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Theorem fmpt2x 5730
 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 DF: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.
StepHypRef Expression
1 vex 2862 . . . . . . . 8 z V
2 vex 2862 . . . . . . . 8 w V
31, 2op1std 5522 . . . . . . 7 (v = z, w → (1stv) = z)
43csbeq1d 3142 . . . . . 6 (v = z, w[(1stv) / x][(2ndv) / y]C = [z / x][(2ndv) / y]C)
51, 2op2ndd 5523 . . . . . . . 8 (v = z, w → (2ndv) = w)
65csbeq1d 3142 . . . . . . 7 (v = z, w[(2ndv) / y]C = [w / y]C)
76csbeq2dv 3161 . . . . . 6 (v = z, w[z / x][(2ndv) / y]C = [z / x][w / y]C)
84, 7eqtrd 2385 . . . . 5 (v = z, w[(1stv) / x][(2ndv) / y]C = [z / x][w / y]C)
98eleq1d 2419 . . . 4 (v = z, w → ([(1stv) / x][(2ndv) / y]C D[z / x][w / y]C D))
109raliunxp 4823 . . 3 (v z A ({z} × [z / x]B)[(1stv) / x][(2ndv) / y]C Dz A w [ z / x]B[z / x][w / y]C D)
11 nfv 1619 . . . . . . 7 z((x A y B) v = C)
12 nfv 1619 . . . . . . 7 w((x A y B) v = C)
13 nfv 1619 . . . . . . . . 9 x z A
14 nfcsb1v 3168 . . . . . . . . . 10 x[z / x]B
1514nfcri 2483 . . . . . . . . 9 x w [z / x]B
1613, 15nfan 1824 . . . . . . . 8 x(z A w [z / x]B)
17 nfcsb1v 3168 . . . . . . . . 9 x[z / x][w / y]C
1817nfeq2 2500 . . . . . . . 8 x v = [z / x][w / y]C
1916, 18nfan 1824 . . . . . . 7 x((z A w [z / x]B) v = [z / x][w / y]C)
20 nfv 1619 . . . . . . . 8 y(z A w [z / x]B)
21 nfcv 2489 . . . . . . . . . 10 yz
22 nfcsb1v 3168 . . . . . . . . . 10 y[w / y]C
2321, 22nfcsb 3170 . . . . . . . . 9 y[z / x][w / y]C
2423nfeq2 2500 . . . . . . . 8 y v = [z / x][w / y]C
2520, 24nfan 1824 . . . . . . 7 y((z A w [z / x]B) v = [z / x][w / y]C)
26 eleq1 2413 . . . . . . . . . 10 (x = z → (x Az A))
2726adantr 451 . . . . . . . . 9 ((x = z y = w) → (x Az A))
28 eleq1 2413 . . . . . . . . . 10 (y = w → (y Bw B))
29 csbeq1a 3144 . . . . . . . . . . 11 (x = zB = [z / x]B)
3029eleq2d 2420 . . . . . . . . . 10 (x = z → (w Bw [z / x]B))
3128, 30sylan9bbr 681 . . . . . . . . 9 ((x = z y = w) → (y Bw [z / x]B))
3227, 31anbi12d 691 . . . . . . . 8 ((x = z y = w) → ((x A y B) ↔ (z A w [z / x]B)))
33 csbeq1a 3144 . . . . . . . . . 10 (y = wC = [w / y]C)
34 csbeq1a 3144 . . . . . . . . . 10 (x = z[w / y]C = [z / x][w / y]C)
3533, 34sylan9eqr 2407 . . . . . . . . 9 ((x = z y = w) → C = [z / x][w / y]C)
3635eqeq2d 2364 . . . . . . . 8 ((x = z y = w) → (v = Cv = [z / x][w / y]C))
3732, 36anbi12d 691 . . . . . . 7 ((x = z y = w) → (((x A y B) v = C) ↔ ((z A w [z / x]B) v = [z / x][w / y]C)))
3811, 12, 19, 25, 37cbvoprab12 5569 . . . . . 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 5654 . . . . . 6 (x A, y B C) = {x, y, v ((x A y B) v = C)}
40 df-mpt2 5654 . . . . . 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)}
4138, 39, 403eqtr4i 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)
438mpt2mptx 5708 . . . . 5 (v z A ({z} × [z / x]B) [(1stv) / x][(2ndv) / y]C) = (z A, w [z / x]B [z / x][w / y]C)
4441, 42, 433eqtr4i 2383 . . . 4 F = (v z A ({z} × [z / x]B) [(1stv) / x][(2ndv) / y]C)
4544fmpt 5692 . . 3 (v z A ({z} × [z / x]B)[(1stv) / x][(2ndv) / y]C DF:z A ({z} × [z / x]B)–→D)
4610, 45bitr3i 242 . 2 (z A w [ z / x]B[z / x][w / y]C DF:z A ({z} × [z / x]B)–→D)
47 nfv 1619 . . 3 zy B C D
4817nfel1 2499 . . . 4 x[z / x][w / y]C D
4914, 48nfral 2667 . . 3 xw [ z / x]B[z / x][w / y]C D
50 nfv 1619 . . . . 5 w C D
5122nfel1 2499 . . . . 5 y[w / y]C D
5233eleq1d 2419 . . . . 5 (y = w → (C D[w / y]C D))
5350, 51, 52cbvral 2831 . . . 4 (y B C Dw B [w / y]C D)
5434eleq1d 2419 . . . . 5 (x = z → ([w / y]C D[z / x][w / y]C D))
5529, 54raleqbidv 2819 . . . 4 (x = z → (w B [w / y]C Dw [ z / x]B[z / x][w / y]C D))
5653, 55syl5bb 248 . . 3 (x = z → (y B C Dw [ z / x]B[z / x][w / y]C D))
5747, 49, 56cbvral 2831 . 2 (x A y B C Dz A w [ z / x]B[z / x][w / y]C D)
58 nfcv 2489 . . . 4 z({x} × B)
59 nfcv 2489 . . . . 5 x{z}
6059, 14nfxp 4810 . . . 4 x({z} × [z / x]B)
61 sneq 3744 . . . . 5 (x = z → {x} = {z})
6261, 29xpeq12d 4809 . . . 4 (x = z → ({x} × B) = ({z} × [z / x]B))
6358, 60, 62cbviun 4003 . . 3 x A ({x} × B) = z A ({z} × [z / x]B)
6463feq2i 5218 . 2 (F:x A ({x} × B)–→DF:z A ({z} × [z / x]B)–→D)
6546, 57, 643bitr4i 268 1 (x A y B C DF:x A ({x} × B)–→D)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2614  [csb 3136  {csn 3737  ∪ciun 3969  ⟨cop 4561  1st c1st 4717   × cxp 4770  –→wf 4777   ‘cfv 4781  2nd c2nd 4783  {coprab 5527   ↦ cmpt 5651   ↦ cmpt2 5653 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-csb 3137  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-iun 3971  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-co 4726  df-ima 4727  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-fo 4793  df-fv 4795  df-2nd 4797  df-oprab 5528  df-mpt 5652  df-mpt2 5654 This theorem is referenced by:  fmpt2  5731
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