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Theorem cbvmpt2x 5582
 Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpt2 5583 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 1421 . . . . 5 𝑧 𝑥𝐴
2 cbvmpt2x.1 . . . . . 6 𝑧𝐵
32nfcri 2172 . . . . 5 𝑧 𝑦𝐵
41, 3nfan 1457 . . . 4 𝑧(𝑥𝐴𝑦𝐵)
5 cbvmpt2x.3 . . . . 5 𝑧𝐶
65nfeq2 2189 . . . 4 𝑧 𝑢 = 𝐶
74, 6nfan 1457 . . 3 𝑧((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)
8 nfv 1421 . . . . 5 𝑤 𝑥𝐴
9 nfcv 2178 . . . . . 6 𝑤𝐵
109nfcri 2172 . . . . 5 𝑤 𝑦𝐵
118, 10nfan 1457 . . . 4 𝑤(𝑥𝐴𝑦𝐵)
12 cbvmpt2x.4 . . . . 5 𝑤𝐶
1312nfeq2 2189 . . . 4 𝑤 𝑢 = 𝐶
1411, 13nfan 1457 . . 3 𝑤((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)
15 nfv 1421 . . . . 5 𝑥 𝑧𝐴
16 cbvmpt2x.2 . . . . . 6 𝑥𝐷
1716nfcri 2172 . . . . 5 𝑥 𝑤𝐷
1815, 17nfan 1457 . . . 4 𝑥(𝑧𝐴𝑤𝐷)
19 cbvmpt2x.5 . . . . 5 𝑥𝐸
2019nfeq2 2189 . . . 4 𝑥 𝑢 = 𝐸
2118, 20nfan 1457 . . 3 𝑥((𝑧𝐴𝑤𝐷) ∧ 𝑢 = 𝐸)
22 nfv 1421 . . . 4 𝑦(𝑧𝐴𝑤𝐷)
23 cbvmpt2x.6 . . . . 5 𝑦𝐸
2423nfeq2 2189 . . . 4 𝑦 𝑢 = 𝐸
2522, 24nfan 1457 . . 3 𝑦((𝑧𝐴𝑤𝐷) ∧ 𝑢 = 𝐸)
26 eleq1 2100 . . . . . 6 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
2726adantr 261 . . . . 5 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑥𝐴𝑧𝐴))
28 cbvmpt2x.7 . . . . . . 7 (𝑥 = 𝑧𝐵 = 𝐷)
2928eleq2d 2107 . . . . . 6 (𝑥 = 𝑧 → (𝑦𝐵𝑦𝐷))
30 eleq1 2100 . . . . . 6 (𝑦 = 𝑤 → (𝑦𝐷𝑤𝐷))
3129, 30sylan9bb 435 . . . . 5 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑦𝐵𝑤𝐷))
3227, 31anbi12d 442 . . . 4 ((𝑥 = 𝑧𝑦 = 𝑤) → ((𝑥𝐴𝑦𝐵) ↔ (𝑧𝐴𝑤𝐷)))
33 cbvmpt2x.8 . . . . 5 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝐶 = 𝐸)
3433eqeq2d 2051 . . . 4 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑢 = 𝐶𝑢 = 𝐸))
3532, 34anbi12d 442 . . 3 ((𝑥 = 𝑧𝑦 = 𝑤) → (((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶) ↔ ((𝑧𝐴𝑤𝐷) ∧ 𝑢 = 𝐸)))
367, 14, 21, 25, 35cbvoprab12 5578 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)} = {⟨⟨𝑧, 𝑤⟩, 𝑢⟩ ∣ ((𝑧𝐴𝑤𝐷) ∧ 𝑢 = 𝐸)}
37 df-mpt2 5517 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)}
38 df-mpt2 5517 . 2 (𝑧𝐴, 𝑤𝐷𝐸) = {⟨⟨𝑧, 𝑤⟩, 𝑢⟩ ∣ ((𝑧𝐴𝑤𝐷) ∧ 𝑢 = 𝐸)}
3936, 37, 383eqtr4i 2070 1 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑤𝐷𝐸)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393  Ⅎwnfc 2165  {coprab 5513   ↦ cmpt2 5514 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-opab 3819  df-oprab 5516  df-mpt2 5517 This theorem is referenced by:  cbvmpt2  5583  mpt2mptsx  5823  dmmpt2ssx  5825
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