Theorem hartogslem1 9487

Description: Lemma for hartogs 9489. (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 5878	. . . 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 5889	. . . 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 1150	. . . . . . . 8 ⊢ ((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) → 𝑟 ⊆ (dom 𝑟 × dom 𝑟))
6		simp1 1148	. . . . . . . . 9 ⊢ ((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) → dom 𝑟 ⊆ 𝐴)
7		xpss12 5660	. . . . . . . . 9 ⊢ ((dom 𝑟 ⊆ 𝐴 ∧ dom 𝑟 ⊆ 𝐴) → (dom 𝑟 × dom 𝑟) ⊆ (𝐴 × 𝐴))
8	6, 6, 7	syl2anc 593	. . . . . . . 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 4559	. . . . . . 7 ⊢ (𝑟 ∈ 𝒫 (𝐴 × 𝐴) ↔ 𝑟 ⊆ (𝐴 × 𝐴))
11	9, 10	sylibr 236	. . . . . 6 ⊢ ((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) → 𝑟 ∈ 𝒫 (𝐴 × 𝐴))
12	11	ad2antrr 736	. . . . 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 1949	. . . 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 4021	. . 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 6554	. . 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 6538	. . 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 233	. 2 ⊢ Fun 𝐹
19	1	rneqi 5911	. . . 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 5928	. . . 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 5102	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ≼ 𝐴 ↔ 𝑦 ≼ 𝐴))
23	22	elrab 3650	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ↔ (𝑦 ∈ On ∧ 𝑦 ≼ 𝐴))
24		f1f 6756	. . . . . . . . . . . . 13 ⊢ (𝑓:𝑦–1-1→𝐴 → 𝑓:𝑦⟶𝐴)
25	24	adantl 485	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → 𝑓:𝑦⟶𝐴)
26	25	frnd 6696	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → ran 𝑓 ⊆ 𝐴)
27		resss 5985	. . . . . . . . . . . . . 14 ⊢ ( I ↾ ran 𝑓) ⊆ I
28		ssun2 4131	. . . . . . . . . . . . . 14 ⊢ I ⊆ (𝑅 ∪ I )
29	27, 28	sstri 3945	. . . . . . . . . . . . 13 ⊢ ( I ↾ ran 𝑓) ⊆ (𝑅 ∪ I )
30		idssxp 6035	. . . . . . . . . . . . 13 ⊢ ( I ↾ ran 𝑓) ⊆ (ran 𝑓 × ran 𝑓)
31	29, 30	ssini 4191	. . . . . . . . . . . 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 4189	. . . . . . . . . . . 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 1140	. . . . . . . . . 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 6352	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ On → Ord 𝑦)
37		ordwe 6355	. . . . . . . . . . . . . 14 ⊢ (Ord 𝑦 → E We 𝑦)
38	36, 37	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ On → E We 𝑦)
39	38	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → E We 𝑦)
40		f1f1orn 6814	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:𝑦–1-1→𝐴 → 𝑓:𝑦–1-1-onto→ran 𝑓)
41	40	adantl 485	. . . . . . . . . . . . . . 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 7331	. . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → 𝑓 Isom E , 𝑅 (𝑦, ran 𝑓))
45		isores2 7313	. . . . . . . . . . . . . 14 ⊢ (𝑓 Isom E , 𝑅 (𝑦, ran 𝑓) ↔ 𝑓 Isom E , (𝑅 ∩ (ran 𝑓 × ran 𝑓))(𝑦, ran 𝑓))
46	44, 45	sylib 220	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → 𝑓 Isom E , (𝑅 ∩ (ran 𝑓 × ran 𝑓))(𝑦, ran 𝑓))
47		isowe 7329	. . . . . . . . . . . . 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 234	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → (𝑅 ∩ (ran 𝑓 × ran 𝑓)) We ran 𝑓)
50		weso 5636	. . . . . . . . . . . . . . . . . 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 4189	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑅 ∩ (ran 𝑓 × ran 𝑓)) ⊆ (ran 𝑓 × ran 𝑓)
53	52	brel 5710	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥(𝑅 ∩ (ran 𝑓 × ran 𝑓))𝑥 → (𝑥 ∈ ran 𝑓 ∧ 𝑥 ∈ ran 𝑓))
54	53	simpld 498	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥(𝑅 ∩ (ran 𝑓 × ran 𝑓))𝑥 → 𝑥 ∈ ran 𝑓)
55		sonr 5577	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∩ (ran 𝑓 × ran 𝑓)) Or ran 𝑓 ∧ 𝑥 ∈ ran 𝑓) → ¬ 𝑥(𝑅 ∩ (ran 𝑓 × ran 𝑓))𝑥)
56	51, 54, 55	syl2an 605	. . . . . . . . . . . . . . . 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 808	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → ¬ 𝑥(𝑅 ∩ (ran 𝑓 × ran 𝑓))𝑥)
58	57	alrimiv 1946	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → ∀𝑥 ¬ 𝑥(𝑅 ∩ (ran 𝑓 × ran 𝑓))𝑥)
59		intirr 6102	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∩ I ) = ∅ ↔ ∀𝑥 ¬ 𝑥(𝑅 ∩ (ran 𝑓 × ran 𝑓))𝑥)
60	58, 59	sylibr 236	. . . . . . . . . . . . 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 220	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → (𝑅 ∩ (ran 𝑓 × ran 𝑓)) = ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ))
63		weeq1 5632	. . . . . . . . . . . 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 234	. . . . . . . . . 10 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) We ran 𝑓)
66	36	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → Ord 𝑦)
67		isoeq3 7299	. . . . . . . . . . . . . 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 234	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → 𝑓 Isom E , ( (𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I )(𝑦, ran 𝑓))
70		vex 3457	. . . . . . . . . . . . . . 15 ⊢ 𝑓 ∈ V
71	70	rnex 7887	. . . . . . . . . . . . . 14 ⊢ ran 𝑓 ∈ V
72		exse 5605	. . . . . . . . . . . . . 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 9484	. . . . . . . . . . . . 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 595	. . . . . . . . . . . 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 722	. . . . . . . . . . 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 498	. . . . . . . . . 10 ⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → 𝑦 = dom OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓))
79	71, 71	xpex 7732	. . . . . . . . . . . 12 ⊢ (ran 𝑓 × ran 𝑓) ∈ V
80	79	inex2 5273	. . . . . . . . . . 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 260	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → 𝑟 ⊆ (ran 𝑓 × ran 𝑓))
83		dmss 5876	. . . . . . . . . . . . . . . . . 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 5904	. . . . . . . . . . . . . . . . 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 6038	. . . . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → ( I ↾ ran 𝑓) ⊆ 𝑟)
90		dmss 5876	. . . . . . . . . . . . . . . . . 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 5963	. . . . . . . . . . . . . . 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 5677	. . . . . . . . . . . . . . 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 1456	. . . . . . . . . . . . 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 4434	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∪ I ) ∖ I ) = (𝑅 ∖ I )
103	102	ineq1i 4168	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∪ I ) ∖ I ) ∩ (ran 𝑓 × ran 𝑓)) = ((𝑅 ∖ I ) ∩ (ran 𝑓 × ran 𝑓))
104		indif1 4234	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∪ I ) ∖ I ) ∩ (ran 𝑓 × ran 𝑓)) = (((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ∖ I )
105		indif1 4234	. . . . . . . . . . . . . . . 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 5634	. . . . . . . . . . . . 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 641	. . . . . . . . . . . 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 9457	. . . . . . . . . . . . . . . 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 9458	. . . . . . . . . . . . . . . 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 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 5879	. . . . . . . . . . . . 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 641	. . . . . . . . . . 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 3565	. . . . . . . . . 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 848	. . . . . . . . 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 416	. . . . . . . 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 1952	. . . . . . 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 8936	. . . . . . 7 ⊢ (𝑦 ≼ 𝐴 → ∃𝑓 𝑓:𝑦–1-1→𝐴)
123	121, 122	impel 513	. . . . . 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 488	. . . . . . . . . . 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 3457	. . . . . . . . . . . . 13 ⊢ 𝑟 ∈ V
126	125	dmex 7886	. . . . . . . . . . . 12 ⊢ dom 𝑟 ∈ V
127		eqid 2761	. . . . . . . . . . . . 13 ⊢ OrdIso((𝑟 ∖ I ), dom 𝑟) = OrdIso((𝑟 ∖ I ), dom 𝑟)
128	127	oion 9481	. . . . . . . . . . . 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 485	. . . . . . . . 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 778	. . . . . . . . . . . 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 9483	. . . . . . . . . . . 12 ⊢ ((dom 𝑟 ∈ V ∧ (𝑟 ∖ I ) We dom 𝑟) → dom OrdIso((𝑟 ∖ I ), dom 𝑟) ≈ dom 𝑟)
134	126, 132, 133	sylancr 596	. . . . . . . . . . 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 5121	. . . . . . . . . 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 8977	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → (dom 𝑟 ⊆ 𝐴 → dom 𝑟 ≼ 𝐴))
137		simpll1 1225	. . . . . . . . . . 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 513	. . . . . . . . . 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 8989	. . . . . . . . . 10 ⊢ ((𝑦 ≈ dom 𝑟 ∧ dom 𝑟 ≼ 𝐴) → 𝑦 ≼ 𝐴)
140	135, 138, 139	syl2an2 696	. . . . . . . . 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 519	. . . . . . . 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 416	. . . . . . 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 1952	. . . . . 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 228	. . . . 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 285	. . . 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 1352	1 ⊢ (dom 𝐹 ⊆ 𝒫 (𝐴 × 𝐴) ∧ Fun 𝐹 ∧ (𝐴 ∈ 𝑉 → ran 𝐹 = {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097  ∀wal 1557   = wceq 1559  ∃wex 1798   ∈ wcel 2141  {cab 2739  ∃wrex 3085  {crab 3413  Vcvv 3453   ∖ cdif 3901   ∪ cun 3902   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  𝒫 cpw 4554   class class class wbr 5099  {copab 5161   I cid 5539   E cep 5544   Or wor 5552   Se wse 5596   We wwe 5597   × cxp 5643  dom cdm 5645  ran crn 5646   ↾ cres 5647  Ord word 6341  Oncon0 6342  Fun wfun 6511  ⟶wf 6513  –1-1→wf1 6514  –1-1-onto→wf1o 6516  ‘cfv 6517   Isom wiso 6518   ≈ cen 8920   ≼ cdom 8921  OrdIsocoi 9454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-se 5599  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-isom 6526  df-riota 7349  df-ov 7395  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-en 8924  df-dom 8925  df-oi 9455

This theorem is referenced by:  hartogslem2  9488  harwdom  9536