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Theorem comffval2 17681
Description: Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
comfffval2.o ๐‘‚ = (compfโ€˜๐ถ)
comfffval2.b ๐ต = (Baseโ€˜๐ถ)
comfffval2.h ๐ป = (Homf โ€˜๐ถ)
comfffval2.x ยท = (compโ€˜๐ถ)
comffval2.x (๐œ‘ โ†’ ๐‘‹ โˆˆ ๐ต)
comffval2.y (๐œ‘ โ†’ ๐‘Œ โˆˆ ๐ต)
comffval2.z (๐œ‘ โ†’ ๐‘ โˆˆ ๐ต)
Assertion
Ref Expression
comffval2 (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‘“)))
Distinct variable groups:   ๐‘“,๐‘”,๐ต   ๐ถ,๐‘“,๐‘”   ยท ,๐‘“,๐‘”   ๐‘“,๐‘‹,๐‘”   ๐‘“,๐‘Œ,๐‘”   ๐œ‘,๐‘“,๐‘”   ๐‘“,๐‘,๐‘”
Allowed substitution hints:   ๐ป(๐‘“,๐‘”)   ๐‘‚(๐‘“,๐‘”)

Proof of Theorem comffval2
StepHypRef Expression
1 comfffval2.o . . 3 ๐‘‚ = (compfโ€˜๐ถ)
2 comfffval2.b . . 3 ๐ต = (Baseโ€˜๐ถ)
3 eqid 2725 . . 3 (Hom โ€˜๐ถ) = (Hom โ€˜๐ถ)
4 comfffval2.x . . 3 ยท = (compโ€˜๐ถ)
5 comffval2.x . . 3 (๐œ‘ โ†’ ๐‘‹ โˆˆ ๐ต)
6 comffval2.y . . 3 (๐œ‘ โ†’ ๐‘Œ โˆˆ ๐ต)
7 comffval2.z . . 3 (๐œ‘ โ†’ ๐‘ โˆˆ ๐ต)
81, 2, 3, 4, 5, 6, 7comffval 17678 . 2 (๐œ‘ โ†’ (โŸจ๐‘‹, ๐‘ŒโŸฉ๐‘‚๐‘) = (๐‘” โˆˆ (๐‘Œ(Hom โ€˜๐ถ)๐‘), ๐‘“ โˆˆ (๐‘‹(Hom โ€˜๐ถ)๐‘Œ) โ†ฆ (๐‘”(โŸจ๐‘‹, ๐‘ŒโŸฉ ยท ๐‘)๐‘“)))
9 comfffval2.h . . . 4 ๐ป = (Homf โ€˜๐ถ)
109, 2, 3, 6, 7homfval 17671 . . 3 (๐œ‘ โ†’ (๐‘Œ๐ป๐‘) = (๐‘Œ(Hom โ€˜๐ถ)๐‘))
119, 2, 3, 5, 6homfval 17671 . . 3 (๐œ‘ โ†’ (๐‘‹๐ป๐‘Œ) = (๐‘‹(Hom โ€˜๐ถ)๐‘Œ))
12 eqidd 2726 . . 3 (๐œ‘ โ†’ (๐‘”(โŸจ๐‘‹, ๐‘ŒโŸฉ ยท ๐‘)๐‘“) = (๐‘”(โŸจ๐‘‹, ๐‘ŒโŸฉ ยท ๐‘)๐‘“))
1310, 11, 12mpoeq123dv 7492 . 2 (๐œ‘ โ†’ (๐‘” โˆˆ (๐‘Œ๐ป๐‘), ๐‘“ โˆˆ (๐‘‹๐ป๐‘Œ) โ†ฆ (๐‘”(โŸจ๐‘‹, ๐‘ŒโŸฉ ยท ๐‘)๐‘“)) = (๐‘” โˆˆ (๐‘Œ(Hom โ€˜๐ถ)๐‘), ๐‘“ โˆˆ (๐‘‹(Hom โ€˜๐ถ)๐‘Œ) โ†ฆ (๐‘”(โŸจ๐‘‹, ๐‘ŒโŸฉ ยท ๐‘)๐‘“)))
148, 13eqtr4d 2768 1 (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‘“)))
Colors of variables: wff setvar class
Syntax hints:   โ†’ wi 4   = wceq 1533   โˆˆ wcel 2098  โŸจcop 4630  โ€˜cfv 6543  (class class class)co 7416   โˆˆ cmpo 7418  Basecbs 17179  Hom chom 17243  compcco 17244  Homf chomf 17645  compfccomf 17646
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 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
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 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7991  df-2nd 7992  df-homf 17649  df-comf 17650
This theorem is referenced by: (None)
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