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Theorem csbmpo123 35062
 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 35061 . . 3 (𝐴𝑉𝐴 / 𝑥{⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ ((𝑦𝑌𝑧𝑍) ∧ 𝑑 = 𝐷)} = {⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ [𝐴 / 𝑥]((𝑦𝑌𝑧𝑍) ∧ 𝑑 = 𝐷)})
2 sbcan 3747 . . . . 5 ([𝐴 / 𝑥]((𝑦𝑌𝑧𝑍) ∧ 𝑑 = 𝐷) ↔ ([𝐴 / 𝑥](𝑦𝑌𝑧𝑍) ∧ [𝐴 / 𝑥]𝑑 = 𝐷))
3 sbcan 3747 . . . . . . 7 ([𝐴 / 𝑥](𝑦𝑌𝑧𝑍) ↔ ([𝐴 / 𝑥]𝑦𝑌[𝐴 / 𝑥]𝑧𝑍))
4 sbcel12 4308 . . . . . . . . 9 ([𝐴 / 𝑥]𝑦𝑌𝐴 / 𝑥𝑦𝐴 / 𝑥𝑌)
5 csbconstg 3826 . . . . . . . . . 10 (𝐴𝑉𝐴 / 𝑥𝑦 = 𝑦)
65eleq1d 2836 . . . . . . . . 9 (𝐴𝑉 → (𝐴 / 𝑥𝑦𝐴 / 𝑥𝑌𝑦𝐴 / 𝑥𝑌))
74, 6syl5bb 286 . . . . . . . 8 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝑌𝑦𝐴 / 𝑥𝑌))
8 sbcel12 4308 . . . . . . . . 9 ([𝐴 / 𝑥]𝑧𝑍𝐴 / 𝑥𝑧𝐴 / 𝑥𝑍)
9 csbconstg 3826 . . . . . . . . . 10 (𝐴𝑉𝐴 / 𝑥𝑧 = 𝑧)
109eleq1d 2836 . . . . . . . . 9 (𝐴𝑉 → (𝐴 / 𝑥𝑧𝐴 / 𝑥𝑍𝑧𝐴 / 𝑥𝑍))
118, 10syl5bb 286 . . . . . . . 8 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝑍𝑧𝐴 / 𝑥𝑍))
127, 11anbi12d 633 . . . . . . 7 (𝐴𝑉 → (([𝐴 / 𝑥]𝑦𝑌[𝐴 / 𝑥]𝑧𝑍) ↔ (𝑦𝐴 / 𝑥𝑌𝑧𝐴 / 𝑥𝑍)))
133, 12syl5bb 286 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥](𝑦𝑌𝑧𝑍) ↔ (𝑦𝐴 / 𝑥𝑌𝑧𝐴 / 𝑥𝑍)))
14 sbceq2g 4316 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝑑 = 𝐷𝑑 = 𝐴 / 𝑥𝐷))
1513, 14anbi12d 633 . . . . 5 (𝐴𝑉 → (([𝐴 / 𝑥](𝑦𝑌𝑧𝑍) ∧ [𝐴 / 𝑥]𝑑 = 𝐷) ↔ ((𝑦𝐴 / 𝑥𝑌𝑧𝐴 / 𝑥𝑍) ∧ 𝑑 = 𝐴 / 𝑥𝐷)))
162, 15syl5bb 286 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]((𝑦𝑌𝑧𝑍) ∧ 𝑑 = 𝐷) ↔ ((𝑦𝐴 / 𝑥𝑌𝑧𝐴 / 𝑥𝑍) ∧ 𝑑 = 𝐴 / 𝑥𝐷)))
1716oprabbidv 7220 . . 3 (𝐴𝑉 → {⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ [𝐴 / 𝑥]((𝑦𝑌𝑧𝑍) ∧ 𝑑 = 𝐷)} = {⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ ((𝑦𝐴 / 𝑥𝑌𝑧𝐴 / 𝑥𝑍) ∧ 𝑑 = 𝐴 / 𝑥𝐷)})
181, 17eqtrd 2793 . 2 (𝐴𝑉𝐴 / 𝑥{⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ ((𝑦𝑌𝑧𝑍) ∧ 𝑑 = 𝐷)} = {⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ ((𝑦𝐴 / 𝑥𝑌𝑧𝐴 / 𝑥𝑍) ∧ 𝑑 = 𝐴 / 𝑥𝐷)})
19 df-mpo 7161 . . 3 (𝑦𝑌, 𝑧𝑍𝐷) = {⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ ((𝑦𝑌𝑧𝑍) ∧ 𝑑 = 𝐷)}
2019csbeq2i 3815 . 2 𝐴 / 𝑥(𝑦𝑌, 𝑧𝑍𝐷) = 𝐴 / 𝑥{⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ ((𝑦𝑌𝑧𝑍) ∧ 𝑑 = 𝐷)}
21 df-mpo 7161 . 2 (𝑦𝐴 / 𝑥𝑌, 𝑧𝐴 / 𝑥𝑍𝐴 / 𝑥𝐷) = {⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ ((𝑦𝐴 / 𝑥𝑌𝑧𝐴 / 𝑥𝑍) ∧ 𝑑 = 𝐴 / 𝑥𝐷)}
2218, 20, 213eqtr4g 2818 1 (𝐴𝑉𝐴 / 𝑥(𝑦𝑌, 𝑧𝑍𝐷) = (𝑦𝐴 / 𝑥𝑌, 𝑧𝐴 / 𝑥𝑍𝐴 / 𝑥𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  [wsbc 3698  ⦋csb 3807  {coprab 7157   ∈ cmpo 7158 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-nul 4228  df-oprab 7160  df-mpo 7161 This theorem is referenced by:  csbfinxpg  35119
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