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Theorem ofc2 7427
Description: Right operation by a constant. (Contributed by NM, 7-Oct-2014.)
Hypotheses
Ref Expression
ofc2.1 (𝜑𝐴𝑉)
ofc2.2 (𝜑𝐵𝑊)
ofc2.3 (𝜑𝐹 Fn 𝐴)
ofc2.4 ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)
Assertion
Ref Expression
ofc2 ((𝜑𝑋𝐴) → ((𝐹f 𝑅(𝐴 × {𝐵}))‘𝑋) = (𝐶𝑅𝐵))

Proof of Theorem ofc2
StepHypRef Expression
1 ofc2.3 . 2 (𝜑𝐹 Fn 𝐴)
2 ofc2.2 . . 3 (𝜑𝐵𝑊)
3 fnconstg 6561 . . 3 (𝐵𝑊 → (𝐴 × {𝐵}) Fn 𝐴)
42, 3syl 17 . 2 (𝜑 → (𝐴 × {𝐵}) Fn 𝐴)
5 ofc2.1 . 2 (𝜑𝐴𝑉)
6 inidm 4194 . 2 (𝐴𝐴) = 𝐴
7 ofc2.4 . 2 ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)
8 fvconst2g 6958 . . 3 ((𝐵𝑊𝑋𝐴) → ((𝐴 × {𝐵})‘𝑋) = 𝐵)
92, 8sylan 582 . 2 ((𝜑𝑋𝐴) → ((𝐴 × {𝐵})‘𝑋) = 𝐵)
101, 4, 5, 5, 6, 7, 9ofval 7412 1 ((𝜑𝑋𝐴) → ((𝐹f 𝑅(𝐴 × {𝐵}))‘𝑋) = (𝐶𝑅𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1533  wcel 2110  {csn 4560   × cxp 5547   Fn wfn 6344  cfv 6349  (class class class)co 7150  f cof 7401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pr 5321
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-of 7403
This theorem is referenced by:  lflvscl  36207  lkrsc  36227  ldualvsval  36268
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