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Theorem comffval 13925
 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
comffval.x
comffval.y
comffval.z
Assertion
Ref Expression
comffval
Distinct variable groups:   ,,   ,,   ,,   ,,   ,,   ,,   ,,
Allowed substitution hints:   (,)   (,)

Proof of Theorem comffval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 comfffval.o . . . 4 compf
2 comfffval.b . . . 4
3 comfffval.h . . . 4
4 comfffval.x . . . 4 comp
51, 2, 3, 4comfffval 13924 . . 3
65a1i 11 . 2
7 simprl 733 . . . . . 6
87fveq2d 5732 . . . . 5
9 comffval.x . . . . . . 7
10 comffval.y . . . . . . 7
11 op2ndg 6360 . . . . . . 7
129, 10, 11syl2anc 643 . . . . . 6
1312adantr 452 . . . . 5
148, 13eqtrd 2468 . . . 4
15 simprr 734 . . . 4
1614, 15oveq12d 6099 . . 3
177fveq2d 5732 . . . 4
18 df-ov 6084 . . . 4
1917, 18syl6eqr 2486 . . 3
207, 15oveq12d 6099 . . . 4
2120oveqd 6098 . . 3
2216, 19, 21mpt2eq123dv 6136 . 2
23 opelxpi 4910 . . 3
249, 10, 23syl2anc 643 . 2
25 comffval.z . 2
26 ovex 6106 . . . 4
27 ovex 6106 . . . 4
2826, 27mpt2ex 6425 . . 3
2928a1i 11 . 2
306, 22, 24, 25, 29ovmpt2d 6201 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725  cvv 2956  cop 3817   cxp 4876  cfv 5454  (class class class)co 6081   cmpt2 6083  c2nd 6348  cbs 13469   chom 13540  compcco 13541  compfccomf 13892 This theorem is referenced by:  comfval  13926  comffval2  13928  comffn  13931 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-comf 13896
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