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Theorem ovmpot 7562
Description: The value of an operation is equal to the value of the same operation expressed in maps-to notation. (Contributed by GG, 16-Mar-2025.) (Revised by GG, 13-Apr-2025.)
Assertion
Ref Expression
ovmpot ((𝐴𝐶𝐵𝐷) → (𝐴(𝑥𝐶, 𝑦𝐷 ↦ (𝑥𝐹𝑦))𝐵) = (𝐴𝐹𝐵))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦   𝑥,𝐹,𝑦

Proof of Theorem ovmpot
StepHypRef Expression
1 oveq12 7411 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝐹𝑦) = (𝐴𝐹𝐵))
2 eqid 2724 . 2 (𝑥𝐶, 𝑦𝐷 ↦ (𝑥𝐹𝑦)) = (𝑥𝐶, 𝑦𝐷 ↦ (𝑥𝐹𝑦))
3 ovex 7435 . 2 (𝐴𝐹𝐵) ∈ V
41, 2, 3ovmpoa 7556 1 ((𝐴𝐶𝐵𝐷) → (𝐴(𝑥𝐶, 𝑦𝐷 ↦ (𝑥𝐹𝑦))𝐵) = (𝐴𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  (class class class)co 7402  cmpo 7404
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 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-iota 6486  df-fun 6536  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407
This theorem is referenced by:  expcn  24734  negcncf  24786  dvcnp2  25793  dvmulbr  25813  dvcobr  25821  cmvth  25867  dvfsumle  25898  dvfsumlem2  25905  mpodvdsmulf1o  27067  fsumdvdsmul  27068  gg-cncrng  35683  gg-taylthlem2  35684  gg-cnfld1  35685
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