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Theorem csbmpo123 34482
Description: Move class substitution in and out of maps-to notation for operations. (Contributed by ML, 25-Oct-2020.)
Assertion
Ref Expression
csbmpo123 (𝐴𝑉𝐴 / 𝑥(𝑦𝑌, 𝑧𝑍𝐷) = (𝑦𝐴 / 𝑥𝑌, 𝑧𝐴 / 𝑥𝑍𝐴 / 𝑥𝐷))
Distinct variable groups:   𝑦,𝐴   𝑧,𝐴   𝑦,𝑉   𝑧,𝑉   𝑥,𝑦   𝑥,𝑧
Allowed substitution hints:   𝐴(𝑥)   𝐷(𝑥,𝑦,𝑧)   𝑉(𝑥)   𝑌(𝑥,𝑦,𝑧)   𝑍(𝑥,𝑦,𝑧)

Proof of Theorem csbmpo123
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 csboprabg 34481 . . 3 (𝐴𝑉𝐴 / 𝑥{⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ ((𝑦𝑌𝑧𝑍) ∧ 𝑑 = 𝐷)} = {⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ [𝐴 / 𝑥]((𝑦𝑌𝑧𝑍) ∧ 𝑑 = 𝐷)})
2 sbcan 3824 . . . . 5 ([𝐴 / 𝑥]((𝑦𝑌𝑧𝑍) ∧ 𝑑 = 𝐷) ↔ ([𝐴 / 𝑥](𝑦𝑌𝑧𝑍) ∧ [𝐴 / 𝑥]𝑑 = 𝐷))
3 sbcan 3824 . . . . . . 7 ([𝐴 / 𝑥](𝑦𝑌𝑧𝑍) ↔ ([𝐴 / 𝑥]𝑦𝑌[𝐴 / 𝑥]𝑧𝑍))
4 sbcel12 4363 . . . . . . . . 9 ([𝐴 / 𝑥]𝑦𝑌𝐴 / 𝑥𝑦𝐴 / 𝑥𝑌)
5 csbconstg 3905 . . . . . . . . . 10 (𝐴𝑉𝐴 / 𝑥𝑦 = 𝑦)
65eleq1d 2901 . . . . . . . . 9 (𝐴𝑉 → (𝐴 / 𝑥𝑦𝐴 / 𝑥𝑌𝑦𝐴 / 𝑥𝑌))
74, 6syl5bb 284 . . . . . . . 8 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝑌𝑦𝐴 / 𝑥𝑌))
8 sbcel12 4363 . . . . . . . . 9 ([𝐴 / 𝑥]𝑧𝑍𝐴 / 𝑥𝑧𝐴 / 𝑥𝑍)
9 csbconstg 3905 . . . . . . . . . 10 (𝐴𝑉𝐴 / 𝑥𝑧 = 𝑧)
109eleq1d 2901 . . . . . . . . 9 (𝐴𝑉 → (𝐴 / 𝑥𝑧𝐴 / 𝑥𝑍𝑧𝐴 / 𝑥𝑍))
118, 10syl5bb 284 . . . . . . . 8 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝑍𝑧𝐴 / 𝑥𝑍))
127, 11anbi12d 630 . . . . . . 7 (𝐴𝑉 → (([𝐴 / 𝑥]𝑦𝑌[𝐴 / 𝑥]𝑧𝑍) ↔ (𝑦𝐴 / 𝑥𝑌𝑧𝐴 / 𝑥𝑍)))
133, 12syl5bb 284 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥](𝑦𝑌𝑧𝑍) ↔ (𝑦𝐴 / 𝑥𝑌𝑧𝐴 / 𝑥𝑍)))
14 sbceq2g 4371 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝑑 = 𝐷𝑑 = 𝐴 / 𝑥𝐷))
1513, 14anbi12d 630 . . . . 5 (𝐴𝑉 → (([𝐴 / 𝑥](𝑦𝑌𝑧𝑍) ∧ [𝐴 / 𝑥]𝑑 = 𝐷) ↔ ((𝑦𝐴 / 𝑥𝑌𝑧𝐴 / 𝑥𝑍) ∧ 𝑑 = 𝐴 / 𝑥𝐷)))
162, 15syl5bb 284 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]((𝑦𝑌𝑧𝑍) ∧ 𝑑 = 𝐷) ↔ ((𝑦𝐴 / 𝑥𝑌𝑧𝐴 / 𝑥𝑍) ∧ 𝑑 = 𝐴 / 𝑥𝐷)))
1716oprabbidv 7215 . . 3 (𝐴𝑉 → {⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ [𝐴 / 𝑥]((𝑦𝑌𝑧𝑍) ∧ 𝑑 = 𝐷)} = {⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ ((𝑦𝐴 / 𝑥𝑌𝑧𝐴 / 𝑥𝑍) ∧ 𝑑 = 𝐴 / 𝑥𝐷)})
181, 17eqtrd 2860 . 2 (𝐴𝑉𝐴 / 𝑥{⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ ((𝑦𝑌𝑧𝑍) ∧ 𝑑 = 𝐷)} = {⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ ((𝑦𝐴 / 𝑥𝑌𝑧𝐴 / 𝑥𝑍) ∧ 𝑑 = 𝐴 / 𝑥𝐷)})
19 df-mpo 7156 . . 3 (𝑦𝑌, 𝑧𝑍𝐷) = {⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ ((𝑦𝑌𝑧𝑍) ∧ 𝑑 = 𝐷)}
2019csbeq2i 3894 . 2 𝐴 / 𝑥(𝑦𝑌, 𝑧𝑍𝐷) = 𝐴 / 𝑥{⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ ((𝑦𝑌𝑧𝑍) ∧ 𝑑 = 𝐷)}
21 df-mpo 7156 . 2 (𝑦𝐴 / 𝑥𝑌, 𝑧𝐴 / 𝑥𝑍𝐴 / 𝑥𝐷) = {⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ ((𝑦𝐴 / 𝑥𝑌𝑧𝐴 / 𝑥𝑍) ∧ 𝑑 = 𝐴 / 𝑥𝐷)}
2218, 20, 213eqtr4g 2885 1 (𝐴𝑉𝐴 / 𝑥(𝑦𝑌, 𝑧𝑍𝐷) = (𝑦𝐴 / 𝑥𝑌, 𝑧𝐴 / 𝑥𝑍𝐴 / 𝑥𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2106  [wsbc 3775  csb 3886  {coprab 7152  cmpo 7153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-nul 4295  df-oprab 7155  df-mpo 7156
This theorem is referenced by:  csbfinxpg  34539
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