Theorem fpwwe2lem5 10720

Description: Lemma for fpwwe2 10728. (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
fpwwe2lem5	⊢ ((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)   𝑉(𝑥, 𝑦, 𝑢, 𝑟)

Proof of Theorem fpwwe2lem5

Step	Hyp	Ref	Expression
1		fpwwe2lem8.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 10717	. . . . . . 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 235	. . . . . 6 ⊢ (𝜑 → ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦)))
6	5	simplrd 782	. . . . 5 ⊢ (𝜑 → 𝑅 ⊆ (𝑋 × 𝑋))
7	6	ssbrd 5148	. . . 4 ⊢ (𝜑 → (𝐶𝑅(𝑀‘𝐵) → 𝐶(𝑋 × 𝑋)(𝑀‘𝐵)))
8		brxp 5700	. . . . 5 ⊢ (𝐶(𝑋 × 𝑋)(𝑀‘𝐵) ↔ (𝐶 ∈ 𝑋 ∧ (𝑀‘𝐵) ∈ 𝑋))
9	8	simplbi 502	. . . 4 ⊢ (𝐶(𝑋 × 𝑋)(𝑀‘𝐵) → 𝐶 ∈ 𝑋)
10	7, 9	syl6 36	. . 3 ⊢ (𝜑 → (𝐶𝑅(𝑀‘𝐵) → 𝐶 ∈ 𝑋))
11	10	imp 412	. 2 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝐶 ∈ 𝑋)
12		imassrn 6197	. . . 4 ⊢ (𝑁 “ 𝐵) ⊆ ran 𝑁
13		fpwwe2lem8.y	. . . . . . . . 9 ⊢ (𝜑 → 𝑌𝑊𝑆)
14	2	relopabiv 5798	. . . . . . . . . 10 ⊢ Rel 𝑊
15	14	brrelex1i 5707	. . . . . . . . 9 ⊢ (𝑌𝑊𝑆 → 𝑌 ∈ V)
16	13, 15	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ V)
17	2, 3	fpwwe2lem2 10717	. . . . . . . . . 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 235	. . . . . . . . 9 ⊢ (𝜑 → ((𝑌 ⊆ 𝐴 ∧ 𝑆 ⊆ (𝑌 × 𝑌)) ∧ (𝑆 We 𝑌 ∧ ∀𝑦 ∈ 𝑌 [(◡𝑆 “ {𝑦}) / 𝑢](𝑢𝐹(𝑆 ∩ (𝑢 × 𝑢))) = 𝑦)))
19	18	simprld 784	. . . . . . . 8 ⊢ (𝜑 → 𝑆 We 𝑌)
20		fpwwe2lem8.n	. . . . . . . . 9 ⊢ 𝑁 = OrdIso(𝑆, 𝑌)
21	20	oiiso 9531	. . . . . . . 8 ⊢ ((𝑌 ∈ V ∧ 𝑆 We 𝑌) → 𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌))
22	16, 19, 21	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → 𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌))
23	22	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌))
24		isof1o 7331	. . . . . 6 ⊢ (𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌) → 𝑁:dom 𝑁–1-1-onto→𝑌)
25	23, 24	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝑁:dom 𝑁–1-1-onto→𝑌)
26		f1ofo 6832	. . . . 5 ⊢ (𝑁:dom 𝑁–1-1-onto→𝑌 → 𝑁:dom 𝑁–onto→𝑌)
27		forn 6799	. . . . 5 ⊢ (𝑁:dom 𝑁–onto→𝑌 → ran 𝑁 = 𝑌)
28	25, 26, 27	3syl 19	. . . 4 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → ran 𝑁 = 𝑌)
29	12, 28	sseqtrid 3973	. . 3 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (𝑁 “ 𝐵) ⊆ 𝑌)
30	14	brrelex1i 5707	. . . . . . . . . . . . . 14 ⊢ (𝑋𝑊𝑅 → 𝑋 ∈ V)
31	1, 30	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ∈ V)
32	5	simprld 784	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 We 𝑋)
33		fpwwe2lem8.m	. . . . . . . . . . . . . 14 ⊢ 𝑀 = OrdIso(𝑅, 𝑋)
34	33	oiiso 9531	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ V ∧ 𝑅 We 𝑋) → 𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋))
35	31, 32, 34	syl2anc 596	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋))
36	35	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋))
37		isof1o 7331	. . . . . . . . . . 11 ⊢ (𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋) → 𝑀:dom 𝑀–1-1-onto→𝑋)
38	36, 37	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝑀:dom 𝑀–1-1-onto→𝑋)
39		f1ocnvfv2 7285	. . . . . . . . . 10 ⊢ ((𝑀:dom 𝑀–1-1-onto→𝑋 ∧ 𝐶 ∈ 𝑋) → (𝑀‘(◡𝑀‘𝐶)) = 𝐶)
40	38, 11, 39	syl2anc 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (𝑀‘(◡𝑀‘𝐶)) = 𝐶)
41		simpr 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝐶𝑅(𝑀‘𝐵))
42	40, 41	eqbrtrd 5127	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (𝑀‘(◡𝑀‘𝐶))𝑅(𝑀‘𝐵))
43		f1ocnv 6837	. . . . . . . . . . 11 ⊢ (𝑀:dom 𝑀–1-1-onto→𝑋 → ◡𝑀:𝑋–1-1-onto→dom 𝑀)
44		f1of 6824	. . . . . . . . . . 11 ⊢ (◡𝑀:𝑋–1-1-onto→dom 𝑀 → ◡𝑀:𝑋⟶dom 𝑀)
45	38, 43, 44	3syl 19	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → ◡𝑀:𝑋⟶dom 𝑀)
46	45, 11	ffvelcdmd 7085	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (◡𝑀‘𝐶) ∈ dom 𝑀)
47		fpwwe2lem5.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ dom 𝑀)
48	47	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝐵 ∈ dom 𝑀)
49		isorel 7334	. . . . . . . . 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 850	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → ((◡𝑀‘𝐶) E 𝐵 ↔ (𝑀‘(◡𝑀‘𝐶))𝑅(𝑀‘𝐵)))
51	42, 50	mpbird 260	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (◡𝑀‘𝐶) E 𝐵)
52		epelg 5552	. . . . . . . 8 ⊢ (𝐵 ∈ dom 𝑀 → ((◡𝑀‘𝐶) E 𝐵 ↔ (◡𝑀‘𝐶) ∈ 𝐵))
53	48, 52	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → ((◡𝑀‘𝐶) E 𝐵 ↔ (◡𝑀‘𝐶) ∈ 𝐵))
54	51, 53	mpbid 235	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (◡𝑀‘𝐶) ∈ 𝐵)
55		ffn 6709	. . . . . . 7 ⊢ (◡𝑀:𝑋⟶dom 𝑀 → ◡𝑀 Fn 𝑋)
56		elpreima 7057	. . . . . . 7 ⊢ (◡𝑀 Fn 𝑋 → (𝐶 ∈ (◡◡𝑀 “ 𝐵) ↔ (𝐶 ∈ 𝑋 ∧ (◡𝑀‘𝐶) ∈ 𝐵)))
57	45, 55, 56	3syl 19	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (𝐶 ∈ (◡◡𝑀 “ 𝐵) ↔ (𝐶 ∈ 𝑋 ∧ (◡𝑀‘𝐶) ∈ 𝐵)))
58	11, 54, 57	mpbir2and 726	. . . . 5 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝐶 ∈ (◡◡𝑀 “ 𝐵))
59		imacnvcnv 6207	. . . . 5 ⊢ (◡◡𝑀 “ 𝐵) = (𝑀 “ 𝐵)
60	58, 59	eleqtrdi 2871	. . . 4 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝐶 ∈ (𝑀 “ 𝐵))
61		fpwwe2lem5.3	. . . . . . 7 ⊢ (𝜑 → (𝑀 ↾ 𝐵) = (𝑁 ↾ 𝐵))
62	61	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (𝑀 ↾ 𝐵) = (𝑁 ↾ 𝐵))
63	62	rneqd 5920	. . . . 5 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → ran (𝑀 ↾ 𝐵) = ran (𝑁 ↾ 𝐵))
64		df-ima 5664	. . . . 5 ⊢ (𝑀 “ 𝐵) = ran (𝑀 ↾ 𝐵)
65		df-ima 5664	. . . . 5 ⊢ (𝑁 “ 𝐵) = ran (𝑁 ↾ 𝐵)
66	63, 64, 65	3eqtr4g 2821	. . . 4 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (𝑀 “ 𝐵) = (𝑁 “ 𝐵))
67	60, 66	eleqtrd 2863	. . 3 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝐶 ∈ (𝑁 “ 𝐵))
68	29, 67	sseldd 3932	. 2 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝐶 ∈ 𝑌)
69	62	cnveqd 5853	. . . . 5 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → ◡(𝑀 ↾ 𝐵) = ◡(𝑁 ↾ 𝐵))
70		dff1o3 6831	. . . . . . 7 ⊢ (𝑀:dom 𝑀–1-1-onto→𝑋 ↔ (𝑀:dom 𝑀–onto→𝑋 ∧ Fun ◡𝑀))
71	70	simprbi 503	. . . . . 6 ⊢ (𝑀:dom 𝑀–1-1-onto→𝑋 → Fun ◡𝑀)
72		funcnvres 6618	. . . . . 6 ⊢ (Fun ◡𝑀 → ◡(𝑀 ↾ 𝐵) = (◡𝑀 ↾ (𝑀 “ 𝐵)))
73	38, 71, 72	3syl 19	. . . . 5 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → ◡(𝑀 ↾ 𝐵) = (◡𝑀 ↾ (𝑀 “ 𝐵)))
74		dff1o3 6831	. . . . . . 7 ⊢ (𝑁:dom 𝑁–1-1-onto→𝑌 ↔ (𝑁:dom 𝑁–onto→𝑌 ∧ Fun ◡𝑁))
75	74	simprbi 503	. . . . . 6 ⊢ (𝑁:dom 𝑁–1-1-onto→𝑌 → Fun ◡𝑁)
76		funcnvres 6618	. . . . . 6 ⊢ (Fun ◡𝑁 → ◡(𝑁 ↾ 𝐵) = (◡𝑁 ↾ (𝑁 “ 𝐵)))
77	25, 75, 76	3syl 19	. . . . 5 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → ◡(𝑁 ↾ 𝐵) = (◡𝑁 ↾ (𝑁 “ 𝐵)))
78	69, 73, 77	3eqtr3d 2804	. . . 4 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (◡𝑀 ↾ (𝑀 “ 𝐵)) = (◡𝑁 ↾ (𝑁 “ 𝐵)))
79	78	fveq1d 6887	. . 3 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → ((◡𝑀 ↾ (𝑀 “ 𝐵))‘𝐶) = ((◡𝑁 ↾ (𝑁 “ 𝐵))‘𝐶))
80	60	fvresd 6905	. . 3 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → ((◡𝑀 ↾ (𝑀 “ 𝐵))‘𝐶) = (◡𝑀‘𝐶))
81	67	fvresd 6905	. . 3 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → ((◡𝑁 ↾ (𝑁 “ 𝐵))‘𝐶) = (◡𝑁‘𝐶))
82	79, 80, 81	3eqtr3d 2804	. 2 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (◡𝑀‘𝐶) = (◡𝑁‘𝐶))
83	11, 68, 82	3jca 1146	1 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (𝐶 ∈ 𝑋 ∧ 𝐶 ∈ 𝑌 ∧ (◡𝑀‘𝐶) = (◡𝑁‘𝐶)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077  Vcvv 3451  [wsbc 3739   ∩ cin 3898   ⊆ wss 3899  {csn 4584   class class class wbr 5103  {copab 5167   E cep 5550   We wwe 5603   × cxp 5649  ^◡ccnv 5650  dom cdm 5651  ran crn 5652   ↾ cres 5653   “ cima 5654  Fun wfun 6532   Fn wfn 6533  ⟶wf 6534  –onto→wfo 6536  –1-1-onto→wf1o 6537  ‘cfv 6538   Isom wiso 6539  (class class class)co 7420  OrdIsocoi 9503

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 7751

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 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-oi 9504

This theorem is used by:  fpwwe2lem6  10721