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Theorem ovmpogad 41527
Description: Value of an operation given by a maps-to rule. Deduction form of ovmpoga 7565. (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 481 . 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 7563 1 (𝜑 → (𝐴𝐹𝐵) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2105  (class class class)co 7412  cmpo 7414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417
This theorem is referenced by: (None)
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