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Theorem ofcfval 28931
 Description: Value of an operation applied to a function and a constant. (Contributed by Thierry Arnoux, 30-Jan-2017.)
Hypotheses
Ref Expression
ofcfval.1
ofcfval.2
ofcfval.3
ofcfval.6
Assertion
Ref Expression
ofcfval 𝑓/𝑐
Distinct variable groups:   ,   ,   ,   ,
Allowed substitution hints:   ()   ()   ()   ()

Proof of Theorem ofcfval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ofc 28929 . . . 4 𝑓/𝑐
21a1i 11 . . 3 𝑓/𝑐
3 simprl 765 . . . . 5
43dmeqd 5040 . . . 4
53fveq1d 5872 . . . . 5
6 simprr 767 . . . . 5
75, 6oveq12d 6313 . . . 4
84, 7mpteq12dv 4484 . . 3
9 ofcfval.1 . . . 4
10 ofcfval.2 . . . 4
11 fnex 6137 . . . 4
129, 10, 11syl2anc 667 . . 3
13 ofcfval.3 . . . 4
14 elex 3056 . . . 4
1513, 14syl 17 . . 3
16 fndm 5680 . . . . . 6
179, 16syl 17 . . . . 5
1817, 10eqeltrd 2531 . . . 4
19 mptexg 6140 . . . 4
2018, 19syl 17 . . 3
212, 8, 12, 15, 20ovmpt2d 6429 . 2 𝑓/𝑐
2217eleq2d 2516 . . . . . 6
2322pm5.32i 643 . . . . 5
24 ofcfval.6 . . . . 5
2523, 24sylbi 199 . . . 4
2625oveq1d 6310 . . 3
2717, 26mpteq12dva 4483 . 2
2821, 27eqtrd 2487 1 𝑓/𝑐
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 371   wceq 1446   wcel 1889  cvv 3047   cmpt 4464   cdm 4837   wfn 5580  cfv 5585  (class class class)co 6295   cmpt2 6297  ∘𝑓/𝑐cofc 28928 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pr 4642 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-ofc 28929 This theorem is referenced by:  ofcval  28932  ofcfn  28933  ofcfeqd2  28934  ofcf  28936  ofcfval2  28937  ofcc  28939  ofcof  28940
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