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Theorem comfval 17687
Description: Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
comfffval.o ๐‘‚ = (compfโ€˜๐ถ)
comfffval.b ๐ต = (Baseโ€˜๐ถ)
comfffval.h ๐ป = (Hom โ€˜๐ถ)
comfffval.x ยท = (compโ€˜๐ถ)
comffval.x (๐œ‘ โ†’ ๐‘‹ โˆˆ ๐ต)
comffval.y (๐œ‘ โ†’ ๐‘Œ โˆˆ ๐ต)
comffval.z (๐œ‘ โ†’ ๐‘ โˆˆ ๐ต)
comfval.f (๐œ‘ โ†’ ๐น โˆˆ (๐‘‹๐ป๐‘Œ))
comfval.g (๐œ‘ โ†’ ๐บ โˆˆ (๐‘Œ๐ป๐‘))
Assertion
Ref Expression
comfval (๐œ‘ โ†’ (๐บ(โŸจ๐‘‹, ๐‘ŒโŸฉ๐‘‚๐‘)๐น) = (๐บ(โŸจ๐‘‹, ๐‘ŒโŸฉ ยท ๐‘)๐น))

Proof of Theorem comfval
Dummy variables ๐‘“ ๐‘” are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 comfffval.o . . 3 ๐‘‚ = (compfโ€˜๐ถ)
2 comfffval.b . . 3 ๐ต = (Baseโ€˜๐ถ)
3 comfffval.h . . 3 ๐ป = (Hom โ€˜๐ถ)
4 comfffval.x . . 3 ยท = (compโ€˜๐ถ)
5 comffval.x . . 3 (๐œ‘ โ†’ ๐‘‹ โˆˆ ๐ต)
6 comffval.y . . 3 (๐œ‘ โ†’ ๐‘Œ โˆˆ ๐ต)
7 comffval.z . . 3 (๐œ‘ โ†’ ๐‘ โˆˆ ๐ต)
81, 2, 3, 4, 5, 6, 7comffval 17686 . 2 (๐œ‘ โ†’ (โŸจ๐‘‹, ๐‘ŒโŸฉ๐‘‚๐‘) = (๐‘” โˆˆ (๐‘Œ๐ป๐‘), ๐‘“ โˆˆ (๐‘‹๐ป๐‘Œ) โ†ฆ (๐‘”(โŸจ๐‘‹, ๐‘ŒโŸฉ ยท ๐‘)๐‘“)))
9 oveq12 7435 . . 3 ((๐‘” = ๐บ โˆง ๐‘“ = ๐น) โ†’ (๐‘”(โŸจ๐‘‹, ๐‘ŒโŸฉ ยท ๐‘)๐‘“) = (๐บ(โŸจ๐‘‹, ๐‘ŒโŸฉ ยท ๐‘)๐น))
109adantl 480 . 2 ((๐œ‘ โˆง (๐‘” = ๐บ โˆง ๐‘“ = ๐น)) โ†’ (๐‘”(โŸจ๐‘‹, ๐‘ŒโŸฉ ยท ๐‘)๐‘“) = (๐บ(โŸจ๐‘‹, ๐‘ŒโŸฉ ยท ๐‘)๐น))
11 comfval.g . 2 (๐œ‘ โ†’ ๐บ โˆˆ (๐‘Œ๐ป๐‘))
12 comfval.f . 2 (๐œ‘ โ†’ ๐น โˆˆ (๐‘‹๐ป๐‘Œ))
13 ovexd 7461 . 2 (๐œ‘ โ†’ (๐บ(โŸจ๐‘‹, ๐‘ŒโŸฉ ยท ๐‘)๐น) โˆˆ V)
148, 10, 11, 12, 13ovmpod 7579 1 (๐œ‘ โ†’ (๐บ(โŸจ๐‘‹, ๐‘ŒโŸฉ๐‘‚๐‘)๐น) = (๐บ(โŸจ๐‘‹, ๐‘ŒโŸฉ ยท ๐‘)๐น))
Colors of variables: wff setvar class
Syntax hints:   โ†’ wi 4   โˆง wa 394   = wceq 1533   โˆˆ wcel 2098  Vcvv 3473  โŸจcop 4638  โ€˜cfv 6553  (class class class)co 7426  Basecbs 17187  Hom chom 17251  compcco 17252  compfccomf 17654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-comf 17658
This theorem is referenced by:  comfval2  17690  comfeqval  17695
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