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Theorem cbvmpt2x2 42642
Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpt2 6900 allows 𝐴 to be a function of 𝑦, analogous to cbvmpt2x 6899. (Contributed by AV, 30-Mar-2019.)
Hypotheses
Ref Expression
cbvmpt2x2.1 𝑧𝐴
cbvmpt2x2.2 𝑦𝐷
cbvmpt2x2.3 𝑧𝐶
cbvmpt2x2.4 𝑤𝐶
cbvmpt2x2.5 𝑥𝐸
cbvmpt2x2.6 𝑦𝐸
cbvmpt2x2.7 (𝑦 = 𝑧𝐴 = 𝐷)
cbvmpt2x2.8 ((𝑦 = 𝑧𝑥 = 𝑤) → 𝐶 = 𝐸)
Assertion
Ref Expression
cbvmpt2x2 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑤𝐷, 𝑧𝐵𝐸)
Distinct variable groups:   𝑥,𝑤,𝑦,𝑧   𝑤,𝐴   𝑤,𝐵,𝑥,𝑦,𝑧   𝑥,𝐷
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧)   𝐶(𝑥,𝑦,𝑧,𝑤)   𝐷(𝑦,𝑧,𝑤)   𝐸(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem cbvmpt2x2
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 nfv 1992 . . . . 5 𝑤 𝑥𝐴
2 nfv 1992 . . . . 5 𝑤 𝑦𝐵
31, 2nfan 1977 . . . 4 𝑤(𝑥𝐴𝑦𝐵)
4 cbvmpt2x2.4 . . . . 5 𝑤𝐶
54nfeq2 2918 . . . 4 𝑤 𝑢 = 𝐶
63, 5nfan 1977 . . 3 𝑤((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)
7 cbvmpt2x2.1 . . . . . 6 𝑧𝐴
87nfcri 2896 . . . . 5 𝑧 𝑥𝐴
9 nfv 1992 . . . . 5 𝑧 𝑦𝐵
108, 9nfan 1977 . . . 4 𝑧(𝑥𝐴𝑦𝐵)
11 cbvmpt2x2.3 . . . . 5 𝑧𝐶
1211nfeq2 2918 . . . 4 𝑧 𝑢 = 𝐶
1310, 12nfan 1977 . . 3 𝑧((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)
14 nfv 1992 . . . . 5 𝑥 𝑤𝐷
15 nfv 1992 . . . . 5 𝑥 𝑧𝐵
1614, 15nfan 1977 . . . 4 𝑥(𝑤𝐷𝑧𝐵)
17 cbvmpt2x2.5 . . . . 5 𝑥𝐸
1817nfeq2 2918 . . . 4 𝑥 𝑢 = 𝐸
1916, 18nfan 1977 . . 3 𝑥((𝑤𝐷𝑧𝐵) ∧ 𝑢 = 𝐸)
20 cbvmpt2x2.2 . . . . . 6 𝑦𝐷
2120nfcri 2896 . . . . 5 𝑦 𝑤𝐷
22 nfv 1992 . . . . 5 𝑦 𝑧𝐵
2321, 22nfan 1977 . . . 4 𝑦(𝑤𝐷𝑧𝐵)
24 cbvmpt2x2.6 . . . . 5 𝑦𝐸
2524nfeq2 2918 . . . 4 𝑦 𝑢 = 𝐸
2623, 25nfan 1977 . . 3 𝑦((𝑤𝐷𝑧𝐵) ∧ 𝑢 = 𝐸)
27 eleq1w 2822 . . . . . 6 (𝑥 = 𝑤 → (𝑥𝐴𝑤𝐴))
28 cbvmpt2x2.7 . . . . . . 7 (𝑦 = 𝑧𝐴 = 𝐷)
2928eleq2d 2825 . . . . . 6 (𝑦 = 𝑧 → (𝑤𝐴𝑤𝐷))
3027, 29sylan9bb 738 . . . . 5 ((𝑥 = 𝑤𝑦 = 𝑧) → (𝑥𝐴𝑤𝐷))
31 simpr 479 . . . . . 6 ((𝑥 = 𝑤𝑦 = 𝑧) → 𝑦 = 𝑧)
3231eleq1d 2824 . . . . 5 ((𝑥 = 𝑤𝑦 = 𝑧) → (𝑦𝐵𝑧𝐵))
3330, 32anbi12d 749 . . . 4 ((𝑥 = 𝑤𝑦 = 𝑧) → ((𝑥𝐴𝑦𝐵) ↔ (𝑤𝐷𝑧𝐵)))
34 cbvmpt2x2.8 . . . . . 6 ((𝑦 = 𝑧𝑥 = 𝑤) → 𝐶 = 𝐸)
3534ancoms 468 . . . . 5 ((𝑥 = 𝑤𝑦 = 𝑧) → 𝐶 = 𝐸)
3635eqeq2d 2770 . . . 4 ((𝑥 = 𝑤𝑦 = 𝑧) → (𝑢 = 𝐶𝑢 = 𝐸))
3733, 36anbi12d 749 . . 3 ((𝑥 = 𝑤𝑦 = 𝑧) → (((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶) ↔ ((𝑤𝐷𝑧𝐵) ∧ 𝑢 = 𝐸)))
386, 13, 19, 26, 37cbvoprab12 6895 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)} = {⟨⟨𝑤, 𝑧⟩, 𝑢⟩ ∣ ((𝑤𝐷𝑧𝐵) ∧ 𝑢 = 𝐸)}
39 df-mpt2 6819 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)}
40 df-mpt2 6819 . 2 (𝑤𝐷, 𝑧𝐵𝐸) = {⟨⟨𝑤, 𝑧⟩, 𝑢⟩ ∣ ((𝑤𝐷𝑧𝐵) ∧ 𝑢 = 𝐸)}
4138, 39, 403eqtr4i 2792 1 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑤𝐷, 𝑧𝐵𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wnfc 2889  {coprab 6815  cmpt2 6816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-opab 4865  df-oprab 6818  df-mpt2 6819
This theorem is referenced by:  dmmpt2ssx2  42643
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