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Theorem csbmpt12 5000
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 4998 . . 3 𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ (𝑦𝑌𝑧 = 𝑍)} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥](𝑦𝑌𝑧 = 𝑍)}
2 sbcan 3472 . . . . 5 ([𝐴 / 𝑥](𝑦𝑌𝑧 = 𝑍) ↔ ([𝐴 / 𝑥]𝑦𝑌[𝐴 / 𝑥]𝑧 = 𝑍))
3 sbcel12 3974 . . . . . . 7 ([𝐴 / 𝑥]𝑦𝑌𝐴 / 𝑥𝑦𝐴 / 𝑥𝑌)
4 csbconstg 3539 . . . . . . . 8 (𝐴𝑉𝐴 / 𝑥𝑦 = 𝑦)
54eleq1d 2684 . . . . . . 7 (𝐴𝑉 → (𝐴 / 𝑥𝑦𝐴 / 𝑥𝑌𝑦𝐴 / 𝑥𝑌))
63, 5syl5bb 272 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝑌𝑦𝐴 / 𝑥𝑌))
7 sbceq2g 3981 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧 = 𝑍𝑧 = 𝐴 / 𝑥𝑍))
86, 7anbi12d 746 . . . . 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 4707 . . 3 (𝐴𝑉 → {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥](𝑦𝑌𝑧 = 𝑍)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴 / 𝑥𝑌𝑧 = 𝐴 / 𝑥𝑍)})
111, 10syl5eq 2666 . 2 (𝐴𝑉𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ (𝑦𝑌𝑧 = 𝑍)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴 / 𝑥𝑌𝑧 = 𝐴 / 𝑥𝑍)})
12 df-mpt 4721 . . 3 (𝑦𝑌𝑍) = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝑌𝑧 = 𝑍)}
1312csbeq2i 3984 . 2 𝐴 / 𝑥(𝑦𝑌𝑍) = 𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ (𝑦𝑌𝑧 = 𝑍)}
14 df-mpt 4721 . 2 (𝑦𝐴 / 𝑥𝑌𝐴 / 𝑥𝑍) = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴 / 𝑥𝑌𝑧 = 𝐴 / 𝑥𝑍)}
1511, 13, 143eqtr4g 2679 1 (𝐴𝑉𝐴 / 𝑥(𝑦𝑌𝑍) = (𝑦𝐴 / 𝑥𝑌𝐴 / 𝑥𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  [wsbc 3429  csb 3526  {copab 4703  cmpt 4720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-opab 4704  df-mpt 4721
This theorem is referenced by:  csbmpt2  5001  esum2dlem  30128
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