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Theorem cbvmpt2x 5521
Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpt2 5522 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 B𝐶) = (z A, w 𝐷𝐸)
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 1418 . . . . 5 z x A
2 cbvmpt2x.1 . . . . . 6 zB
32nfcri 2169 . . . . 5 z y B
41, 3nfan 1454 . . . 4 z(x A y B)
5 cbvmpt2x.3 . . . . 5 z𝐶
65nfeq2 2186 . . . 4 z u = 𝐶
74, 6nfan 1454 . . 3 z((x A y B) u = 𝐶)
8 nfv 1418 . . . . 5 w x A
9 nfcv 2175 . . . . . 6 wB
109nfcri 2169 . . . . 5 w y B
118, 10nfan 1454 . . . 4 w(x A y B)
12 cbvmpt2x.4 . . . . 5 w𝐶
1312nfeq2 2186 . . . 4 w u = 𝐶
1411, 13nfan 1454 . . 3 w((x A y B) u = 𝐶)
15 nfv 1418 . . . . 5 x z A
16 cbvmpt2x.2 . . . . . 6 x𝐷
1716nfcri 2169 . . . . 5 x w 𝐷
1815, 17nfan 1454 . . . 4 x(z A w 𝐷)
19 cbvmpt2x.5 . . . . 5 x𝐸
2019nfeq2 2186 . . . 4 x u = 𝐸
2118, 20nfan 1454 . . 3 x((z A w 𝐷) u = 𝐸)
22 nfv 1418 . . . 4 y(z A w 𝐷)
23 cbvmpt2x.6 . . . . 5 y𝐸
2423nfeq2 2186 . . . 4 y u = 𝐸
2522, 24nfan 1454 . . 3 y((z A w 𝐷) u = 𝐸)
26 eleq1 2097 . . . . . 6 (x = z → (x Az A))
2726adantr 261 . . . . 5 ((x = z y = w) → (x Az A))
28 cbvmpt2x.7 . . . . . . 7 (x = zB = 𝐷)
2928eleq2d 2104 . . . . . 6 (x = z → (y By 𝐷))
30 eleq1 2097 . . . . . 6 (y = w → (y 𝐷w 𝐷))
3129, 30sylan9bb 435 . . . . 5 ((x = z y = w) → (y Bw 𝐷))
3227, 31anbi12d 442 . . . 4 ((x = z y = w) → ((x A y B) ↔ (z A w 𝐷)))
33 cbvmpt2x.8 . . . . 5 ((x = z y = w) → 𝐶 = 𝐸)
3433eqeq2d 2048 . . . 4 ((x = z y = w) → (u = 𝐶u = 𝐸))
3532, 34anbi12d 442 . . 3 ((x = z y = w) → (((x A y B) u = 𝐶) ↔ ((z A w 𝐷) u = 𝐸)))
367, 14, 21, 25, 35cbvoprab12 5517 . 2 {⟨⟨x, y⟩, u⟩ ∣ ((x A y B) u = 𝐶)} = {⟨⟨z, w⟩, u⟩ ∣ ((z A w 𝐷) u = 𝐸)}
37 df-mpt2 5457 . 2 (x A, y B𝐶) = {⟨⟨x, y⟩, u⟩ ∣ ((x A y B) u = 𝐶)}
38 df-mpt2 5457 . 2 (z A, w 𝐷𝐸) = {⟨⟨z, w⟩, u⟩ ∣ ((z A w 𝐷) u = 𝐸)}
3936, 37, 383eqtr4i 2067 1 (x A, y B𝐶) = (z A, w 𝐷𝐸)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  wnfc 2162  {coprab 5453  cmpt2 5454
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3865  ax-pow 3917  ax-pr 3934
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3352  df-sn 3372  df-pr 3373  df-op 3375  df-opab 3809  df-oprab 5456  df-mpt2 5457
This theorem is referenced by:  cbvmpt2  5522  mpt2mptsx  5762  dmmpt2ssx  5764
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