Theorem cofmpt 5633

Description: Express composition of a maps-to function with another function in a maps-to notation. (Contributed by Thierry Arnoux, 29-Jun-2017.)

Hypotheses

Ref	Expression
cofmpt.1	|- ( ph -> F : C --> D )
cofmpt.2	|- ( ( ph /\ x e. A ) -> B e. C )

Assertion

Ref	Expression
cofmpt	|- ( ph -> ( F o. ( x e. A |-> B ) ) = ( x e. A |-> ( F ` B ) ) )

Distinct variable groups:   x, A   x, C   x, F   ph, x

Allowed substitution hints:   B( x)   D( x)

Proof of Theorem cofmpt

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cofmpt.2	. 2 |- ( ( ph /\ x e. A ) -> B e. C )
2		eqidd 2158	. 2 |- ( ph -> ( x e. A |-> B ) = ( x e. A |-> B ) )
3		cofmpt.1	. . 3 |- ( ph -> F : C --> D )
4	3	feqmptd 5518	. 2 |- ( ph -> F = ( y e. C |-> ( F ` y ) ) )
5		fveq2 5465	. 2 |- ( y = B -> ( F ` y ) = ( F ` B ) )
6	1, 2, 4, 5	fmptco 5630	1 |- ( ph -> ( F o. ( x e. A |-> B ) ) = ( x e. A |-> ( F ` B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1335   e. wcel 2128   |-> cmpt 4025   o. ccom 4587   -->wf 5163   ` cfv 5167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4252  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-fv 5175

This theorem is referenced by:  dvcjbr  13032  dvmptcjx  13046  dvef  13048