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Theorem ovmpogad 41757
Description: Value of an operation given by a maps-to rule. Deduction form of ovmpoga 7581. (Contributed by SN, 14-Mar-2025.)
Hypotheses
Ref Expression
ovmpogad.f 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
ovmpogad.s ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)
ovmpogad.1 (𝜑𝐴𝐶)
ovmpogad.2 (𝜑𝐵𝐷)
ovmpogad.v (𝜑𝑆𝑉)
Assertion
Ref Expression
ovmpogad (𝜑 → (𝐴𝐹𝐵) = 𝑆)
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑆,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem ovmpogad
StepHypRef Expression
1 ovmpogad.f . . 3 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
21a1i 11 . 2 (𝜑𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))
3 ovmpogad.s . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)
43adantl 480 . 2 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
5 ovmpogad.1 . 2 (𝜑𝐴𝐶)
6 ovmpogad.2 . 2 (𝜑𝐵𝐷)
7 ovmpogad.v . 2 (𝜑𝑆𝑉)
82, 4, 5, 6, 7ovmpod 7579 1 (𝜑 → (𝐴𝐹𝐵) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  (class class class)co 7426  cmpo 7428
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431
This theorem is referenced by: (None)
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