9	5, 8	sstrd 3946	. . . . . . 7 ⊢ ((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) → 𝑟 ⊆ (𝐴 × 𝐴))
10		velpw 4561	. . . . . . 7 ⊢ (𝑟 ∈ 𝒫 (𝐴 × 𝐴) ↔ 𝑟 ⊆ (𝐴 × 𝐴))
11	9, 10	sylibr 234	. . . . . 6 ⊢ ((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) → 𝑟 ∈ 𝒫 (𝐴 × 𝐴))
12	11	ad2antrr 727	. . . . 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 1932	. . . 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 4022	. . 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 3982	. 2 ⊢ dom 𝐹 ⊆ 𝒫 (𝐴 × 𝐴)
16		funopab4 6537	. . 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 6521	. . 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 231	. 2 ⊢ Fun 𝐹
19	1	rneqi 5894	. . . 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 5911	. . . 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 2760	. . 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 5103	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ≼ 𝐴 ↔ 𝑦 ≼ 𝐴))
23	22	elrab 3648	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ↔ (𝑦 ∈ On ∧ 𝑦 ≼ 𝐴))
24		f1f 6738	. . . . . . . . . . . . 13 ⊢ (𝑓:𝑦–1-1→𝐴 → 𝑓:𝑦⟶𝐴)
25	24	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → 𝑓:𝑦⟶𝐴)
26	25	frnd 6678	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → ran 𝑓 ⊆ 𝐴)
27		resss 5968	. . . . . . . . . . . . . 14 ⊢ ( I ↾ ran 𝑓) ⊆ I
28		ssun2 4133	. . . . . . . . . . . . . 14 ⊢ I ⊆ (𝑅 ∪ I )
29	27, 28	sstri 3945	. . . . . . . . . . . . 13 ⊢ ( I ↾ ran 𝑓) ⊆ (𝑅 ∪ I )
30		idssxp 6016	. . . . . . . . . . . . 13 ⊢ ( I ↾ ran 𝑓) ⊆ (ran 𝑓 × ran 𝑓)
31	29, 30	ssini 4194	. . . . . . . . . . . 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 4192	. . . . . . . . . . . 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 1129	. . . . . . . . . 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 6335	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ On → Ord 𝑦)
37		ordwe 6338	. . . . . . . . . . . . . 14 ⊢ (Ord 𝑦 → E We 𝑦)
38	36, 37	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ On → E We 𝑦)
39	38	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → E We 𝑦)
40		f1f1orn 6793	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:𝑦–1-1→𝐴 → 𝑓:𝑦–1-1-onto→ran 𝑓)
41	40	adantl 481	. . . . . . . . . . . . . . 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 7307	. . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → 𝑓 Isom E , 𝑅 (𝑦, ran 𝑓))
45		isores2 7289	. . . . . . . . . . . . . 14 ⊢ (𝑓 Isom E , 𝑅 (𝑦, ran 𝑓) ↔ 𝑓 Isom E , (𝑅 ∩ (ran 𝑓 × ran 𝑓))(𝑦, ran 𝑓))
46	44, 45	sylib 218	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → 𝑓 Isom E , (𝑅 ∩ (ran 𝑓 × ran 𝑓))(𝑦, ran 𝑓))
47		isowe 7305	. . . . . . . . . . . . 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 17	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → ( E We 𝑦 ↔ (𝑅 ∩ (ran 𝑓 × ran 𝑓)) We ran 𝑓))
49	39, 48	mpbid 232	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → (𝑅 ∩ (ran 𝑓 × ran 𝑓)) We ran 𝑓)
50		weso 5623	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) We ran 𝑓 → (𝑅 ∩ (ran 𝑓 × ran 𝑓)) Or ran 𝑓)
51	49, 50	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → (𝑅 ∩ (ran 𝑓 × ran 𝑓)) Or ran 𝑓)
52		inss2 4192	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑅 ∩ (ran 𝑓 × ran 𝑓)) ⊆ (ran 𝑓 × ran 𝑓)
53	52	brel 5697	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥(𝑅 ∩ (ran 𝑓 × ran 𝑓))𝑥 → (𝑥 ∈ ran 𝑓 ∧ 𝑥 ∈ ran 𝑓))
54	53	simpld 494	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥(𝑅 ∩ (ran 𝑓 × ran 𝑓))𝑥 → 𝑥 ∈ ran 𝑓)
55		sonr 5564	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∩ (ran 𝑓 × ran 𝑓)) Or ran 𝑓 ∧ 𝑥 ∈ ran 𝑓) → ¬ 𝑥(𝑅 ∩ (ran 𝑓 × ran 𝑓))𝑥)
56	51, 54, 55	syl2an 597	. . . . . . . . . . . . . . . 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 799	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → ¬ 𝑥(𝑅 ∩ (ran 𝑓 × ran 𝑓))𝑥)
58	57	alrimiv 1929	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → ∀𝑥 ¬ 𝑥(𝑅 ∩ (ran 𝑓 × ran 𝑓))𝑥)
59		intirr 6083	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∩ I ) = ∅ ↔ ∀𝑥 ¬ 𝑥(𝑅 ∩ (ran 𝑓 × ran 𝑓))𝑥)
60	58, 59	sylibr 234	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∩ I ) = ∅)
61		disj3 4408	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∩ I ) = ∅ ↔ (𝑅 ∩ (ran 𝑓 × ran 𝑓)) = ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ))
62	60, 61	sylib 218	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → (𝑅 ∩ (ran 𝑓 × ran 𝑓)) = ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ))
63		weeq1 5619	. . . . . . . . . . . 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 17	. . . . . . . . . . 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 232	. . . . . . . . . 10 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) We ran 𝑓)
66	36	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → Ord 𝑦)
67		isoeq3 7275	. . . . . . . . . . . . . 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 17	. . . . . . . . . . . . 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 232	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → 𝑓 Isom E , ( (𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I )(𝑦, ran 𝑓))
70		vex 3446	. . . . . . . . . . . . . . 15 ⊢ 𝑓 ∈ V
71	70	rnex 7862	. . . . . . . . . . . . . 14 ⊢ ran 𝑓 ∈ V
72		exse 5592	. . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . 14 ⊢ OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓) = OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓)
75	74	oieu 9456	. . . . . . . . . . . . 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 587	. . . . . . . . . . . 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 713	. . . . . . . . . . 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 494	. . . . . . . . . 10 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → 𝑦 = dom OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓))
79	71, 71	xpex 7708	. . . . . . . . . . . 12 ⊢ (ran 𝑓 × ran 𝑓) ∈ V
80	79	inex2 5265	. . . . . . . . . . 11 ⊢ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ∈ V
81		sseq1 3961	. . . . . . . . . . . . . . . . . . 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 258	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → 𝑟 ⊆ (ran 𝑓 × ran 𝑓))
83		dmss 5859	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 ⊆ (ran 𝑓 × ran 𝑓) → dom 𝑟 ⊆ dom (ran 𝑓 × ran 𝑓))
84	82, 83	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → dom 𝑟 ⊆ dom (ran 𝑓 × ran 𝑓))
85		dmxpid 5887	. . . . . . . . . . . . . . . . 17 ⊢ dom (ran 𝑓 × ran 𝑓) = ran 𝑓
86	84, 85	sseqtrdi 3976	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → dom 𝑟 ⊆ ran 𝑓)
87		dmresi 6019	. . . . . . . . . . . . . . . . 17 ⊢ dom ( I ↾ ran 𝑓) = ran 𝑓
88		sseq2 3962	. . . . . . . . . . . . . . . . . . 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 258	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → ( I ↾ ran 𝑓) ⊆ 𝑟)
90		dmss 5859	. . . . . . . . . . . . . . . . . 18 ⊢ (( I ↾ ran 𝑓) ⊆ 𝑟 → dom ( I ↾ ran 𝑓) ⊆ dom 𝑟)
91	89, 90	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → dom ( I ↾ ran 𝑓) ⊆ dom 𝑟)
92	87, 91	eqsstrrid 3975	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → ran 𝑓 ⊆ dom 𝑟)
93	86, 92	eqssd 3953	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → dom 𝑟 = ran 𝑓)
94	93	sseq1d 3967	. . . . . . . . . . . . . 14 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → (dom 𝑟 ⊆ 𝐴 ↔ ran 𝑓 ⊆ 𝐴))
95	93	reseq2d 5946	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → ( I ↾ dom 𝑟) = ( I ↾ ran 𝑓))
96		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → 𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)))
97	95, 96	sseq12d 3969	. . . . . . . . . . . . . 14 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → (( I ↾ dom 𝑟) ⊆ 𝑟 ↔ ( I ↾ ran 𝑓) ⊆ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓))))
98	93	sqxpeqd 5664	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → (dom 𝑟 × dom 𝑟) = (ran 𝑓 × ran 𝑓))
99	96, 98	sseq12d 3969	. . . . . . . . . . . . . 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 1439	. . . . . . . . . . . . 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 4073	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → (𝑟 ∖ I ) = (((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ∖ I ))
102		difun2 4435	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∪ I ) ∖ I ) = (𝑅 ∖ I )
103	102	ineq1i 4170	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∪ I ) ∖ I ) ∩ (ran 𝑓 × ran 𝑓)) = ((𝑅 ∖ I ) ∩ (ran 𝑓 × ran 𝑓))
104		indif1 4236	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∪ I ) ∖ I ) ∩ (ran 𝑓 × ran 𝑓)) = (((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ∖ I )
105		indif1 4236	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∖ I ) ∩ (ran 𝑓 × ran 𝑓)) = ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I )
106	103, 104, 105	3eqtr3i 2768	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) = ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I )
107	101, 106	eqtrdi 2788	. . . . . . . . . . . . . 14 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → (𝑟 ∖ I ) = ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ))
108	107, 93	weeq12d 5621	. . . . . . . . . . . . 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 633	. . . . . . . . . . . 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 9429	. . . . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → OrdIso((𝑟 ∖ I ), dom 𝑟) = OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), dom 𝑟))
112		oieq2 9430	. . . . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . 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 5862	. . . . . . . . . . . . 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 2748	. . . . . . . . . . . 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 633	. . . . . . . . . . 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 3562	. . . . . . . . . 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 838	. . . . . . . . 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 412	. . . . . . . 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 1935	. . . . . . 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 8908	. . . . . . 7 ⊢ (𝑦 ≼ 𝐴 → ∃𝑓 𝑓:𝑦–1-1→𝐴)
123	121, 122	impel 505	. . . . . 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 484	. . . . . . . . . . 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 3446	. . . . . . . . . . . . 13 ⊢ 𝑟 ∈ V
126	125	dmex 7861	. . . . . . . . . . . 12 ⊢ dom 𝑟 ∈ V
127		eqid 2737	. . . . . . . . . . . . 13 ⊢ OrdIso((𝑟 ∖ I ), dom 𝑟) = OrdIso((𝑟 ∖ I ), dom 𝑟)
128	127	oion 9453	. . . . . . . . . . . 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 2845	. . . . . . . . . 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 481	. . . . . . . . 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 769	. . . . . . . . . . . 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 9455	. . . . . . . . . . . 12 ⊢ ((dom 𝑟 ∈ V ∧ (𝑟 ∖ I ) We dom 𝑟) → dom OrdIso((𝑟 ∖ I ), dom 𝑟) ≈ dom 𝑟)
134	126, 132, 133	sylancr 588	. . . . . . . . . . 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 5122	. . . . . . . . . 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 8949	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → (dom 𝑟 ⊆ 𝐴 → dom 𝑟 ≼ 𝐴))
137		simpll1 1214	. . . . . . . . . . 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 505	. . . . . . . . . 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 8961	. . . . . . . . . 10 ⊢ ((𝑦 ≈ dom 𝑟 ∧ dom 𝑟 ≼ 𝐴) → 𝑦 ≼ 𝐴)
140	135, 138, 139	syl2an2 687	. . . . . . . . 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 511	. . . . . . . 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 412	. . . . . . 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 1935	. . . . . 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 226	. . . . 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 283	. . . 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 2870	. . 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 2791	. 2 ⊢ (𝐴 ∈ 𝑉 → ran 𝐹 = {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴})
148	15, 18, 147	3pm3.2i 1341	1 ⊢ (dom 𝐹 ⊆ 𝒫 (𝐴 × 𝐴) ∧ Fun 𝐹 ∧ (𝐴 ∈ 𝑉 → ran 𝐹 = {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715  ∃wrex 3062  {crab 3401  Vcvv 3442   ∖ cdif 3900   ∪ cun 3901   ∩ cin 3902   ⊆ wss 3903  ∅c0 4287  𝒫 cpw 4556   class class class wbr 5100  {copab 5162   I cid 5526   E cep 5531   Or wor 5539   Se wse 5583   We wwe 5584   × cxp 5630  dom cdm 5632  ran crn 5633   ↾ cres 5634  Ord word 6324  Oncon0 6325  Fun wfun 6494  ⟶wf 6496  –1-1→wf1 6497  –1-1-onto→wf1o 6499  ‘cfv 6500   Isom wiso 6501   ≈ cen 8892   ≼ cdom 8893  OrdIsocoi 9426

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-en 8896  df-dom 8897  df-oi 9427

This theorem is referenced by:  hartogslem2  9460  harwdom  9508