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Theorem ovmpod 5907
 Description: Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 7-Dec-2014.)
Hypotheses
Ref Expression
ovmpod.1 (𝜑𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))
ovmpod.2 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
ovmpod.3 (𝜑𝐴𝐶)
ovmpod.4 (𝜑𝐵𝐷)
ovmpod.5 (𝜑𝑆𝑋)
Assertion
Ref Expression
ovmpod (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 ovmpod
StepHypRef Expression
1 ovmpod.1 . 2 (𝜑𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))
2 ovmpod.2 . 2 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
3 eqidd 2141 . 2 ((𝜑𝑥 = 𝐴) → 𝐷 = 𝐷)
4 ovmpod.3 . 2 (𝜑𝐴𝐶)
5 ovmpod.4 . 2 (𝜑𝐵𝐷)
6 ovmpod.5 . 2 (𝜑𝑆𝑋)
71, 2, 3, 4, 5, 6ovmpodx 5906 1 (𝜑 → (𝐴𝐹𝐵) = 𝑆)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332   ∈ wcel 1481  (class class class)co 5783   ∈ cmpo 5785 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4055  ax-pow 4107  ax-pr 4140  ax-setind 4461 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-v 2692  df-sbc 2915  df-dif 3079  df-un 3081  df-in 3083  df-ss 3090  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-br 3939  df-opab 3999  df-id 4224  df-xp 4554  df-rel 4555  df-cnv 4556  df-co 4557  df-dm 4558  df-iota 5097  df-fun 5134  df-fv 5140  df-ov 5786  df-oprab 5787  df-mpo 5788 This theorem is referenced by:  ovmpoga  5909  iseqovex  10280  seqvalcd  10283  resqrexlemp1rp  10830  resqrexlemfp1  10833  lcmval  11800  ennnfonelemg  11972  cnfval  12422  cnpfval  12423  blvalps  12616  blval  12617
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