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Theorem csbmpt12 5205
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 5203 . . 3 𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ (𝑦𝑌𝑧 = 𝑍)} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥](𝑦𝑌𝑧 = 𝑍)}
2 sbcan 3676 . . . . 5 ([𝐴 / 𝑥](𝑦𝑌𝑧 = 𝑍) ↔ ([𝐴 / 𝑥]𝑦𝑌[𝐴 / 𝑥]𝑧 = 𝑍))
3 sbcel12 4180 . . . . . . 7 ([𝐴 / 𝑥]𝑦𝑌𝐴 / 𝑥𝑦𝐴 / 𝑥𝑌)
4 csbconstg 3741 . . . . . . . 8 (𝐴𝑉𝐴 / 𝑥𝑦 = 𝑦)
54eleq1d 2870 . . . . . . 7 (𝐴𝑉 → (𝐴 / 𝑥𝑦𝐴 / 𝑥𝑌𝑦𝐴 / 𝑥𝑌))
63, 5syl5bb 274 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝑌𝑦𝐴 / 𝑥𝑌))
7 sbceq2g 4187 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧 = 𝑍𝑧 = 𝐴 / 𝑥𝑍))
86, 7anbi12d 618 . . . . 5 (𝐴𝑉 → (([𝐴 / 𝑥]𝑦𝑌[𝐴 / 𝑥]𝑧 = 𝑍) ↔ (𝑦𝐴 / 𝑥𝑌𝑧 = 𝐴 / 𝑥𝑍)))
92, 8syl5bb 274 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥](𝑦𝑌𝑧 = 𝑍) ↔ (𝑦𝐴 / 𝑥𝑌𝑧 = 𝐴 / 𝑥𝑍)))
109opabbidv 4910 . . 3 (𝐴𝑉 → {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥](𝑦𝑌𝑧 = 𝑍)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴 / 𝑥𝑌𝑧 = 𝐴 / 𝑥𝑍)})
111, 10syl5eq 2852 . 2 (𝐴𝑉𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ (𝑦𝑌𝑧 = 𝑍)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴 / 𝑥𝑌𝑧 = 𝐴 / 𝑥𝑍)})
12 df-mpt 4924 . . 3 (𝑦𝑌𝑍) = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝑌𝑧 = 𝑍)}
1312csbeq2i 4190 . 2 𝐴 / 𝑥(𝑦𝑌𝑍) = 𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ (𝑦𝑌𝑧 = 𝑍)}
14 df-mpt 4924 . 2 (𝑦𝐴 / 𝑥𝑌𝐴 / 𝑥𝑍) = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴 / 𝑥𝑌𝑧 = 𝐴 / 𝑥𝑍)}
1511, 13, 143eqtr4g 2865 1 (𝐴𝑉𝐴 / 𝑥(𝑦𝑌𝑍) = (𝑦𝐴 / 𝑥𝑌𝐴 / 𝑥𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1637  wcel 2156  [wsbc 3633  csb 3728  {copab 4906  cmpt 4923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-opab 4907  df-mpt 4924
This theorem is referenced by:  csbmpt2  5206  esum2dlem  30475
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