Theorem fmpox 6395

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 2815	. . . . . . . 8 ⊢ 𝑧 ∈ V
2		vex 2815	. . . . . . . 8 ⊢ 𝑤 ∈ V
3	1, 2	op1std 6341	. . . . . . 7 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (1st ‘𝑣) = 𝑧)
4	3	csbeq1d 3144	. . . . . 6 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → ⦋(1st ‘𝑣) / 𝑥⦌⦋(2nd ‘𝑣) / 𝑦⦌𝐶 = ⦋𝑧 / 𝑥⦌⦋(2nd ‘𝑣) / 𝑦⦌𝐶)
5	1, 2	op2ndd 6342	. . . . . . . 8 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (2nd ‘𝑣) = 𝑤)
6	5	csbeq1d 3144	. . . . . . 7 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → ⦋(2nd ‘𝑣) / 𝑦⦌𝐶 = ⦋𝑤 / 𝑦⦌𝐶)
7	6	csbeq2dv 3163	. . . . . 6 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → ⦋𝑧 / 𝑥⦌⦋(2nd ‘𝑣) / 𝑦⦌𝐶 = ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶)
8	4, 7	eqtrd 2265	. . . . 5 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → ⦋(1st ‘𝑣) / 𝑥⦌⦋(2nd ‘𝑣) / 𝑦⦌𝐶 = ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶)
9	8	eleq1d 2301	. . . 4 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (⦋(1st ‘𝑣) / 𝑥⦌⦋(2nd ‘𝑣) / 𝑦⦌𝐶 ∈ 𝐷 ↔ ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷))
10	9	raliunxp 4895	. . 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 1577	. . . . . . 7 ⊢ Ⅎ𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶)
12		nfv 1577	. . . . . . 7 ⊢ Ⅎ𝑤((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶)
13		nfv 1577	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
14		nfcsb1v 3170	. . . . . . . . . 10 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵
15	14	nfcri 2378	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵
16	13, 15	nfan 1614	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐴 ∧ 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵)
17		nfcsb1v 3170	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶
18	17	nfeq2 2396	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑣 = ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶
19	16, 18	nfan 1614	. . . . . . 7 ⊢ Ⅎ𝑥((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵) ∧ 𝑣 = ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶)
20		nfv 1577	. . . . . . . 8 ⊢ Ⅎ𝑦(𝑧 ∈ 𝐴 ∧ 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵)
21		nfcv 2384	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝑧
22		nfcsb1v 3170	. . . . . . . . . 10 ⊢ Ⅎ𝑦⦋𝑤 / 𝑦⦌𝐶
23	21, 22	nfcsb 3175	. . . . . . . . 9 ⊢ Ⅎ𝑦⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶
24	23	nfeq2 2396	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑣 = ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶
25	20, 24	nfan 1614	. . . . . . 7 ⊢ Ⅎ𝑦((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵) ∧ 𝑣 = ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶)
26		eleq1 2295	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
27	26	adantr 276	. . . . . . . . 9 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
28		eleq1 2295	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐵 ↔ 𝑤 ∈ 𝐵))
29		csbeq1a 3146	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
30	29	eleq2d 2302	. . . . . . . . . 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 3146	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → 𝐶 = ⦋𝑤 / 𝑦⦌𝐶)
34		csbeq1a 3146	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → ⦋𝑤 / 𝑦⦌𝐶 = ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶)
35	33, 34	sylan9eqr 2287	. . . . . . . . 9 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶)
36	35	eqeq2d 2244	. . . . . . . 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 6126	. . . . . 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 6054	. . . . . 6 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑣⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶)}
40		df-mpo 6054	. . . . . 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 2263	. . . . 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 6143	. . . . 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 2263	. . . 4 ⊢ 𝐹 = (𝑣 ∈ ∪ 𝑧 ∈ 𝐴 ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵) ↦ ⦋(1st ‘𝑣) / 𝑥⦌⦋(2nd ‘𝑣) / 𝑦⦌𝐶)
45	44	fmpt 5826	. . 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 1577	. . 3 ⊢ Ⅎ𝑧∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷
48	17	nfel1 2395	. . . 4 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷
49	14, 48	nfralxy 2580	. . 3 ⊢ Ⅎ𝑥∀𝑤 ∈ ⦋ 𝑧 / 𝑥⦌𝐵⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷
50		nfv 1577	. . . . 5 ⊢ Ⅎ𝑤 𝐶 ∈ 𝐷
51	22	nfel1 2395	. . . . 5 ⊢ Ⅎ𝑦⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷
52	33	eleq1d 2301	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝐶 ∈ 𝐷 ↔ ⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷))
53	50, 51, 52	cbvral 2773	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 ↔ ∀𝑤 ∈ 𝐵 ⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷)
54	34	eleq1d 2301	. . . . 5 ⊢ (𝑥 = 𝑧 → (⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷 ↔ ⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷))
55	29, 54	raleqbidv 2756	. . . 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 2773	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ ⦋ 𝑧 / 𝑥⦌𝐵⦋𝑧 / 𝑥⦌⦋𝑤 / 𝑦⦌𝐶 ∈ 𝐷)
58		nfcv 2384	. . . 4 ⊢ Ⅎ𝑧({𝑥} × 𝐵)
59		nfcv 2384	. . . . 5 ⊢ Ⅎ𝑥{𝑧}
60	59, 14	nfxp 4775	. . . 4 ⊢ Ⅎ𝑥({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)
61		sneq 3699	. . . . 5 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
62	61, 29	xpeq12d 4773	. . . 4 ⊢ (𝑥 = 𝑧 → ({𝑥} × 𝐵) = ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵))
63	58, 60, 62	cbviun 4027	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = ∪ 𝑧 ∈ 𝐴 ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)
64	63	feq2i 5501	. 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 1398   ∈ wcel 2203  ∀wral 2520  ⦋csb 3137  {csn 3688  ⟨cop 3691  ∪ ciun 3990   ↦ cmpt 4170   × cxp 4746  ⟶wf 5347  ‘cfv 5351  {coprab 6050   ∈ cmpo 6051  1^st c1st 6331  2^nd c2nd 6332

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-fv 5359  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334

This theorem is referenced by:  fmpo  6396