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Theorem cbvmpt2x 5678
 Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpt2 5679 allows B to be a function of x. (Contributed by NM, 29-Dec-2014.)
Hypotheses
Ref Expression
cbvmpt2x.1 zB
cbvmpt2x.2 xD
cbvmpt2x.3 zC
cbvmpt2x.4 wC
cbvmpt2x.5 xE
cbvmpt2x.6 yE
cbvmpt2x.7 (x = zB = D)
cbvmpt2x.8 ((x = z y = w) → C = E)
Assertion
Ref Expression
cbvmpt2x (x A, y B C) = (z A, w D E)
Distinct variable groups:   x,w,y,z,A   w,B   y,D
Allowed substitution hints:   B(x,y,z)   C(x,y,z,w)   D(x,z,w)   E(x,y,z,w)

Proof of Theorem cbvmpt2x
Dummy variable u is distinct from all other variables.
StepHypRef Expression
1 nfv 1619 . . . . 5 z x A
2 cbvmpt2x.1 . . . . . 6 zB
32nfcri 2483 . . . . 5 z y B
41, 3nfan 1824 . . . 4 z(x A y B)
5 cbvmpt2x.3 . . . . 5 zC
65nfeq2 2500 . . . 4 z u = C
74, 6nfan 1824 . . 3 z((x A y B) u = C)
8 nfv 1619 . . . . 5 w x A
9 nfcv 2489 . . . . . 6 wB
109nfcri 2483 . . . . 5 w y B
118, 10nfan 1824 . . . 4 w(x A y B)
12 cbvmpt2x.4 . . . . 5 wC
1312nfeq2 2500 . . . 4 w u = C
1411, 13nfan 1824 . . 3 w((x A y B) u = C)
15 nfv 1619 . . . . 5 x z A
16 cbvmpt2x.2 . . . . . 6 xD
1716nfcri 2483 . . . . 5 x w D
1815, 17nfan 1824 . . . 4 x(z A w D)
19 cbvmpt2x.5 . . . . 5 xE
2019nfeq2 2500 . . . 4 x u = E
2118, 20nfan 1824 . . 3 x((z A w D) u = E)
22 nfv 1619 . . . 4 y(z A w D)
23 cbvmpt2x.6 . . . . 5 yE
2423nfeq2 2500 . . . 4 y u = E
2522, 24nfan 1824 . . 3 y((z A w D) u = E)
26 eleq1 2413 . . . . . 6 (x = z → (x Az A))
2726adantr 451 . . . . 5 ((x = z y = w) → (x Az A))
28 cbvmpt2x.7 . . . . . . 7 (x = zB = D)
2928eleq2d 2420 . . . . . 6 (x = z → (y By D))
30 eleq1 2413 . . . . . 6 (y = w → (y Dw D))
3129, 30sylan9bb 680 . . . . 5 ((x = z y = w) → (y Bw D))
3227, 31anbi12d 691 . . . 4 ((x = z y = w) → ((x A y B) ↔ (z A w D)))
33 cbvmpt2x.8 . . . . 5 ((x = z y = w) → C = E)
3433eqeq2d 2364 . . . 4 ((x = z y = w) → (u = Cu = E))
3532, 34anbi12d 691 . . 3 ((x = z y = w) → (((x A y B) u = C) ↔ ((z A w D) u = E)))
367, 14, 21, 25, 35cbvoprab12 5569 . 2 {x, y, u ((x A y B) u = C)} = {z, w, u ((z A w D) u = E)}
37 df-mpt2 5654 . 2 (x A, y B C) = {x, y, u ((x A y B) u = C)}
38 df-mpt2 5654 . 2 (z A, w D E) = {z, w, u ((z A w D) u = E)}
3936, 37, 383eqtr4i 2383 1 (x A, y B C) = (z A, w D E)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2476  {coprab 5527   ↦ cmpt2 5653 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-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-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  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-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-opab 4623  df-oprab 5528  df-mpt2 5654 This theorem is referenced by:  cbvmpt2  5679
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