Theorem fmpox 8092

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 3484	. . . . . . . 8 ⊢ 𝑧 ∈ V
2		vex 3484	. . . . . . . 8 ⊢ 𝑤 ∈ V
3	1, 2	op1std 8024	. . . . . . 7 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (1st ‘𝑣) = 𝑧)
4	3	csbeq1d 3903	. . . . . 6 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → ⦋(1st ‘𝑣) / 𝑥⦌⦋(2nd ‘𝑣) / 𝑦⦌𝐶 = ⦋𝑧 / 𝑥⦌⦋(2nd ‘𝑣) / 𝑦⦌𝐶)
5	1, 2	op2ndd 8025	. . . . . . . 8 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (2nd ‘𝑣) = 𝑤)
6	5	csbeq1d 3903	. . . . . . 7 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → ⦋(2nd ‘𝑣) / 𝑦⦌𝐶 = ⦋𝑤 / 𝑦⦌𝐶)
7	6	csbeq2dv 3906	. . . . . 6 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → ⦋𝑧 / 𝑥⦌⦋(2nd ‘𝑣) / 𝑦⦌𝐶 = ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶)
8	4, 7	eqtrd 2777	. . . . 5 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → ⦋(1st ‘𝑣) / 𝑥⦌⦋(2nd ‘𝑣) / 𝑦⦌𝐶 = ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶)
9	8	eleq1d 2826	. . . 4 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (⦋(1st ‘𝑣) / 𝑥⦌⦋(2nd ‘𝑣) / 𝑦⦌𝐶 ∈ 𝐷 ↔ ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷))
10	9	raliunxp 5850	. . 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 1914	. . . . . . 7 ⊢ Ⅎ𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶)
12		nfv 1914	. . . . . . 7 ⊢ Ⅎ𝑤((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶)
13		nfv 1914	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
14		nfcsb1v 3923	. . . . . . . . . 10 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵
15	14	nfcri 2897	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵
16	13, 15	nfan 1899	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐴 ∧ 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵)
17		nfcsb1v 3923	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶
18	17	nfeq2 2923	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑣 = ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶
19	16, 18	nfan 1899	. . . . . . 7 ⊢ Ⅎ𝑥((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵) ∧ 𝑣 = ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶)
20		nfv 1914	. . . . . . . 8 ⊢ Ⅎ𝑦(𝑧 ∈ 𝐴 ∧ 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵)
21		nfcv 2905	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝑧
22		nfcsb1v 3923	. . . . . . . . . 10 ⊢ Ⅎ𝑦⦋𝑤 / 𝑦⦌𝐶
23	21, 22	nfcsbw 3925	. . . . . . . . 9 ⊢ Ⅎ𝑦⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶
24	23	nfeq2 2923	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑣 = ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶
25	20, 24	nfan 1899	. . . . . . 7 ⊢ Ⅎ𝑦((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵) ∧ 𝑣 = ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶)
26		eleq1w 2824	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
27	26	adantr 480	. . . . . . . . 9 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
28		eleq1w 2824	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐵 ↔ 𝑤 ∈ 𝐵))
29		csbeq1a 3913	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
30	29	eleq2d 2827	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑤 ∈ 𝐵 ↔ 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵))
31	28, 30	sylan9bbr 510	. . . . . . . . 9 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑦 ∈ 𝐵 ↔ 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵))
32	27, 31	anbi12d 632	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵)))
33		csbeq1a 3913	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → 𝐶 = ⦋𝑤 / 𝑦⦌𝐶)
34		csbeq1a 3913	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → ⦋𝑤 / 𝑦⦌𝐶 = ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶)
35	33, 34	sylan9eqr 2799	. . . . . . . . 9 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶)
36	35	eqeq2d 2748	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑣 = 𝐶 ↔ 𝑣 = ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶))
37	32, 36	anbi12d 632	. . . . . . 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 7522	. . . . . 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 7436	. . . . . 6 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑣⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶)}
40		df-mpo 7436	. . . . . 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 2775	. . . . 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 7546	. . . . 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 2775	. . . 4 ⊢ 𝐹 = (𝑣 ∈ ∪ 𝑧 ∈ 𝐴 ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵) ↦ ⦋(1st ‘𝑣) / 𝑥⦌⦋(2nd ‘𝑣) / 𝑦⦌𝐶)
45	44	fmpt 7130	. . 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 277	. 2 ⊢ (∀𝑧 ∈ 𝐴 ∀𝑤 ∈ ⦋ 𝑧 / 𝑥⦌𝐵⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷 ↔ 𝐹:∪ 𝑧 ∈ 𝐴 ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)⟶𝐷)
47		nfv 1914	. . 3 ⊢ Ⅎ𝑧∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷
48	17	nfel1 2922	. . . 4 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷
49	14, 48	nfralw 3311	. . 3 ⊢ Ⅎ𝑥∀𝑤 ∈ ⦋ 𝑧 / 𝑥⦌𝐵⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷
50		nfv 1914	. . . . 5 ⊢ Ⅎ𝑤 𝐶 ∈ 𝐷
51	22	nfel1 2922	. . . . 5 ⊢ Ⅎ𝑦⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷
52	33	eleq1d 2826	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝐶 ∈ 𝐷 ↔ ⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷))
53	50, 51, 52	cbvralw 3306	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 ↔ ∀𝑤 ∈ 𝐵 ⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷)
54	34	eleq1d 2826	. . . . 5 ⊢ (𝑥 = 𝑧 → (⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷 ↔ ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷))
55	29, 54	raleqbidv 3346	. . . 4 ⊢ (𝑥 = 𝑧 → (∀𝑤 ∈ 𝐵 ⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷 ↔ ∀𝑤 ∈ ⦋ 𝑧 / 𝑥⦌𝐵⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷))
56	53, 55	bitrid 283	. . 3 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 ↔ ∀𝑤 ∈ ⦋ 𝑧 / 𝑥⦌𝐵⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷))
57	47, 49, 56	cbvralw 3306	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ ⦋ 𝑧 / 𝑥⦌𝐵⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷)
58		nfcv 2905	. . . 4 ⊢ Ⅎ𝑧({𝑥} × 𝐵)
59		nfcv 2905	. . . . 5 ⊢ Ⅎ𝑥{𝑧}
60	59, 14	nfxp 5718	. . . 4 ⊢ Ⅎ𝑥({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)
61		sneq 4636	. . . . 5 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
62	61, 29	xpeq12d 5716	. . . 4 ⊢ (𝑥 = 𝑧 → ({𝑥} × 𝐵) = ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵))
63	58, 60, 62	cbviun 5036	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = ∪ 𝑧 ∈ 𝐴 ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)
64	63	feq2i 6728	. 2 ⊢ (𝐹:∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)⟶𝐷 ↔ 𝐹:∪ 𝑧 ∈ 𝐴 ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)⟶𝐷)
65	46, 57, 64	3bitr4i 303	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 ↔ 𝐹:∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)⟶𝐷)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ⦋csb 3899  {csn 4626  ⟨cop 4632  ∪ ciun 4991   ↦ cmpt 5225   × cxp 5683  ⟶wf 6557  ‘cfv 6561  {coprab 7432   ∈ cmpo 7433  1^st c1st 8012  2^nd c2nd 8013

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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015

This theorem is referenced by:  fmpo  8093  eldmcoa  18110  gsum2d2lem  19991  gsum2d2  19992  gsumcom2  19993  dmdprd  20018  dprdval  20023  dprd2d2  20064  ablfaclem2  20106  ptbasfi  23589  ptcmplem1  24060  prdsxmslem2  24542  tglnfn  28555