Theorem fmpox 8067

Description: Functionality, domain and codomain of a class given by the maps-to notation, where 𝐵(𝑥) is not constant but depends on 𝑥. (Contributed by NM, 29-Dec-2014.)

Hypothesis

Ref	Expression
fmpox.1	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Assertion

Ref	Expression
fmpox	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 ↔ 𝐹:∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)⟶𝐷)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑥,𝐷,𝑦

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥, 𝑦)   𝐹(𝑥, 𝑦)

Proof of Theorem fmpox

Dummy variables 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3455	. . . . . . . 8 ⊢ 𝑧 ∈ V
2		vex 3455	. . . . . . . 8 ⊢ 𝑤 ∈ V
3	1, 2	op1std 8000	. . . . . . 7 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (1st ‘𝑣) = 𝑧)
4	3	csbeq1d 3851	. . . . . 6 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → ⦋(1st ‘𝑣) / 𝑥⦌⦋(2nd ‘𝑣) / 𝑦⦌𝐶 = ⦋𝑧 / 𝑥⦌⦋(2nd ‘𝑣) / 𝑦⦌𝐶)
5	1, 2	op2ndd 8001	. . . . . . . 8 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (2nd ‘𝑣) = 𝑤)
6	5	csbeq1d 3851	. . . . . . 7 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → ⦋(2nd ‘𝑣) / 𝑦⦌𝐶 = ⦋𝑤 / 𝑦⦌𝐶)
7	6	csbeq2dv 3854	. . . . . 6 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → ⦋𝑧 / 𝑥⦌⦋(2nd ‘𝑣) / 𝑦⦌𝐶 = ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶)
8	4, 7	eqtrd 2796	. . . . 5 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → ⦋(1st ‘𝑣) / 𝑥⦌⦋(2nd ‘𝑣) / 𝑦⦌𝐶 = ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶)
9	8	eleq1d 2846	. . . 4 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (⦋(1st ‘𝑣) / 𝑥⦌⦋(2nd ‘𝑣) / 𝑦⦌𝐶 ∈ 𝐷 ↔ ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷))
10	9	raliunxp 5816	. . 3 ⊢ (∀𝑣 ∈ ∪ 𝑧 ∈ 𝐴 ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)⦋(1st ‘𝑣) / 𝑥⦌⦋(2nd ‘𝑣) / 𝑦⦌𝐶 ∈ 𝐷 ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ ⦋ 𝑧 / 𝑥⦌𝐵⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷)
11		nfv 1947	. . . . . . 7 ⊢ Ⅎ𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶)
12		nfv 1947	. . . . . . 7 ⊢ Ⅎ𝑤((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶)
13		nfv 1947	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
14		nfcsb1v 3871	. . . . . . . . . 10 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵
15	14	nfcri 2915	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵
16	13, 15	nfan 1932	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐴 ∧ 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵)
17		nfcsb1v 3871	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶
18	17	nfeq2 2940	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑣 = ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶
19	16, 18	nfan 1932	. . . . . . 7 ⊢ Ⅎ𝑥((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵) ∧ 𝑣 = ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶)
20		nfv 1947	. . . . . . . 8 ⊢ Ⅎ𝑦(𝑧 ∈ 𝐴 ∧ 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵)
21		nfcv 2923	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝑧
22		nfcsb1v 3871	. . . . . . . . . 10 ⊢ Ⅎ𝑦⦋𝑤 / 𝑦⦌𝐶
23	21, 22	nfcsbw 3873	. . . . . . . . 9 ⊢ Ⅎ𝑦⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶
24	23	nfeq2 2940	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑣 = ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶
25	20, 24	nfan 1932	. . . . . . 7 ⊢ Ⅎ𝑦((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵) ∧ 𝑣 = ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶)
26		eleq1w 2844	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
27	26	adantr 486	. . . . . . . . 9 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
28		eleq1w 2844	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐵 ↔ 𝑤 ∈ 𝐵))
29		csbeq1a 3861	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
30	29	eleq2d 2847	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑤 ∈ 𝐵 ↔ 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵))
31	28, 30	sylan9bbr 520	. . . . . . . . 9 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑦 ∈ 𝐵 ↔ 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵))
32	27, 31	anbi12d 644	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵)))
33		csbeq1a 3861	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → 𝐶 = ⦋𝑤 / 𝑦⦌𝐶)
34		csbeq1a 3861	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → ⦋𝑤 / 𝑦⦌𝐶 = ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶)
35	33, 34	sylan9eqr 2818	. . . . . . . . 9 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶)
36	35	eqeq2d 2772	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑣 = 𝐶 ↔ 𝑣 = ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶))
37	32, 36	anbi12d 644	. . . . . . 7 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶) ↔ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵) ∧ 𝑣 = ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶)))
38	11, 12, 19, 25, 37	cbvoprab12 7501	. . . . . 6 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑣⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶)} = {⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∣ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵) ∧ 𝑣 = ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶)}
39		df-mpo 7417	. . . . . 6 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑣⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶)}
40		df-mpo 7417	. . . . . 6 ⊢ (𝑧 ∈ 𝐴, 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵 ↦ ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶) = {⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∣ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵) ∧ 𝑣 = ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶)}
41	38, 39, 40	3eqtr4i 2794	. . . . 5 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵 ↦ ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶)
42		fmpox.1	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
43	8	mpomptx 7525	. . . . 5 ⊢ (𝑣 ∈ ∪ 𝑧 ∈ 𝐴 ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵) ↦ ⦋(1st ‘𝑣) / 𝑥⦌⦋(2nd ‘𝑣) / 𝑦⦌𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵 ↦ ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶)
44	41, 42, 43	3eqtr4i 2794	. . . 4 ⊢ 𝐹 = (𝑣 ∈ ∪ 𝑧 ∈ 𝐴 ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵) ↦ ⦋(1st ‘𝑣) / 𝑥⦌⦋(2nd ‘𝑣) / 𝑦⦌𝐶)
45	44	fmpt 7102	. . 3 ⊢ (∀𝑣 ∈ ∪ 𝑧 ∈ 𝐴 ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)⦋(1st ‘𝑣) / 𝑥⦌⦋(2nd ‘𝑣) / 𝑦⦌𝐶 ∈ 𝐷 ↔ 𝐹:∪ 𝑧 ∈ 𝐴 ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)⟶𝐷)
46	10, 45	bitr3i 280	. 2 ⊢ (∀𝑧 ∈ 𝐴 ∀𝑤 ∈ ⦋ 𝑧 / 𝑥⦌𝐵⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷 ↔ 𝐹:∪ 𝑧 ∈ 𝐴 ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)⟶𝐷)
47		nfv 1947	. . 3 ⊢ Ⅎ𝑧∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷
48	17	nfel1 2939	. . . 4 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷
49	14, 48	nfralw 3310	. . 3 ⊢ Ⅎ𝑥∀𝑤 ∈ ⦋ 𝑧 / 𝑥⦌𝐵⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷
50		nfv 1947	. . . . 5 ⊢ Ⅎ𝑤 𝐶 ∈ 𝐷
51	22	nfel1 2939	. . . . 5 ⊢ Ⅎ𝑦⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷
52	33	eleq1d 2846	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝐶 ∈ 𝐷 ↔ ⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷))
53	50, 51, 52	cbvralw 3305	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 ↔ ∀𝑤 ∈ 𝐵 ⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷)
54	34	eleq1d 2846	. . . . 5 ⊢ (𝑥 = 𝑧 → (⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷 ↔ ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷))
55	29, 54	raleqbidv 3335	. . . 4 ⊢ (𝑥 = 𝑧 → (∀𝑤 ∈ 𝐵 ⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷 ↔ ∀𝑤 ∈ ⦋ 𝑧 / 𝑥⦌𝐵⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷))
56	53, 55	bitrid 286	. . 3 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 ↔ ∀𝑤 ∈ ⦋ 𝑧 / 𝑥⦌𝐵⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷))
57	47, 49, 56	cbvralw 3305	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ ⦋ 𝑧 / 𝑥⦌𝐵⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷)
58		nfcv 2923	. . . 4 ⊢ Ⅎ𝑧({𝑥} × 𝐵)
59		nfcv 2923	. . . . 5 ⊢ Ⅎ𝑥{𝑧}
60	59, 14	nfxp 5684	. . . 4 ⊢ Ⅎ𝑥({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)
61		sneq 4594	. . . . 5 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
62	61, 29	xpeq12d 5682	. . . 4 ⊢ (𝑥 = 𝑧 → ({𝑥} × 𝐵) = ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵))
63	58, 60, 62	cbviun 4993	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = ∪ 𝑧 ∈ 𝐴 ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)
64	63	feq2i 6693	. 2 ⊢ (𝐹:∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)⟶𝐷 ↔ 𝐹:∪ 𝑧 ∈ 𝐴 ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)⟶𝐷)
65	46, 57, 64	3bitr4i 306	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 ↔ 𝐹:∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)⟶𝐷)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ⦋csb 3847  {csn 4584  ⟨cop 4590  ∪ ciun 4951   ↦ cmpt 5186   × cxp 5649  ⟶wf 6527  ‘cfv 6531  {coprab 7413   ∈ cmpo 7414  1^st c1st 7988  2^nd c2nd 7989

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-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  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-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  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-id 5546  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-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539  df-oprab 7416  df-mpo 7417  df-1st 7990  df-2nd 7991

This theorem is used by:  fmpo  8068  eldmcoa  18220  gsum2d2lem  20167  gsum2d2  20168  gsumcom2  20169  dmdprd  20194  dprdval  20199  dprd2d2  20240  ablfaclem2  20282  ptbasfi  23880  ptcmplem1  24351  prdsxmslem2  24828  tglnfn  28992  tgplnfn  29235