Users' Mathboxes Mathbox for ML < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  csbmpt22g Structured version   Visualization version   GIF version

Theorem csbmpt22g 33307
Description: Move class substitution in and out of maps-to notation for operations. (Contributed by ML, 25-Oct-2020.)
Assertion
Ref Expression
csbmpt22g (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 csbmpt22g
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 csboprabg 33306 . . 3 (𝐴𝑉𝐴 / 𝑥{⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ ((𝑦𝑌𝑧𝑍) ∧ 𝑑 = 𝐷)} = {⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ [𝐴 / 𝑥]((𝑦𝑌𝑧𝑍) ∧ 𝑑 = 𝐷)})
2 sbcan 3511 . . . . 5 ([𝐴 / 𝑥]((𝑦𝑌𝑧𝑍) ∧ 𝑑 = 𝐷) ↔ ([𝐴 / 𝑥](𝑦𝑌𝑧𝑍) ∧ [𝐴 / 𝑥]𝑑 = 𝐷))
3 sbcan 3511 . . . . . . 7 ([𝐴 / 𝑥](𝑦𝑌𝑧𝑍) ↔ ([𝐴 / 𝑥]𝑦𝑌[𝐴 / 𝑥]𝑧𝑍))
4 sbcel12 4016 . . . . . . . . 9 ([𝐴 / 𝑥]𝑦𝑌𝐴 / 𝑥𝑦𝐴 / 𝑥𝑌)
5 csbconstg 3579 . . . . . . . . . 10 (𝐴𝑉𝐴 / 𝑥𝑦 = 𝑦)
65eleq1d 2715 . . . . . . . . 9 (𝐴𝑉 → (𝐴 / 𝑥𝑦𝐴 / 𝑥𝑌𝑦𝐴 / 𝑥𝑌))
74, 6syl5bb 272 . . . . . . . 8 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝑌𝑦𝐴 / 𝑥𝑌))
8 sbcel12 4016 . . . . . . . . 9 ([𝐴 / 𝑥]𝑧𝑍𝐴 / 𝑥𝑧𝐴 / 𝑥𝑍)
9 csbconstg 3579 . . . . . . . . . 10 (𝐴𝑉𝐴 / 𝑥𝑧 = 𝑧)
109eleq1d 2715 . . . . . . . . 9 (𝐴𝑉 → (𝐴 / 𝑥𝑧𝐴 / 𝑥𝑍𝑧𝐴 / 𝑥𝑍))
118, 10syl5bb 272 . . . . . . . 8 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝑍𝑧𝐴 / 𝑥𝑍))
127, 11anbi12d 747 . . . . . . 7 (𝐴𝑉 → (([𝐴 / 𝑥]𝑦𝑌[𝐴 / 𝑥]𝑧𝑍) ↔ (𝑦𝐴 / 𝑥𝑌𝑧𝐴 / 𝑥𝑍)))
133, 12syl5bb 272 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥](𝑦𝑌𝑧𝑍) ↔ (𝑦𝐴 / 𝑥𝑌𝑧𝐴 / 𝑥𝑍)))
14 sbceq2g 4023 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝑑 = 𝐷𝑑 = 𝐴 / 𝑥𝐷))
1513, 14anbi12d 747 . . . . 5 (𝐴𝑉 → (([𝐴 / 𝑥](𝑦𝑌𝑧𝑍) ∧ [𝐴 / 𝑥]𝑑 = 𝐷) ↔ ((𝑦𝐴 / 𝑥𝑌𝑧𝐴 / 𝑥𝑍) ∧ 𝑑 = 𝐴 / 𝑥𝐷)))
162, 15syl5bb 272 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]((𝑦𝑌𝑧𝑍) ∧ 𝑑 = 𝐷) ↔ ((𝑦𝐴 / 𝑥𝑌𝑧𝐴 / 𝑥𝑍) ∧ 𝑑 = 𝐴 / 𝑥𝐷)))
1716oprabbidv 6751 . . 3 (𝐴𝑉 → {⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ [𝐴 / 𝑥]((𝑦𝑌𝑧𝑍) ∧ 𝑑 = 𝐷)} = {⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ ((𝑦𝐴 / 𝑥𝑌𝑧𝐴 / 𝑥𝑍) ∧ 𝑑 = 𝐴 / 𝑥𝐷)})
181, 17eqtrd 2685 . 2 (𝐴𝑉𝐴 / 𝑥{⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ ((𝑦𝑌𝑧𝑍) ∧ 𝑑 = 𝐷)} = {⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ ((𝑦𝐴 / 𝑥𝑌𝑧𝐴 / 𝑥𝑍) ∧ 𝑑 = 𝐴 / 𝑥𝐷)})
19 df-mpt2 6695 . . 3 (𝑦𝑌, 𝑧𝑍𝐷) = {⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ ((𝑦𝑌𝑧𝑍) ∧ 𝑑 = 𝐷)}
2019csbeq2i 4026 . 2 𝐴 / 𝑥(𝑦𝑌, 𝑧𝑍𝐷) = 𝐴 / 𝑥{⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ ((𝑦𝑌𝑧𝑍) ∧ 𝑑 = 𝐷)}
21 df-mpt2 6695 . 2 (𝑦𝐴 / 𝑥𝑌, 𝑧𝐴 / 𝑥𝑍𝐴 / 𝑥𝐷) = {⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ ((𝑦𝐴 / 𝑥𝑌𝑧𝐴 / 𝑥𝑍) ∧ 𝑑 = 𝐴 / 𝑥𝐷)}
2218, 20, 213eqtr4g 2710 1 (𝐴𝑉𝐴 / 𝑥(𝑦𝑌, 𝑧𝑍𝐷) = (𝑦𝐴 / 𝑥𝑌, 𝑧𝐴 / 𝑥𝑍𝐴 / 𝑥𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  [wsbc 3468  csb 3566  {coprab 6691  cmpt2 6692
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
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  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-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-nul 3949  df-oprab 6694  df-mpt2 6695
This theorem is referenced by:  csbfinxpg  33355
  Copyright terms: Public domain W3C validator