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Theorem cbvmpt2x 5609
Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpt2 5610 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 = 𝑧𝑦 = 𝑤) → 𝐶 = 𝐸)
Assertion
Ref Expression
cbvmpt2x (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑤𝐷𝐸)
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 1437 . . . . 5 𝑧 𝑥𝐴
2 cbvmpt2x.1 . . . . . 6 𝑧𝐵
32nfcri 2188 . . . . 5 𝑧 𝑦𝐵
41, 3nfan 1473 . . . 4 𝑧(𝑥𝐴𝑦𝐵)
5 cbvmpt2x.3 . . . . 5 𝑧𝐶
65nfeq2 2205 . . . 4 𝑧 𝑢 = 𝐶
74, 6nfan 1473 . . 3 𝑧((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)
8 nfv 1437 . . . . 5 𝑤 𝑥𝐴
9 nfcv 2194 . . . . . 6 𝑤𝐵
109nfcri 2188 . . . . 5 𝑤 𝑦𝐵
118, 10nfan 1473 . . . 4 𝑤(𝑥𝐴𝑦𝐵)
12 cbvmpt2x.4 . . . . 5 𝑤𝐶
1312nfeq2 2205 . . . 4 𝑤 𝑢 = 𝐶
1411, 13nfan 1473 . . 3 𝑤((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)
15 nfv 1437 . . . . 5 𝑥 𝑧𝐴
16 cbvmpt2x.2 . . . . . 6 𝑥𝐷
1716nfcri 2188 . . . . 5 𝑥 𝑤𝐷
1815, 17nfan 1473 . . . 4 𝑥(𝑧𝐴𝑤𝐷)
19 cbvmpt2x.5 . . . . 5 𝑥𝐸
2019nfeq2 2205 . . . 4 𝑥 𝑢 = 𝐸
2118, 20nfan 1473 . . 3 𝑥((𝑧𝐴𝑤𝐷) ∧ 𝑢 = 𝐸)
22 nfv 1437 . . . 4 𝑦(𝑧𝐴𝑤𝐷)
23 cbvmpt2x.6 . . . . 5 𝑦𝐸
2423nfeq2 2205 . . . 4 𝑦 𝑢 = 𝐸
2522, 24nfan 1473 . . 3 𝑦((𝑧𝐴𝑤𝐷) ∧ 𝑢 = 𝐸)
26 eleq1 2116 . . . . . 6 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
2726adantr 265 . . . . 5 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑥𝐴𝑧𝐴))
28 cbvmpt2x.7 . . . . . . 7 (𝑥 = 𝑧𝐵 = 𝐷)
2928eleq2d 2123 . . . . . 6 (𝑥 = 𝑧 → (𝑦𝐵𝑦𝐷))
30 eleq1 2116 . . . . . 6 (𝑦 = 𝑤 → (𝑦𝐷𝑤𝐷))
3129, 30sylan9bb 443 . . . . 5 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑦𝐵𝑤𝐷))
3227, 31anbi12d 450 . . . 4 ((𝑥 = 𝑧𝑦 = 𝑤) → ((𝑥𝐴𝑦𝐵) ↔ (𝑧𝐴𝑤𝐷)))
33 cbvmpt2x.8 . . . . 5 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝐶 = 𝐸)
3433eqeq2d 2067 . . . 4 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑢 = 𝐶𝑢 = 𝐸))
3532, 34anbi12d 450 . . 3 ((𝑥 = 𝑧𝑦 = 𝑤) → (((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶) ↔ ((𝑧𝐴𝑤𝐷) ∧ 𝑢 = 𝐸)))
367, 14, 21, 25, 35cbvoprab12 5605 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)} = {⟨⟨𝑧, 𝑤⟩, 𝑢⟩ ∣ ((𝑧𝐴𝑤𝐷) ∧ 𝑢 = 𝐸)}
37 df-mpt2 5544 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)}
38 df-mpt2 5544 . 2 (𝑧𝐴, 𝑤𝐷𝐸) = {⟨⟨𝑧, 𝑤⟩, 𝑢⟩ ∣ ((𝑧𝐴𝑤𝐷) ∧ 𝑢 = 𝐸)}
3936, 37, 383eqtr4i 2086 1 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑤𝐷𝐸)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  wnfc 2181  {coprab 5540  cmpt2 5541
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-opab 3846  df-oprab 5543  df-mpt2 5544
This theorem is referenced by:  cbvmpt2  5610  mpt2mptsx  5850  dmmpt2ssx  5852
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