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Theorem ovmpt2d 5656
Description: Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 7-Dec-2014.)
Hypotheses
Ref Expression
ovmpt2d.1 (𝜑𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))
ovmpt2d.2 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
ovmpt2d.3 (𝜑𝐴𝐶)
ovmpt2d.4 (𝜑𝐵𝐷)
ovmpt2d.5 (𝜑𝑆𝑋)
Assertion
Ref Expression
ovmpt2d (𝜑 → (𝐴𝐹𝐵) = 𝑆)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝑆,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝑋(𝑥,𝑦)

Proof of Theorem ovmpt2d
StepHypRef Expression
1 ovmpt2d.1 . 2 (𝜑𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))
2 ovmpt2d.2 . 2 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
3 eqidd 2057 . 2 ((𝜑𝑥 = 𝐴) → 𝐷 = 𝐷)
4 ovmpt2d.3 . 2 (𝜑𝐴𝐶)
5 ovmpt2d.4 . 2 (𝜑𝐵𝐷)
6 ovmpt2d.5 . 2 (𝜑𝑆𝑋)
71, 2, 3, 4, 5, 6ovmpt2dx 5655 1 (𝜑 → (𝐴𝐹𝐵) = 𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wcel 1409  (class class class)co 5540  cmpt2 5542
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-setind 4290
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-iota 4895  df-fun 4932  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545
This theorem is referenced by:  ovmpt2ga  5658  sprmpt2  5888  iseqovex  9383  resqrexlemp1rp  9833  resqrexlemfp1  9836
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