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Theorem csbmpt12 5039
Description: Move substitution into a maps-to notation. (Contributed by AV, 26-Sep-2019.)
Assertion
Ref Expression
csbmpt12 (𝐴𝑉𝐴 / 𝑥(𝑦𝑌𝑍) = (𝑦𝐴 / 𝑥𝑌𝐴 / 𝑥𝑍))
Distinct variable groups:   𝑦,𝐴   𝑦,𝑉   𝑦,𝑌   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥)   𝑉(𝑥)   𝑌(𝑥)   𝑍(𝑥,𝑦)

Proof of Theorem csbmpt12
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 csbopab 5037 . . 3 𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ (𝑦𝑌𝑧 = 𝑍)} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥](𝑦𝑌𝑧 = 𝑍)}
2 sbcan 3511 . . . . 5 ([𝐴 / 𝑥](𝑦𝑌𝑧 = 𝑍) ↔ ([𝐴 / 𝑥]𝑦𝑌[𝐴 / 𝑥]𝑧 = 𝑍))
3 sbcel12 4016 . . . . . . 7 ([𝐴 / 𝑥]𝑦𝑌𝐴 / 𝑥𝑦𝐴 / 𝑥𝑌)
4 csbconstg 3579 . . . . . . . 8 (𝐴𝑉𝐴 / 𝑥𝑦 = 𝑦)
54eleq1d 2715 . . . . . . 7 (𝐴𝑉 → (𝐴 / 𝑥𝑦𝐴 / 𝑥𝑌𝑦𝐴 / 𝑥𝑌))
63, 5syl5bb 272 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝑌𝑦𝐴 / 𝑥𝑌))
7 sbceq2g 4023 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧 = 𝑍𝑧 = 𝐴 / 𝑥𝑍))
86, 7anbi12d 747 . . . . 5 (𝐴𝑉 → (([𝐴 / 𝑥]𝑦𝑌[𝐴 / 𝑥]𝑧 = 𝑍) ↔ (𝑦𝐴 / 𝑥𝑌𝑧 = 𝐴 / 𝑥𝑍)))
92, 8syl5bb 272 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥](𝑦𝑌𝑧 = 𝑍) ↔ (𝑦𝐴 / 𝑥𝑌𝑧 = 𝐴 / 𝑥𝑍)))
109opabbidv 4749 . . 3 (𝐴𝑉 → {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥](𝑦𝑌𝑧 = 𝑍)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴 / 𝑥𝑌𝑧 = 𝐴 / 𝑥𝑍)})
111, 10syl5eq 2697 . 2 (𝐴𝑉𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ (𝑦𝑌𝑧 = 𝑍)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴 / 𝑥𝑌𝑧 = 𝐴 / 𝑥𝑍)})
12 df-mpt 4763 . . 3 (𝑦𝑌𝑍) = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝑌𝑧 = 𝑍)}
1312csbeq2i 4026 . 2 𝐴 / 𝑥(𝑦𝑌𝑍) = 𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ (𝑦𝑌𝑧 = 𝑍)}
14 df-mpt 4763 . 2 (𝑦𝐴 / 𝑥𝑌𝐴 / 𝑥𝑍) = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴 / 𝑥𝑌𝑧 = 𝐴 / 𝑥𝑍)}
1511, 13, 143eqtr4g 2710 1 (𝐴𝑉𝐴 / 𝑥(𝑦𝑌𝑍) = (𝑦𝐴 / 𝑥𝑌𝐴 / 𝑥𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  [wsbc 3468  csb 3566  {copab 4745  cmpt 4762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-opab 4746  df-mpt 4763
This theorem is referenced by:  csbmpt2  5040  esum2dlem  30282
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