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Theorem comfffval2 15316
 Description: Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
comfffval2.o compf
comfffval2.b
comfffval2.h f
comfffval2.x comp
Assertion
Ref Expression
comfffval2
Distinct variable groups:   ,,,,   ,,,,   ,,,
Allowed substitution hints:   ()   (,,,)   (,,,)

Proof of Theorem comfffval2
StepHypRef Expression
1 comfffval2.o . . 3 compf
2 comfffval2.b . . 3
3 eqid 2404 . . 3
4 comfffval2.x . . 3 comp
51, 2, 3, 4comfffval 15313 . 2
6 comfffval2.h . . . . 5 f
7 xp2nd 6817 . . . . . 6
87adantr 465 . . . . 5
9 simpr 461 . . . . 5
106, 2, 3, 8, 9homfval 15307 . . . 4
11 xp1st 6816 . . . . . . . 8
1211adantr 465 . . . . . . 7
136, 2, 3, 12, 8homfval 15307 . . . . . 6
14 df-ov 6283 . . . . . 6
15 df-ov 6283 . . . . . 6
1613, 14, 153eqtr3g 2468 . . . . 5
17 1st2nd2 6823 . . . . . . 7
1817adantr 465 . . . . . 6
1918fveq2d 5855 . . . . 5
2018fveq2d 5855 . . . . 5
2116, 19, 203eqtr4d 2455 . . . 4
22 eqidd 2405 . . . 4
2310, 21, 22mpt2eq123dv 6342 . . 3
2423mpt2eq3ia 6345 . 2
255, 24eqtr4i 2436 1
 Colors of variables: wff setvar class Syntax hints:   wa 369   wceq 1407   wcel 1844  cop 3980   cxp 4823  cfv 5571  (class class class)co 6280   cmpt2 6282  c1st 6784  c2nd 6785  cbs 14843   chom 14922  compcco 14923   f chomf 15282  compfccomf 15283 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576 This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-1st 6786  df-2nd 6787  df-homf 15286  df-comf 15287 This theorem is referenced by: (None)
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