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Theorem ovmpt2rdx 43133
 Description: Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpt2dxf 7063. (Contributed by AV, 30-Mar-2019.)
Hypotheses
Ref Expression
ovmpt2rdx.1 (𝜑𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))
ovmpt2rdx.2 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
ovmpt2rdx.3 ((𝜑𝑦 = 𝐵) → 𝐶 = 𝐿)
ovmpt2rdx.4 (𝜑𝐴𝐿)
ovmpt2rdx.5 (𝜑𝐵𝐷)
ovmpt2rdx.6 (𝜑𝑆𝑋)
Assertion
Ref Expression
ovmpt2rdx (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 ovmpt2rdx
StepHypRef Expression
1 ovmpt2rdx.1 . 2 (𝜑𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))
2 ovmpt2rdx.2 . 2 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
3 ovmpt2rdx.3 . 2 ((𝜑𝑦 = 𝐵) → 𝐶 = 𝐿)
4 ovmpt2rdx.4 . 2 (𝜑𝐴𝐿)
5 ovmpt2rdx.5 . 2 (𝜑𝐵𝐷)
6 ovmpt2rdx.6 . 2 (𝜑𝑆𝑋)
7 nfv 1957 . 2 𝑥𝜑
8 nfv 1957 . 2 𝑦𝜑
9 nfcv 2934 . 2 𝑦𝐴
10 nfcv 2934 . 2 𝑥𝐵
11 nfcv 2934 . 2 𝑥𝑆
12 nfcv 2934 . 2 𝑦𝑆
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12ovmpt2rdxf 43132 1 (𝜑 → (𝐴𝐹𝐵) = 𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107  (class class class)co 6922   ↦ cmpt2 6924 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927 This theorem is referenced by:  ovmpt2x2  43134
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