Theorem fmpt2x 5731

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
fmpt2x.1	⊢ F = (x ∈ A, y ∈ B ↦ C)

Assertion

Ref	Expression
fmpt2x	⊢ (∀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 fmpt2x

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

Step	Hyp	Ref	Expression
1		vex 2863	. . . . . . . 8 ⊢ z ∈ V
2		vex 2863	. . . . . . . 8 ⊢ w ∈ V
3	1, 2	op1std 5523	. . . . . . 7 ⊢ (v = ⟨z, w⟩ → (1st ‘v) = z)
4	3	csbeq1d 3143	. . . . . 6 ⊢ (v = ⟨z, w⟩ → [(1st ‘v) / x][(2nd ‘v) / y]C = [z / x][(2nd ‘v) / y]C)
5	1, 2	op2ndd 5524	. . . . . . . 8 ⊢ (v = ⟨z, w⟩ → (2nd ‘v) = w)
6	5	csbeq1d 3143	. . . . . . 7 ⊢ (v = ⟨z, w⟩ → [(2nd ‘v) / y]C = [w / y]C)
7	6	csbeq2dv 3162	. . . . . 6 ⊢ (v = ⟨z, w⟩ → [z / x][(2nd ‘v) / y]C = [z / x][w / y]C)
8	4, 7	eqtrd 2385	. . . . 5 ⊢ (v = ⟨z, w⟩ → [(1st ‘v) / x][(2nd ‘v) / y]C = [z / x][w / y]C)
9	8	eleq1d 2419	. . . 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 1619	. . . . . . 7 ⊢ Ⅎz((x ∈ A ∧ y ∈ B) ∧ v = C)
12		nfv 1619	. . . . . . 7 ⊢ Ⅎw((x ∈ A ∧ y ∈ B) ∧ v = C)
13		nfv 1619	. . . . . . . . 9 ⊢ Ⅎx z ∈ A
14		nfcsb1v 3169	. . . . . . . . . 10 ⊢ Ⅎx[z / x]B
15	14	nfcri 2484	. . . . . . . . 9 ⊢ Ⅎx w ∈ [z / x]B
16	13, 15	nfan 1824	. . . . . . . 8 ⊢ Ⅎx(z ∈ A ∧ w ∈ [z / x]B)
17		nfcsb1v 3169	. . . . . . . . 9 ⊢ Ⅎx[z / x][w / y]C
18	17	nfeq2 2501	. . . . . . . 8 ⊢ Ⅎx v = [z / x][w / y]C
19	16, 18	nfan 1824	. . . . . . 7 ⊢ Ⅎx((z ∈ A ∧ w ∈ [z / x]B) ∧ v = [z / x][w / y]C)
20		nfv 1619	. . . . . . . 8 ⊢ Ⅎy(z ∈ A ∧ w ∈ [z / x]B)
21		nfcv 2490	. . . . . . . . . 10 ⊢ Ⅎyz
22		nfcsb1v 3169	. . . . . . . . . 10 ⊢ Ⅎy[w / y]C
23	21, 22	nfcsb 3171	. . . . . . . . 9 ⊢ Ⅎy[z / x][w / y]C
24	23	nfeq2 2501	. . . . . . . 8 ⊢ Ⅎy v = [z / x][w / y]C
25	20, 24	nfan 1824	. . . . . . 7 ⊢ Ⅎy((z ∈ A ∧ w ∈ [z / x]B) ∧ v = [z / x][w / y]C)
26		eleq1 2413	. . . . . . . . . 10 ⊢ (x = z → (x ∈ A ↔ z ∈ A))
27	26	adantr 451	. . . . . . . . 9 ⊢ ((x = z ∧ y = w) → (x ∈ A ↔ z ∈ A))
28		eleq1 2413	. . . . . . . . . 10 ⊢ (y = w → (y ∈ B ↔ w ∈ B))
29		csbeq1a 3145	. . . . . . . . . . 11 ⊢ (x = z → B = [z / x]B)
30	29	eleq2d 2420	. . . . . . . . . 10 ⊢ (x = z → (w ∈ B ↔ w ∈ [z / x]B))
31	28, 30	sylan9bbr 681	. . . . . . . . 9 ⊢ ((x = z ∧ y = w) → (y ∈ B ↔ w ∈ [z / x]B))
32	27, 31	anbi12d 691	. . . . . . . 8 ⊢ ((x = z ∧ y = w) → ((x ∈ A ∧ y ∈ B) ↔ (z ∈ A ∧ w ∈ [z / x]B)))
33		csbeq1a 3145	. . . . . . . . . 10 ⊢ (y = w → C = [w / y]C)
34		csbeq1a 3145	. . . . . . . . . 10 ⊢ (x = z → [w / y]C = [z / x][w / y]C)
35	33, 34	sylan9eqr 2407	. . . . . . . . 9 ⊢ ((x = z ∧ y = w) → C = [z / x][w / y]C)
36	35	eqeq2d 2364	. . . . . . . 8 ⊢ ((x = z ∧ y = w) → (v = C ↔ v = [z / x][w / y]C))
37	32, 36	anbi12d 691	. . . . . . 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 5570	. . . . . 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-mpt2 5655	. . . . . 6 ⊢ (x ∈ A, y ∈ B ↦ C) = {⟨⟨x, y⟩, v⟩ ∣ ((x ∈ A ∧ y ∈ B) ∧ v = C)}
40		df-mpt2 5655	. . . . . 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 2383	. . . . 5 ⊢ (x ∈ A, y ∈ B ↦ C) = (z ∈ A, w ∈ [z / x]B ↦ [z / x][w / y]C)
42		fmpt2x.1	. . . . 5 ⊢ F = (x ∈ A, y ∈ B ↦ C)
43	8	mpt2mptx 5709	. . . . 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 2383	. . . 4 ⊢ F = (v ∈ ∪z ∈ A ({z} × [z / x]B) ↦ [(1st ‘v) / x][(2nd ‘v) / y]C)
45	44	fmpt 5693	. . 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 242	. 2 ⊢ (∀z ∈ A ∀w ∈ [ z / x]B[z / x][w / y]C ∈ D ↔ F:∪z ∈ A ({z} × [z / x]B)–→D)
47		nfv 1619	. . 3 ⊢ Ⅎz∀y ∈ B C ∈ D
48	17	nfel1 2500	. . . 4 ⊢ Ⅎx[z / x][w / y]C ∈ D
49	14, 48	nfral 2668	. . 3 ⊢ Ⅎx∀w ∈ [ z / x]B[z / x][w / y]C ∈ D
50		nfv 1619	. . . . 5 ⊢ Ⅎw C ∈ D
51	22	nfel1 2500	. . . . 5 ⊢ Ⅎy[w / y]C ∈ D
52	33	eleq1d 2419	. . . . 5 ⊢ (y = w → (C ∈ D ↔ [w / y]C ∈ D))
53	50, 51, 52	cbvral 2832	. . . 4 ⊢ (∀y ∈ B C ∈ D ↔ ∀w ∈ B [w / y]C ∈ D)
54	34	eleq1d 2419	. . . . 5 ⊢ (x = z → ([w / y]C ∈ D ↔ [z / x][w / y]C ∈ D))
55	29, 54	raleqbidv 2820	. . . 4 ⊢ (x = z → (∀w ∈ B [w / y]C ∈ D ↔ ∀w ∈ [ z / x]B[z / x][w / y]C ∈ D))
56	53, 55	syl5bb 248	. . 3 ⊢ (x = z → (∀y ∈ B C ∈ D ↔ ∀w ∈ [ z / x]B[z / x][w / y]C ∈ D))
57	47, 49, 56	cbvral 2832	. 2 ⊢ (∀x ∈ A ∀y ∈ B C ∈ D ↔ ∀z ∈ A ∀w ∈ [ z / x]B[z / x][w / y]C ∈ D)
58		nfcv 2490	. . . 4 ⊢ Ⅎz({x} × B)
59		nfcv 2490	. . . . 5 ⊢ Ⅎx{z}
60	59, 14	nfxp 4811	. . . 4 ⊢ Ⅎx({z} × [z / x]B)
61		sneq 3745	. . . . 5 ⊢ (x = z → {x} = {z})
62	61, 29	xpeq12d 4810	. . . 4 ⊢ (x = z → ({x} × B) = ({z} × [z / x]B))
63	58, 60, 62	cbviun 4004	. . 3 ⊢ ∪x ∈ A ({x} × B) = ∪z ∈ A ({z} × [z / x]B)
64	63	feq2i 5219	. 2 ⊢ (F:∪x ∈ A ({x} × B)–→D ↔ F:∪z ∈ A ({z} × [z / x]B)–→D)
65	46, 57, 64	3bitr4i 268	1 ⊢ (∀x ∈ A ∀y ∈ B C ∈ D ↔ F:∪x ∈ A ({x} × B)–→D)

Syntax hints:   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2615  [csb 3137  {csn 3738  ∪ciun 3970  ⟨cop 4562  1^st c1st 4718   × cxp 4771  –→wf 4778   ‘cfv 4782  2^nd c2nd 4784  {coprab 5528   ↦ cmpt 5652   ↦ cmpt2 5654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-csb 3138  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-iun 3972  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-co 4727  df-ima 4728  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-fo 4794  df-fv 4796  df-2nd 4798  df-oprab 5529  df-mpt 5653  df-mpt2 5655

This theorem is referenced by:  fmpt2  5732