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Theorem comffval2 17542
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 2737 . . 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 17539 . 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 17532 . . 3 (๐œ‘ โ†’ (๐‘Œ๐ป๐‘) = (๐‘Œ(Hom โ€˜๐ถ)๐‘))
119, 2, 3, 5, 6homfval 17532 . . 3 (๐œ‘ โ†’ (๐‘‹๐ป๐‘Œ) = (๐‘‹(Hom โ€˜๐ถ)๐‘Œ))
12 eqidd 2738 . . 3 (๐œ‘ โ†’ (๐‘”(โŸจ๐‘‹, ๐‘ŒโŸฉ ยท ๐‘)๐‘“) = (๐‘”(โŸจ๐‘‹, ๐‘ŒโŸฉ ยท ๐‘)๐‘“))
1310, 11, 12mpoeq123dv 7426 . 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 2780 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 1541   โˆˆ wcel 2106  โŸจcop 4590  โ€˜cfv 6493  (class class class)co 7351   โˆˆ cmpo 7353  Basecbs 17043  Hom chom 17104  compcco 17105  Homf chomf 17506  compfccomf 17507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-1st 7913  df-2nd 7914  df-homf 17510  df-comf 17511
This theorem is referenced by: (None)
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