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Theorem comfffval 15615
 Description: Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
comfffval.o compf
comfffval.b
comfffval.h
comfffval.x comp
Assertion
Ref Expression
comfffval
Distinct variable groups:   ,,   ,,,,   ,,,   ,,,
Allowed substitution hints:   (,)   ()   ()   (,,,)

Proof of Theorem comfffval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 comfffval.o . 2 compf
2 fveq2 5870 . . . . . . 7
3 comfffval.b . . . . . . 7
42, 3syl6eqr 2505 . . . . . 6
54sqxpeqd 4863 . . . . 5
6 fveq2 5870 . . . . . . . 8
7 comfffval.h . . . . . . . 8
86, 7syl6eqr 2505 . . . . . . 7
98oveqd 6312 . . . . . 6
108fveq1d 5872 . . . . . 6
11 fveq2 5870 . . . . . . . . 9 comp comp
12 comfffval.x . . . . . . . . 9 comp
1311, 12syl6eqr 2505 . . . . . . . 8 comp
1413oveqd 6312 . . . . . . 7 comp
1514oveqd 6312 . . . . . 6 comp
169, 10, 15mpt2eq123dv 6358 . . . . 5 comp
175, 4, 16mpt2eq123dv 6358 . . . 4 comp
18 df-comf 15589 . . . 4 compf comp
19 fvex 5880 . . . . . . 7
203, 19eqeltri 2527 . . . . . 6
2120, 20xpex 6600 . . . . 5
2221, 20mpt2ex 6875 . . . 4
2317, 18, 22fvmpt 5953 . . 3 compf
24 fvprc 5864 . . . 4 compf
25 fvprc 5864 . . . . . . . . 9
263, 25syl5eq 2499 . . . . . . . 8
2726xpeq2d 4861 . . . . . . 7
28 xp0 5258 . . . . . . 7
2927, 28syl6eq 2503 . . . . . 6
30 mpt2eq12 6356 . . . . . 6
3129, 26, 30syl2anc 667 . . . . 5
32 mpt20 6366 . . . . 5
3331, 32syl6eq 2503 . . . 4
3424, 33eqtr4d 2490 . . 3 compf
3523, 34pm2.61i 168 . 2 compf
361, 35eqtri 2475 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wceq 1446   wcel 1889  cvv 3047  c0 3733   cxp 4835  cfv 5585  (class class class)co 6295   cmpt2 6297  c2nd 6797  cbs 15133   chom 15213  compcco 15214  compfccomf 15585 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-8 1891  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-pow 4584  ax-pr 4642  ax-un 6588 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-pw 3955  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-1st 6798  df-2nd 6799  df-comf 15589 This theorem is referenced by:  comffval  15616  comfffval2  15618  comfffn  15621  comfeq  15623
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