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Theorem ovmpt2dx 6685
 Description: Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 29-Dec-2014.)
Hypotheses
Ref Expression
ovmpt2dx.1 (𝜑𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))
ovmpt2dx.2 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
ovmpt2dx.3 ((𝜑𝑥 = 𝐴) → 𝐷 = 𝐿)
ovmpt2dx.4 (𝜑𝐴𝐶)
ovmpt2dx.5 (𝜑𝐵𝐿)
ovmpt2dx.6 (𝜑𝑆𝑋)
Assertion
Ref Expression
ovmpt2dx (𝜑 → (𝐴𝐹𝐵) = 𝑆)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑦,𝐴   𝑥,𝐵   𝑥,𝑆,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐿(𝑥,𝑦)   𝑋(𝑥,𝑦)

Proof of Theorem ovmpt2dx
StepHypRef Expression
1 ovmpt2dx.1 . 2 (𝜑𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))
2 ovmpt2dx.2 . 2 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
3 ovmpt2dx.3 . 2 ((𝜑𝑥 = 𝐴) → 𝐷 = 𝐿)
4 ovmpt2dx.4 . 2 (𝜑𝐴𝐶)
5 ovmpt2dx.5 . 2 (𝜑𝐵𝐿)
6 ovmpt2dx.6 . 2 (𝜑𝑆𝑋)
7 nfv 1830 . 2 𝑥𝜑
8 nfv 1830 . 2 𝑦𝜑
9 nfcv 2751 . 2 𝑦𝐴
10 nfcv 2751 . 2 𝑥𝐵
11 nfcv 2751 . 2 𝑥𝑆
12 nfcv 2751 . 2 𝑦𝑆
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12ovmpt2dxf 6684 1 (𝜑 → (𝐴𝐹𝐵) = 𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  (class class class)co 6549   ↦ cmpt2 6551 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554 This theorem is referenced by:  ovmpt2d  6686  ovmpt2x  6687  dpjfval  18277  fgval  21484  om1val  22638  pi1val  22645  dvfval  23467  dvnfval  23491  taylfval  23917
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