Theorem fpwwe2lem6 10060

Description: Lemma for fpwwe2 10068. (Contributed by Mario Carneiro, 18-May-2015.)

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)
fpwwe2lem9.x	⊢ (𝜑 → 𝑋𝑊𝑅)
fpwwe2lem9.y	⊢ (𝜑 → 𝑌𝑊𝑆)
fpwwe2lem9.m	⊢ 𝑀 = OrdIso(𝑅, 𝑋)
fpwwe2lem9.n	⊢ 𝑁 = OrdIso(𝑆, 𝑌)
fpwwe2lem7.1	⊢ (𝜑 → 𝐵 ∈ dom 𝑀)
fpwwe2lem7.2	⊢ (𝜑 → 𝐵 ∈ dom 𝑁)
fpwwe2lem7.3	⊢ (𝜑 → (𝑀 ↾ 𝐵) = (𝑁 ↾ 𝐵))

Assertion

Ref	Expression
fpwwe2lem6	⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (𝐶 ∈ 𝑋 ∧ 𝐶 ∈ 𝑌 ∧ (◡𝑀‘𝐶) = (◡𝑁‘𝐶)))

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)

Proof of Theorem fpwwe2lem6

Step	Hyp	Ref	Expression
1		fpwwe2lem9.x	. . . . . . 7 ⊢ (𝜑 → 𝑋𝑊𝑅)
2		fpwwe2.1	. . . . . . . 8 ⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
3		fpwwe2.2	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ V)
4	2, 3	fpwwe2lem2 10057	. . . . . . 7 ⊢ (𝜑 → (𝑋𝑊𝑅 ↔ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))
5	1, 4	mpbid 234	. . . . . 6 ⊢ (𝜑 → ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦)))
6	5	simplrd 768	. . . . 5 ⊢ (𝜑 → 𝑅 ⊆ (𝑋 × 𝑋))
7	6	ssbrd 5112	. . . 4 ⊢ (𝜑 → (𝐶𝑅(𝑀‘𝐵) → 𝐶(𝑋 × 𝑋)(𝑀‘𝐵)))
8		brxp 5604	. . . . 5 ⊢ (𝐶(𝑋 × 𝑋)(𝑀‘𝐵) ↔ (𝐶 ∈ 𝑋 ∧ (𝑀‘𝐵) ∈ 𝑋))
9	8	simplbi 500	. . . 4 ⊢ (𝐶(𝑋 × 𝑋)(𝑀‘𝐵) → 𝐶 ∈ 𝑋)
10	7, 9	syl6 35	. . 3 ⊢ (𝜑 → (𝐶𝑅(𝑀‘𝐵) → 𝐶 ∈ 𝑋))
11	10	imp 409	. 2 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝐶 ∈ 𝑋)
12		imassrn 5943	. . . 4 ⊢ (𝑁 “ 𝐵) ⊆ ran 𝑁
13		fpwwe2lem9.y	. . . . . . . . 9 ⊢ (𝜑 → 𝑌𝑊𝑆)
14	2	relopabi 5697	. . . . . . . . . 10 ⊢ Rel 𝑊
15	14	brrelex1i 5611	. . . . . . . . 9 ⊢ (𝑌𝑊𝑆 → 𝑌 ∈ V)
16	13, 15	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ V)
17	2, 3	fpwwe2lem2 10057	. . . . . . . . . 10 ⊢ (𝜑 → (𝑌𝑊𝑆 ↔ ((𝑌 ⊆ 𝐴 ∧ 𝑆 ⊆ (𝑌 × 𝑌)) ∧ (𝑆 We 𝑌 ∧ ∀𝑦 ∈ 𝑌 [(◡𝑆 “ {𝑦}) / 𝑢](𝑢𝐹(𝑆 ∩ (𝑢 × 𝑢))) = 𝑦))))
18	13, 17	mpbid 234	. . . . . . . . 9 ⊢ (𝜑 → ((𝑌 ⊆ 𝐴 ∧ 𝑆 ⊆ (𝑌 × 𝑌)) ∧ (𝑆 We 𝑌 ∧ ∀𝑦 ∈ 𝑌 [(◡𝑆 “ {𝑦}) / 𝑢](𝑢𝐹(𝑆 ∩ (𝑢 × 𝑢))) = 𝑦)))
19	18	simprld 770	. . . . . . . 8 ⊢ (𝜑 → 𝑆 We 𝑌)
20		fpwwe2lem9.n	. . . . . . . . 9 ⊢ 𝑁 = OrdIso(𝑆, 𝑌)
21	20	oiiso 9004	. . . . . . . 8 ⊢ ((𝑌 ∈ V ∧ 𝑆 We 𝑌) → 𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌))
22	16, 19, 21	syl2anc 586	. . . . . . 7 ⊢ (𝜑 → 𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌))
23	22	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌))
24		isof1o 7079	. . . . . 6 ⊢ (𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌) → 𝑁:dom 𝑁–1-1-onto→𝑌)
25	23, 24	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝑁:dom 𝑁–1-1-onto→𝑌)
26		f1ofo 6625	. . . . 5 ⊢ (𝑁:dom 𝑁–1-1-onto→𝑌 → 𝑁:dom 𝑁–onto→𝑌)
27		forn 6596	. . . . 5 ⊢ (𝑁:dom 𝑁–onto→𝑌 → ran 𝑁 = 𝑌)
28	25, 26, 27	3syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → ran 𝑁 = 𝑌)
29	12, 28	sseqtrid 4022	. . 3 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (𝑁 “ 𝐵) ⊆ 𝑌)
30	14	brrelex1i 5611	. . . . . . . . . . . . . 14 ⊢ (𝑋𝑊𝑅 → 𝑋 ∈ V)
31	1, 30	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ∈ V)
32	5	simprld 770	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 We 𝑋)
33		fpwwe2lem9.m	. . . . . . . . . . . . . 14 ⊢ 𝑀 = OrdIso(𝑅, 𝑋)
34	33	oiiso 9004	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ V ∧ 𝑅 We 𝑋) → 𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋))
35	31, 32, 34	syl2anc 586	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋))
36	35	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋))
37		isof1o 7079	. . . . . . . . . . 11 ⊢ (𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋) → 𝑀:dom 𝑀–1-1-onto→𝑋)
38	36, 37	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝑀:dom 𝑀–1-1-onto→𝑋)
39		f1ocnvfv2 7037	. . . . . . . . . 10 ⊢ ((𝑀:dom 𝑀–1-1-onto→𝑋 ∧ 𝐶 ∈ 𝑋) → (𝑀‘(◡𝑀‘𝐶)) = 𝐶)
40	38, 11, 39	syl2anc 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (𝑀‘(◡𝑀‘𝐶)) = 𝐶)
41		simpr 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝐶𝑅(𝑀‘𝐵))
42	40, 41	eqbrtrd 5091	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (𝑀‘(◡𝑀‘𝐶))𝑅(𝑀‘𝐵))
43		f1ocnv 6630	. . . . . . . . . . 11 ⊢ (𝑀:dom 𝑀–1-1-onto→𝑋 → ◡𝑀:𝑋–1-1-onto→dom 𝑀)
44		f1of 6618	. . . . . . . . . . 11 ⊢ (◡𝑀:𝑋–1-1-onto→dom 𝑀 → ◡𝑀:𝑋⟶dom 𝑀)
45	38, 43, 44	3syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → ◡𝑀:𝑋⟶dom 𝑀)
46	45, 11	ffvelrnd 6855	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (◡𝑀‘𝐶) ∈ dom 𝑀)
47		fpwwe2lem7.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ dom 𝑀)
48	47	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝐵 ∈ dom 𝑀)
49		isorel 7082	. . . . . . . . 9 ⊢ ((𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋) ∧ ((◡𝑀‘𝐶) ∈ dom 𝑀 ∧ 𝐵 ∈ dom 𝑀)) → ((◡𝑀‘𝐶) E 𝐵 ↔ (𝑀‘(◡𝑀‘𝐶))𝑅(𝑀‘𝐵)))
50	36, 46, 48, 49	syl12anc 834	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → ((◡𝑀‘𝐶) E 𝐵 ↔ (𝑀‘(◡𝑀‘𝐶))𝑅(𝑀‘𝐵)))
51	42, 50	mpbird 259	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (◡𝑀‘𝐶) E 𝐵)
52		epelg 5469	. . . . . . . 8 ⊢ (𝐵 ∈ dom 𝑀 → ((◡𝑀‘𝐶) E 𝐵 ↔ (◡𝑀‘𝐶) ∈ 𝐵))
53	48, 52	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → ((◡𝑀‘𝐶) E 𝐵 ↔ (◡𝑀‘𝐶) ∈ 𝐵))
54	51, 53	mpbid 234	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (◡𝑀‘𝐶) ∈ 𝐵)
55		ffn 6517	. . . . . . 7 ⊢ (◡𝑀:𝑋⟶dom 𝑀 → ◡𝑀 Fn 𝑋)
56		elpreima 6831	. . . . . . 7 ⊢ (◡𝑀 Fn 𝑋 → (𝐶 ∈ (◡◡𝑀 “ 𝐵) ↔ (𝐶 ∈ 𝑋 ∧ (◡𝑀‘𝐶) ∈ 𝐵)))
57	45, 55, 56	3syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (𝐶 ∈ (◡◡𝑀 “ 𝐵) ↔ (𝐶 ∈ 𝑋 ∧ (◡𝑀‘𝐶) ∈ 𝐵)))
58	11, 54, 57	mpbir2and 711	. . . . 5 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝐶 ∈ (◡◡𝑀 “ 𝐵))
59		imacnvcnv 6066	. . . . 5 ⊢ (◡◡𝑀 “ 𝐵) = (𝑀 “ 𝐵)
60	58, 59	eleqtrdi 2926	. . . 4 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝐶 ∈ (𝑀 “ 𝐵))
61		fpwwe2lem7.3	. . . . . . 7 ⊢ (𝜑 → (𝑀 ↾ 𝐵) = (𝑁 ↾ 𝐵))
62	61	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (𝑀 ↾ 𝐵) = (𝑁 ↾ 𝐵))
63	62	rneqd 5811	. . . . 5 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → ran (𝑀 ↾ 𝐵) = ran (𝑁 ↾ 𝐵))
64		df-ima 5571	. . . . 5 ⊢ (𝑀 “ 𝐵) = ran (𝑀 ↾ 𝐵)
65		df-ima 5571	. . . . 5 ⊢ (𝑁 “ 𝐵) = ran (𝑁 ↾ 𝐵)
66	63, 64, 65	3eqtr4g 2884	. . . 4 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (𝑀 “ 𝐵) = (𝑁 “ 𝐵))
67	60, 66	eleqtrd 2918	. . 3 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝐶 ∈ (𝑁 “ 𝐵))
68	29, 67	sseldd 3971	. 2 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝐶 ∈ 𝑌)
69	62	cnveqd 5749	. . . . 5 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → ◡(𝑀 ↾ 𝐵) = ◡(𝑁 ↾ 𝐵))
70		dff1o3 6624	. . . . . . 7 ⊢ (𝑀:dom 𝑀–1-1-onto→𝑋 ↔ (𝑀:dom 𝑀–onto→𝑋 ∧ Fun ◡𝑀))
71	70	simprbi 499	. . . . . 6 ⊢ (𝑀:dom 𝑀–1-1-onto→𝑋 → Fun ◡𝑀)
72		funcnvres 6435	. . . . . 6 ⊢ (Fun ◡𝑀 → ◡(𝑀 ↾ 𝐵) = (◡𝑀 ↾ (𝑀 “ 𝐵)))
73	38, 71, 72	3syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → ◡(𝑀 ↾ 𝐵) = (◡𝑀 ↾ (𝑀 “ 𝐵)))
74		dff1o3 6624	. . . . . . 7 ⊢ (𝑁:dom 𝑁–1-1-onto→𝑌 ↔ (𝑁:dom 𝑁–onto→𝑌 ∧ Fun ◡𝑁))
75	74	simprbi 499	. . . . . 6 ⊢ (𝑁:dom 𝑁–1-1-onto→𝑌 → Fun ◡𝑁)
76		funcnvres 6435	. . . . . 6 ⊢ (Fun ◡𝑁 → ◡(𝑁 ↾ 𝐵) = (◡𝑁 ↾ (𝑁 “ 𝐵)))
77	25, 75, 76	3syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → ◡(𝑁 ↾ 𝐵) = (◡𝑁 ↾ (𝑁 “ 𝐵)))
78	69, 73, 77	3eqtr3d 2867	. . . 4 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (◡𝑀 ↾ (𝑀 “ 𝐵)) = (◡𝑁 ↾ (𝑁 “ 𝐵)))
79	78	fveq1d 6675	. . 3 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → ((◡𝑀 ↾ (𝑀 “ 𝐵))‘𝐶) = ((◡𝑁 ↾ (𝑁 “ 𝐵))‘𝐶))
80	60	fvresd 6693	. . 3 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → ((◡𝑀 ↾ (𝑀 “ 𝐵))‘𝐶) = (◡𝑀‘𝐶))
81	67	fvresd 6693	. . 3 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → ((◡𝑁 ↾ (𝑁 “ 𝐵))‘𝐶) = (◡𝑁‘𝐶))
82	79, 80, 81	3eqtr3d 2867	. 2 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (◡𝑀‘𝐶) = (◡𝑁‘𝐶))
83	11, 68, 82	3jca 1124	1 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (𝐶 ∈ 𝑋 ∧ 𝐶 ∈ 𝑌 ∧ (◡𝑀‘𝐶) = (◡𝑁‘𝐶)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1536   ∈ wcel 2113  ∀wral 3141  Vcvv 3497  [wsbc 3775   ∩ cin 3938   ⊆ wss 3939  {csn 4570   class class class wbr 5069  {copab 5131   E cep 5467   We wwe 5516   × cxp 5556  ^◡ccnv 5557  dom cdm 5558  ran crn 5559   ↾ cres 5560   “ cima 5561  Fun wfun 6352   Fn wfn 6353  ⟶wf 6354  –onto→wfo 6356  –1-1-onto→wf1o 6357  ‘cfv 6358   Isom wiso 6359  (class class class)co 7159  OrdIsocoi 8976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rmo 3149  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-se 5518  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-isom 6367  df-riota 7117  df-ov 7162  df-wrecs 7950  df-recs 8011  df-oi 8977

This theorem is referenced by:  fpwwe2lem7  10061