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 e. A, y e. B |-> C)

Assertion

Ref	Expression
fmpt2x	|- (A.x e. A A.y e. B C e. D <-> F:U_x e. A ({x} 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 e. _V
2		vex 2863	. . . . . . . 8 |- w e. _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 e. D <-> [_z / x]_[_w / y]_C e. D))
10	9	raliunxp 4824	. . 3 |- (A.v e. U_ z e. A ({z} X. [_z / x]_B)[_(1st` v) / x]_[_(2nd` v) / y]_C e. D <-> A.z e. A A.w e. [_ z / x]_B[_z / x]_[_w / y]_C e. D)
11		nfv 1619	. . . . . . 7 |- F/z((x e. A /\ y e. B) /\ v = C)
12		nfv 1619	. . . . . . 7 |- F/w((x e. A /\ y e. B) /\ v = C)
13		nfv 1619	. . . . . . . . 9 |- F/x z e. A
14		nfcsb1v 3169	. . . . . . . . . 10 |- F/_x[_z / x]_B
15	14	nfcri 2484	. . . . . . . . 9 |- F/x w e. [_z / x]_B
16	13, 15	nfan 1824	. . . . . . . 8 |- F/x(z e. A /\ w e. [_z / x]_B)
17		nfcsb1v 3169	. . . . . . . . 9 |- F/_x[_z / x]_[_w / y]_C
18	17	nfeq2 2501	. . . . . . . 8 |- F/x v = [_z / x]_[_w / y]_C
19	16, 18	nfan 1824	. . . . . . 7 |- F/x((z e. A /\ w e. [_z / x]_B) /\ v = [_z / x]_[_w / y]_C)
20		nfv 1619	. . . . . . . 8 |- F/y(z e. A /\ w e. [_z / x]_B)
21		nfcv 2490	. . . . . . . . . 10 |- F/_yz
22		nfcsb1v 3169	. . . . . . . . . 10 |- F/_y[_w / y]_C
23	21, 22	nfcsb 3171	. . . . . . . . 9 |- F/_y[_z / x]_[_w / y]_C
24	23	nfeq2 2501	. . . . . . . 8 |- F/y v = [_z / x]_[_w / y]_C
25	20, 24	nfan 1824	. . . . . . 7 |- F/y((z e. A /\ w e. [_z / x]_B) /\ v = [_z / x]_[_w / y]_C)
26		eleq1 2413	. . . . . . . . . 10 |- (x = z -> (x e. A <-> z e. A))
27	26	adantr 451	. . . . . . . . 9 |- ((x = z /\ y = w) -> (x e. A <-> z e. A))
28		eleq1 2413	. . . . . . . . . 10 |- (y = w -> (y e. B <-> w e. B))
29		csbeq1a 3145	. . . . . . . . . . 11 |- (x = z -> B = [_z / x]_B)
30	29	eleq2d 2420	. . . . . . . . . 10 |- (x = z -> (w e. B <-> w e. [_z / x]_B))
31	28, 30	sylan9bbr 681	. . . . . . . . 9 |- ((x = z /\ y = w) -> (y e. B <-> w e. [_z / x]_B))
32	27, 31	anbi12d 691	. . . . . . . 8 |- ((x = z /\ y = w) -> ((x e. A /\ y e. B) <-> (z e. A /\ w e. [_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 e. A /\ y e. B) /\ v = C) <-> ((z e. A /\ w e. [_z / x]_B) /\ v = [_z / x]_[_w / y]_C)))
38	11, 12, 19, 25, 37	cbvoprab12 5570	. . . . . 6 |- {<.<.x, y>., v>. | ((x e. A /\ y e. B) /\ v = C)} = {<.<.z, w>., v>. | ((z e. A /\ w e. [_z / x]_B) /\ v = [_z / x]_[_w / y]_C)}
39		df-mpt2 5655	. . . . . 6 |- (x e. A, y e. B |-> C) = {<.<.x, y>., v>. | ((x e. A /\ y e. B) /\ v = C)}
40		df-mpt2 5655	. . . . . 6 |- (z e. A, w e. [_z / x]_B |-> [_z / x]_[_w / y]_C) = {<.<.z, w>., v>. | ((z e. A /\ w e. [_z / x]_B) /\ v = [_z / x]_[_w / y]_C)}
41	38, 39, 40	3eqtr4i 2383	. . . . 5 |- (x e. A, y e. B |-> C) = (z e. A, w e. [_z / x]_B |-> [_z / x]_[_w / y]_C)
42		fmpt2x.1	. . . . 5 |- F = (x e. A, y e. B |-> C)
43	8	mpt2mptx 5709	. . . . 5 |- (v e. U_z e. A ({z} X. [_z / x]_B) |-> [_(1st` v) / x]_[_(2nd` v) / y]_C) = (z e. A, w e. [_z / x]_B |-> [_z / x]_[_w / y]_C)
44	41, 42, 43	3eqtr4i 2383	. . . 4 |- F = (v e. U_z e. A ({z} X. [_z / x]_B) |-> [_(1st` v) / x]_[_(2nd` v) / y]_C)
45	44	fmpt 5693	. . 3 |- (A.v e. U_ z e. A ({z} X. [_z / x]_B)[_(1st` v) / x]_[_(2nd` v) / y]_C e. D <-> F:U_z e. A ({z} X. [_z / x]_B)-->D)
46	10, 45	bitr3i 242	. 2 |- (A.z e. A A.w e. [_ z / x]_B[_z / x]_[_w / y]_C e. D <-> F:U_z e. A ({z} X. [_z / x]_B)-->D)
47		nfv 1619	. . 3 |- F/zA.y e. B C e. D
48	17	nfel1 2500	. . . 4 |- F/x[_z / x]_[_w / y]_C e. D
49	14, 48	nfral 2668	. . 3 |- F/xA.w e. [_ z / x]_B[_z / x]_[_w / y]_C e. D
50		nfv 1619	. . . . 5 |- F/w C e. D
51	22	nfel1 2500	. . . . 5 |- F/y[_w / y]_C e. D
52	33	eleq1d 2419	. . . . 5 |- (y = w -> (C e. D <-> [_w / y]_C e. D))
53	50, 51, 52	cbvral 2832	. . . 4 |- (A.y e. B C e. D <-> A.w e. B [_w / y]_C e. D)
54	34	eleq1d 2419	. . . . 5 |- (x = z -> ([_w / y]_C e. D <-> [_z / x]_[_w / y]_C e. D))
55	29, 54	raleqbidv 2820	. . . 4 |- (x = z -> (A.w e. B [_w / y]_C e. D <-> A.w e. [_ z / x]_B[_z / x]_[_w / y]_C e. D))
56	53, 55	syl5bb 248	. . 3 |- (x = z -> (A.y e. B C e. D <-> A.w e. [_ z / x]_B[_z / x]_[_w / y]_C e. D))
57	47, 49, 56	cbvral 2832	. 2 |- (A.x e. A A.y e. B C e. D <-> A.z e. A A.w e. [_ z / x]_B[_z / x]_[_w / y]_C e. D)
58		nfcv 2490	. . . 4 |- F/_z({x} X. B)
59		nfcv 2490	. . . . 5 |- F/_x{z}
60	59, 14	nfxp 4811	. . . 4 |- F/_x({z} X. [_z / x]_B)
61		sneq 3745	. . . . 5 |- (x = z -> {x} = {z})
62	61, 29	xpeq12d 4810	. . . 4 |- (x = z -> ({x} X. B) = ({z} X. [_z / x]_B))
63	58, 60, 62	cbviun 4004	. . 3 |- U_x e. A ({x} X. B) = U_z e. A ({z} X. [_z / x]_B)
64	63	feq2i 5219	. 2 |- (F:U_x e. A ({x} X. B)-->D <-> F:U_z e. A ({z} X. [_z / x]_B)-->D)
65	46, 57, 64	3bitr4i 268	1 |- (A.x e. A A.y e. B C e. D <-> F:U_x e. A ({x} X. B)-->D)

Syntax hints:   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710  A.wral 2615  [_csb 3137  {csn 3738  U_ciun 3970  <.cop 4562  1stc1st 4718   X. cxp 4771  -->wf 4778  ` cfv 4782  2ndc2nd 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