Theorem cofmpt 5880

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 2239	. 2 |- ( ph -> ( x e. A |-> B ) = ( x e. A |-> B ) )
3		cofmpt.1	. . 3 |- ( ph -> F : C --> D )
4	3	feqmptd 5759	. 2 |- ( ph -> F = ( y e. C |-> ( F ` y ) ) )
5		fveq2 5698	. 2 |- ( y = B -> ( F ` y ) = ( F ` B ) )
6	1, 2, 4, 5	fmptco 5877	1 |- ( ph -> ( F o. ( x e. A |-> B ) ) = ( x e. A |-> ( F ` B ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   |-> cmpt 4193   o. ccom 4780   -->wf 5376   ` cfv 5380

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-fv 5388

This theorem is used by:  gsumsncmn  14246  dvcjbr  15934  dvmptcjx  15950  dvef  15953  lgseisenlem4  16397