Theorem fpwwe2lem6 10561

Description: Lemma for fpwwe2 10568. (Contributed by Mario Carneiro, 18-May-2015.) (Revised by AV, 20-Jul-2024.)

Hypotheses

Ref	Expression
fpwwe2.1	⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
fpwwe2.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)
fpwwe2.3	⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
fpwwe2lem8.x	⊢ (𝜑 → 𝑋𝑊𝑅)
fpwwe2lem8.y	⊢ (𝜑 → 𝑌𝑊𝑆)
fpwwe2lem8.m	⊢ 𝑀 = OrdIso(𝑅, 𝑋)
fpwwe2lem8.n	⊢ 𝑁 = OrdIso(𝑆, 𝑌)
fpwwe2lem5.1	⊢ (𝜑 → 𝐵 ∈ dom 𝑀)
fpwwe2lem5.2	⊢ (𝜑 → 𝐵 ∈ dom 𝑁)
fpwwe2lem5.3	⊢ (𝜑 → (𝑀 ↾ 𝐵) = (𝑁 ↾ 𝐵))

Assertion

Ref	Expression
fpwwe2lem6	⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (𝐶𝑆(𝑁‘𝐵) ∧ (𝐷𝑅(𝑀‘𝐵) → (𝐶𝑅𝐷 ↔ 𝐶𝑆𝐷))))

Distinct variable groups:   𝑦,𝑢,𝐵   𝑢,𝑟,𝑥,𝑦,𝐹   𝑋,𝑟,𝑢,𝑥,𝑦   𝑀,𝑟,𝑢,𝑥,𝑦   𝑁,𝑟,𝑢,𝑥,𝑦   𝜑,𝑟,𝑢,𝑥,𝑦   𝐴,𝑟,𝑥   𝑅,𝑟,𝑢,𝑥,𝑦   𝑌,𝑟,𝑢,𝑥,𝑦   𝑆,𝑟,𝑢,𝑥,𝑦   𝑊,𝑟,𝑢,𝑥,𝑦

Allowed substitution hints:   𝐴(𝑦,𝑢)   𝐵(𝑥,𝑟)   𝐶(𝑥,𝑦,𝑢,𝑟)   𝐷(𝑥,𝑦,𝑢,𝑟)   𝑉(𝑥,𝑦,𝑢,𝑟)

Proof of Theorem fpwwe2lem6

Step	Hyp	Ref	Expression
1		fpwwe2lem8.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌𝑊𝑆)
2		fpwwe2.1	. . . . . . . . . 10 ⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
3	2	relopabiv 5779	. . . . . . . . 9 ⊢ Rel 𝑊
4	3	brrelex1i 5690	. . . . . . . 8 ⊢ (𝑌𝑊𝑆 → 𝑌 ∈ V)
5	1, 4	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ V)
6		fpwwe2.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
7	2, 6	fpwwe2lem2 10557	. . . . . . . . 9 ⊢ (𝜑 → (𝑌𝑊𝑆 ↔ ((𝑌 ⊆ 𝐴 ∧ 𝑆 ⊆ (𝑌 × 𝑌)) ∧ (𝑆 We 𝑌 ∧ ∀𝑦 ∈ 𝑌 [(◡𝑆 “ {𝑦}) / 𝑢](𝑢𝐹(𝑆 ∩ (𝑢 × 𝑢))) = 𝑦))))
8	1, 7	mpbid 232	. . . . . . . 8 ⊢ (𝜑 → ((𝑌 ⊆ 𝐴 ∧ 𝑆 ⊆ (𝑌 × 𝑌)) ∧ (𝑆 We 𝑌 ∧ ∀𝑦 ∈ 𝑌 [(◡𝑆 “ {𝑦}) / 𝑢](𝑢𝐹(𝑆 ∩ (𝑢 × 𝑢))) = 𝑦)))
9	8	simprld 772	. . . . . . 7 ⊢ (𝜑 → 𝑆 We 𝑌)
10		fpwwe2lem8.n	. . . . . . . 8 ⊢ 𝑁 = OrdIso(𝑆, 𝑌)
11	10	oiiso 9456	. . . . . . 7 ⊢ ((𝑌 ∈ V ∧ 𝑆 We 𝑌) → 𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌))
12	5, 9, 11	syl2anc 585	. . . . . 6 ⊢ (𝜑 → 𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌))
13	12	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌))
14		isof1o 7281	. . . . 5 ⊢ (𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌) → 𝑁:dom 𝑁–1-1-onto→𝑌)
15	13, 14	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝑁:dom 𝑁–1-1-onto→𝑌)
16		fpwwe2.3	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
17		fpwwe2lem8.x	. . . . . 6 ⊢ (𝜑 → 𝑋𝑊𝑅)
18		fpwwe2lem8.m	. . . . . 6 ⊢ 𝑀 = OrdIso(𝑅, 𝑋)
19		fpwwe2lem5.1	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ dom 𝑀)
20		fpwwe2lem5.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ dom 𝑁)
21		fpwwe2lem5.3	. . . . . 6 ⊢ (𝜑 → (𝑀 ↾ 𝐵) = (𝑁 ↾ 𝐵))
22	2, 6, 16, 17, 1, 18, 10, 19, 20, 21	fpwwe2lem5 10560	. . . . 5 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (𝐶 ∈ 𝑋 ∧ 𝐶 ∈ 𝑌 ∧ (◡𝑀‘𝐶) = (◡𝑁‘𝐶)))
23	22	simp2d 1144	. . . 4 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝐶 ∈ 𝑌)
24		f1ocnvfv2 7235	. . . 4 ⊢ ((𝑁:dom 𝑁–1-1-onto→𝑌 ∧ 𝐶 ∈ 𝑌) → (𝑁‘(◡𝑁‘𝐶)) = 𝐶)
25	15, 23, 24	syl2anc 585	. . 3 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (𝑁‘(◡𝑁‘𝐶)) = 𝐶)
26	22	simp3d 1145	. . . . 5 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (◡𝑀‘𝐶) = (◡𝑁‘𝐶))
27	3	brrelex1i 5690	. . . . . . . . . . . 12 ⊢ (𝑋𝑊𝑅 → 𝑋 ∈ V)
28	17, 27	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ V)
29	2, 6	fpwwe2lem2 10557	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋𝑊𝑅 ↔ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))
30	17, 29	mpbid 232	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦)))
31	30	simprld 772	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 We 𝑋)
32	18	oiiso 9456	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ V ∧ 𝑅 We 𝑋) → 𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋))
33	28, 31, 32	syl2anc 585	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋))
34	33	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋))
35		isof1o 7281	. . . . . . . . 9 ⊢ (𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋) → 𝑀:dom 𝑀–1-1-onto→𝑋)
36	34, 35	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝑀:dom 𝑀–1-1-onto→𝑋)
37	22	simp1d 1143	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝐶 ∈ 𝑋)
38		f1ocnvfv2 7235	. . . . . . . 8 ⊢ ((𝑀:dom 𝑀–1-1-onto→𝑋 ∧ 𝐶 ∈ 𝑋) → (𝑀‘(◡𝑀‘𝐶)) = 𝐶)
39	36, 37, 38	syl2anc 585	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (𝑀‘(◡𝑀‘𝐶)) = 𝐶)
40		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝐶𝑅(𝑀‘𝐵))
41	39, 40	eqbrtrd 5122	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (𝑀‘(◡𝑀‘𝐶))𝑅(𝑀‘𝐵))
42		f1ocnv 6796	. . . . . . . . 9 ⊢ (𝑀:dom 𝑀–1-1-onto→𝑋 → ◡𝑀:𝑋–1-1-onto→dom 𝑀)
43		f1of 6784	. . . . . . . . 9 ⊢ (◡𝑀:𝑋–1-1-onto→dom 𝑀 → ◡𝑀:𝑋⟶dom 𝑀)
44	36, 42, 43	3syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → ◡𝑀:𝑋⟶dom 𝑀)
45	44, 37	ffvelcdmd 7041	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (◡𝑀‘𝐶) ∈ dom 𝑀)
46	19	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝐵 ∈ dom 𝑀)
47		isorel 7284	. . . . . . 7 ⊢ ((𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋) ∧ ((◡𝑀‘𝐶) ∈ dom 𝑀 ∧ 𝐵 ∈ dom 𝑀)) → ((◡𝑀‘𝐶) E 𝐵 ↔ (𝑀‘(◡𝑀‘𝐶))𝑅(𝑀‘𝐵)))
48	34, 45, 46, 47	syl12anc 837	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → ((◡𝑀‘𝐶) E 𝐵 ↔ (𝑀‘(◡𝑀‘𝐶))𝑅(𝑀‘𝐵)))
49	41, 48	mpbird 257	. . . . 5 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (◡𝑀‘𝐶) E 𝐵)
50	26, 49	eqbrtrrd 5124	. . . 4 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (◡𝑁‘𝐶) E 𝐵)
51		f1ocnv 6796	. . . . . . 7 ⊢ (𝑁:dom 𝑁–1-1-onto→𝑌 → ◡𝑁:𝑌–1-1-onto→dom 𝑁)
52		f1of 6784	. . . . . . 7 ⊢ (◡𝑁:𝑌–1-1-onto→dom 𝑁 → ◡𝑁:𝑌⟶dom 𝑁)
53	15, 51, 52	3syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → ◡𝑁:𝑌⟶dom 𝑁)
54	53, 23	ffvelcdmd 7041	. . . . 5 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (◡𝑁‘𝐶) ∈ dom 𝑁)
55	20	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝐵 ∈ dom 𝑁)
56		isorel 7284	. . . . 5 ⊢ ((𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌) ∧ ((◡𝑁‘𝐶) ∈ dom 𝑁 ∧ 𝐵 ∈ dom 𝑁)) → ((◡𝑁‘𝐶) E 𝐵 ↔ (𝑁‘(◡𝑁‘𝐶))𝑆(𝑁‘𝐵)))
57	13, 54, 55, 56	syl12anc 837	. . . 4 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → ((◡𝑁‘𝐶) E 𝐵 ↔ (𝑁‘(◡𝑁‘𝐶))𝑆(𝑁‘𝐵)))
58	50, 57	mpbid 232	. . 3 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (𝑁‘(◡𝑁‘𝐶))𝑆(𝑁‘𝐵))
59	25, 58	eqbrtrrd 5124	. 2 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝐶𝑆(𝑁‘𝐵))
60	26	adantrr 718	. . . . 5 ⊢ ((𝜑 ∧ (𝐶𝑅(𝑀‘𝐵) ∧ 𝐷𝑅(𝑀‘𝐵))) → (◡𝑀‘𝐶) = (◡𝑁‘𝐶))
61	2, 6, 16, 17, 1, 18, 10, 19, 20, 21	fpwwe2lem5 10560	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐷𝑅(𝑀‘𝐵)) → (𝐷 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌 ∧ (◡𝑀‘𝐷) = (◡𝑁‘𝐷)))
62	61	simp3d 1145	. . . . . 6 ⊢ ((𝜑 ∧ 𝐷𝑅(𝑀‘𝐵)) → (◡𝑀‘𝐷) = (◡𝑁‘𝐷))
63	62	adantrl 717	. . . . 5 ⊢ ((𝜑 ∧ (𝐶𝑅(𝑀‘𝐵) ∧ 𝐷𝑅(𝑀‘𝐵))) → (◡𝑀‘𝐷) = (◡𝑁‘𝐷))
64	60, 63	breq12d 5113	. . . 4 ⊢ ((𝜑 ∧ (𝐶𝑅(𝑀‘𝐵) ∧ 𝐷𝑅(𝑀‘𝐵))) → ((◡𝑀‘𝐶) E (◡𝑀‘𝐷) ↔ (◡𝑁‘𝐶) E (◡𝑁‘𝐷)))
65	33	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝐶𝑅(𝑀‘𝐵) ∧ 𝐷𝑅(𝑀‘𝐵))) → 𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋))
66		isocnv 7288	. . . . . 6 ⊢ (𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋) → ◡𝑀 Isom 𝑅, E (𝑋, dom 𝑀))
67	65, 66	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝐶𝑅(𝑀‘𝐵) ∧ 𝐷𝑅(𝑀‘𝐵))) → ◡𝑀 Isom 𝑅, E (𝑋, dom 𝑀))
68	37	adantrr 718	. . . . 5 ⊢ ((𝜑 ∧ (𝐶𝑅(𝑀‘𝐵) ∧ 𝐷𝑅(𝑀‘𝐵))) → 𝐶 ∈ 𝑋)
69	30	simplrd 770	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ⊆ (𝑋 × 𝑋))
70	69	ssbrd 5143	. . . . . . . 8 ⊢ (𝜑 → (𝐷𝑅(𝑀‘𝐵) → 𝐷(𝑋 × 𝑋)(𝑀‘𝐵)))
71	70	imp 406	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐷𝑅(𝑀‘𝐵)) → 𝐷(𝑋 × 𝑋)(𝑀‘𝐵))
72		brxp 5683	. . . . . . . 8 ⊢ (𝐷(𝑋 × 𝑋)(𝑀‘𝐵) ↔ (𝐷 ∈ 𝑋 ∧ (𝑀‘𝐵) ∈ 𝑋))
73	72	simplbi 496	. . . . . . 7 ⊢ (𝐷(𝑋 × 𝑋)(𝑀‘𝐵) → 𝐷 ∈ 𝑋)
74	71, 73	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝐷𝑅(𝑀‘𝐵)) → 𝐷 ∈ 𝑋)
75	74	adantrl 717	. . . . 5 ⊢ ((𝜑 ∧ (𝐶𝑅(𝑀‘𝐵) ∧ 𝐷𝑅(𝑀‘𝐵))) → 𝐷 ∈ 𝑋)
76		isorel 7284	. . . . 5 ⊢ ((◡𝑀 Isom 𝑅, E (𝑋, dom 𝑀) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (𝐶𝑅𝐷 ↔ (◡𝑀‘𝐶) E (◡𝑀‘𝐷)))
77	67, 68, 75, 76	syl12anc 837	. . . 4 ⊢ ((𝜑 ∧ (𝐶𝑅(𝑀‘𝐵) ∧ 𝐷𝑅(𝑀‘𝐵))) → (𝐶𝑅𝐷 ↔ (◡𝑀‘𝐶) E (◡𝑀‘𝐷)))
78	12	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝐶𝑅(𝑀‘𝐵) ∧ 𝐷𝑅(𝑀‘𝐵))) → 𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌))
79		isocnv 7288	. . . . . 6 ⊢ (𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌) → ◡𝑁 Isom 𝑆, E (𝑌, dom 𝑁))
80	78, 79	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝐶𝑅(𝑀‘𝐵) ∧ 𝐷𝑅(𝑀‘𝐵))) → ◡𝑁 Isom 𝑆, E (𝑌, dom 𝑁))
81	23	adantrr 718	. . . . 5 ⊢ ((𝜑 ∧ (𝐶𝑅(𝑀‘𝐵) ∧ 𝐷𝑅(𝑀‘𝐵))) → 𝐶 ∈ 𝑌)
82	61	simp2d 1144	. . . . . 6 ⊢ ((𝜑 ∧ 𝐷𝑅(𝑀‘𝐵)) → 𝐷 ∈ 𝑌)
83	82	adantrl 717	. . . . 5 ⊢ ((𝜑 ∧ (𝐶𝑅(𝑀‘𝐵) ∧ 𝐷𝑅(𝑀‘𝐵))) → 𝐷 ∈ 𝑌)
84		isorel 7284	. . . . 5 ⊢ ((◡𝑁 Isom 𝑆, E (𝑌, dom 𝑁) ∧ (𝐶 ∈ 𝑌 ∧ 𝐷 ∈ 𝑌)) → (𝐶𝑆𝐷 ↔ (◡𝑁‘𝐶) E (◡𝑁‘𝐷)))
85	80, 81, 83, 84	syl12anc 837	. . . 4 ⊢ ((𝜑 ∧ (𝐶𝑅(𝑀‘𝐵) ∧ 𝐷𝑅(𝑀‘𝐵))) → (𝐶𝑆𝐷 ↔ (◡𝑁‘𝐶) E (◡𝑁‘𝐷)))
86	64, 77, 85	3bitr4d 311	. . 3 ⊢ ((𝜑 ∧ (𝐶𝑅(𝑀‘𝐵) ∧ 𝐷𝑅(𝑀‘𝐵))) → (𝐶𝑅𝐷 ↔ 𝐶𝑆𝐷))
87	86	expr 456	. 2 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (𝐷𝑅(𝑀‘𝐵) → (𝐶𝑅𝐷 ↔ 𝐶𝑆𝐷)))
88	59, 87	jca 511	1 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (𝐶𝑆(𝑁‘𝐵) ∧ (𝐷𝑅(𝑀‘𝐵) → (𝐶𝑅𝐷 ↔ 𝐶𝑆𝐷))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3442  [wsbc 3742   ∩ cin 3902   ⊆ wss 3903  {csn 4582   class class class wbr 5100  {copab 5162   E cep 5533   We wwe 5586   × cxp 5632  ^◡ccnv 5633  dom cdm 5634   ↾ cres 5636   “ cima 5637  ⟶wf 6498  –1-1-onto→wf1o 6501  ‘cfv 6502   Isom wiso 6503  (class class class)co 7370  OrdIsocoi 9428

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 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692

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 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-se 5588  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-isom 6511  df-riota 7327  df-ov 7373  df-2nd 7946  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-oi 9429

This theorem is referenced by:  fpwwe2lem7  10562