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Theorem comfval 17454
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 17453 . 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 7316 . . 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 483 . 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 7342 . 2 (๐œ‘ โ†’ (๐บ(โŸจ๐‘‹, ๐‘ŒโŸฉ ยท ๐‘)๐น) โˆˆ V)
148, 10, 11, 12, 13ovmpod 7457 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 397   = wceq 1539   โˆˆ wcel 2104  Vcvv 3437  โŸจcop 4571  โ€˜cfv 6458  (class class class)co 7307  Basecbs 16957  Hom chom 17018  compcco 17019  compfccomf 17421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-ov 7310  df-oprab 7311  df-mpo 7312  df-1st 7863  df-2nd 7864  df-comf 17425
This theorem is referenced by:  comfval2  17457  comfeqval  17462
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