Theorem fmpox 6296

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 2776	. . . . . . . 8 ⊢ 𝑧 ∈ V
2		vex 2776	. . . . . . . 8 ⊢ 𝑤 ∈ V
3	1, 2	op1std 6244	. . . . . . 7 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (1st ‘𝑣) = 𝑧)
4	3	csbeq1d 3102	. . . . . 6 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → ⦋(1st ‘𝑣) / 𝑥⦌⦋(2nd ‘𝑣) / 𝑦⦌𝐶 = ⦋𝑧 / 𝑥⦌⦋(2nd ‘𝑣) / 𝑦⦌𝐶)
5	1, 2	op2ndd 6245	. . . . . . . 8 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (2nd ‘𝑣) = 𝑤)
6	5	csbeq1d 3102	. . . . . . 7 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → ⦋(2nd ‘𝑣) / 𝑦⦌𝐶 = ⦋𝑤 / 𝑦⦌𝐶)
7	6	csbeq2dv 3121	. . . . . 6 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → ⦋𝑧 / 𝑥⦌⦋(2nd ‘𝑣) / 𝑦⦌𝐶 = ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶)
8	4, 7	eqtrd 2239	. . . . 5 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → ⦋(1st ‘𝑣) / 𝑥⦌⦋(2nd ‘𝑣) / 𝑦⦌𝐶 = ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶)
9	8	eleq1d 2275	. . . 4 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (⦋(1st ‘𝑣) / 𝑥⦌⦋(2nd ‘𝑣) / 𝑦⦌𝐶 ∈ 𝐷 ↔ ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷))
10	9	raliunxp 4824	. . 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 1552	. . . . . . 7 ⊢ Ⅎ𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶)
12		nfv 1552	. . . . . . 7 ⊢ Ⅎ𝑤((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶)
13		nfv 1552	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
14		nfcsb1v 3128	. . . . . . . . . 10 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵
15	14	nfcri 2343	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵
16	13, 15	nfan 1589	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐴 ∧ 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵)
17		nfcsb1v 3128	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶
18	17	nfeq2 2361	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑣 = ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶
19	16, 18	nfan 1589	. . . . . . 7 ⊢ Ⅎ𝑥((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵) ∧ 𝑣 = ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶)
20		nfv 1552	. . . . . . . 8 ⊢ Ⅎ𝑦(𝑧 ∈ 𝐴 ∧ 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵)
21		nfcv 2349	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝑧
22		nfcsb1v 3128	. . . . . . . . . 10 ⊢ Ⅎ𝑦⦋𝑤 / 𝑦⦌𝐶
23	21, 22	nfcsb 3133	. . . . . . . . 9 ⊢ Ⅎ𝑦⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶
24	23	nfeq2 2361	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑣 = ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶
25	20, 24	nfan 1589	. . . . . . 7 ⊢ Ⅎ𝑦((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵) ∧ 𝑣 = ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶)
26		eleq1 2269	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
27	26	adantr 276	. . . . . . . . 9 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
28		eleq1 2269	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐵 ↔ 𝑤 ∈ 𝐵))
29		csbeq1a 3104	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
30	29	eleq2d 2276	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑤 ∈ 𝐵 ↔ 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵))
31	28, 30	sylan9bbr 463	. . . . . . . . 9 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑦 ∈ 𝐵 ↔ 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵))
32	27, 31	anbi12d 473	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵)))
33		csbeq1a 3104	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → 𝐶 = ⦋𝑤 / 𝑦⦌𝐶)
34		csbeq1a 3104	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → ⦋𝑤 / 𝑦⦌𝐶 = ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶)
35	33, 34	sylan9eqr 2261	. . . . . . . . 9 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶)
36	35	eqeq2d 2218	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑣 = 𝐶 ↔ 𝑣 = ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶))
37	32, 36	anbi12d 473	. . . . . . 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 6029	. . . . . 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 5959	. . . . . 6 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑣⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶)}
40		df-mpo 5959	. . . . . 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 2237	. . . . 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 6046	. . . . 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 2237	. . . 4 ⊢ 𝐹 = (𝑣 ∈ ∪ 𝑧 ∈ 𝐴 ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵) ↦ ⦋(1st ‘𝑣) / 𝑥⦌⦋(2nd ‘𝑣) / 𝑦⦌𝐶)
45	44	fmpt 5740	. . 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 186	. 2 ⊢ (∀𝑧 ∈ 𝐴 ∀𝑤 ∈ ⦋ 𝑧 / 𝑥⦌𝐵⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷 ↔ 𝐹:∪ 𝑧 ∈ 𝐴 ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)⟶𝐷)
47		nfv 1552	. . 3 ⊢ Ⅎ𝑧∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷
48	17	nfel1 2360	. . . 4 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷
49	14, 48	nfralxy 2545	. . 3 ⊢ Ⅎ𝑥∀𝑤 ∈ ⦋ 𝑧 / 𝑥⦌𝐵⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷
50		nfv 1552	. . . . 5 ⊢ Ⅎ𝑤 𝐶 ∈ 𝐷
51	22	nfel1 2360	. . . . 5 ⊢ Ⅎ𝑦⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷
52	33	eleq1d 2275	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝐶 ∈ 𝐷 ↔ ⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷))
53	50, 51, 52	cbvral 2735	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 ↔ ∀𝑤 ∈ 𝐵 ⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷)
54	34	eleq1d 2275	. . . . 5 ⊢ (𝑥 = 𝑧 → (⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷 ↔ ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷))
55	29, 54	raleqbidv 2719	. . . 4 ⊢ (𝑥 = 𝑧 → (∀𝑤 ∈ 𝐵 ⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷 ↔ ∀𝑤 ∈ ⦋ 𝑧 / 𝑥⦌𝐵⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷))
56	53, 55	bitrid 192	. . 3 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 ↔ ∀𝑤 ∈ ⦋ 𝑧 / 𝑥⦌𝐵⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷))
57	47, 49, 56	cbvral 2735	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ ⦋ 𝑧 / 𝑥⦌𝐵⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷)
58		nfcv 2349	. . . 4 ⊢ Ⅎ𝑧({𝑥} × 𝐵)
59		nfcv 2349	. . . . 5 ⊢ Ⅎ𝑥{𝑧}
60	59, 14	nfxp 4707	. . . 4 ⊢ Ⅎ𝑥({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)
61		sneq 3646	. . . . 5 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
62	61, 29	xpeq12d 4705	. . . 4 ⊢ (𝑥 = 𝑧 → ({𝑥} × 𝐵) = ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵))
63	58, 60, 62	cbviun 3967	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = ∪ 𝑧 ∈ 𝐴 ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)
64	63	feq2i 5426	. 2 ⊢ (𝐹:∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)⟶𝐷 ↔ 𝐹:∪ 𝑧 ∈ 𝐴 ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)⟶𝐷)
65	46, 57, 64	3bitr4i 212	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 ↔ 𝐹:∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)⟶𝐷)

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1373   ∈ wcel 2177  ∀wral 2485  ⦋csb 3095  {csn 3635  ⟨cop 3638  ∪ ciun 3930   ↦ cmpt 4110   × cxp 4678  ⟶wf 5273  ‘cfv 5277  {coprab 5955   ∈ cmpo 5956  1^st c1st 6234  2^nd c2nd 6235

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4167  ax-pow 4223  ax-pr 4258  ax-un 4485

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3001  df-csb 3096  df-un 3172  df-in 3174  df-ss 3181  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-iun 3932  df-br 4049  df-opab 4111  df-mpt 4112  df-id 4345  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-rn 4691  df-res 4692  df-ima 4693  df-iota 5238  df-fun 5279  df-fn 5280  df-f 5281  df-fv 5285  df-oprab 5958  df-mpo 5959  df-1st 6236  df-2nd 6237

This theorem is referenced by:  fmpo  6297