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Theorem ofc2 7182
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 6331 . . 3 (𝐵𝑊 → (𝐴 × {𝐵}) Fn 𝐴)
42, 3syl 17 . 2 (𝜑 → (𝐴 × {𝐵}) Fn 𝐴)
5 ofc2.1 . 2 (𝜑𝐴𝑉)
6 inidm 4048 . 2 (𝐴𝐴) = 𝐴
7 ofc2.4 . 2 ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)
8 fvconst2g 6724 . . 3 ((𝐵𝑊𝑋𝐴) → ((𝐴 × {𝐵})‘𝑋) = 𝐵)
92, 8sylan 577 . 2 ((𝜑𝑋𝐴) → ((𝐴 × {𝐵})‘𝑋) = 𝐵)
101, 4, 5, 5, 6, 7, 9ofval 7167 1 ((𝜑𝑋𝐴) → ((𝐹𝑓 𝑅(𝐴 × {𝐵}))‘𝑋) = (𝐶𝑅𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1658  wcel 2166  {csn 4398   × cxp 5341   Fn wfn 6119  cfv 6124  (class class class)co 6906  𝑓 cof 7156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pr 5128
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-reu 3125  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-of 7158
This theorem is referenced by:  lflvscl  35153  lkrsc  35173  ldualvsval  35214
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