Theorem cofmpt 5824

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 2232	. 2 |- ( ph -> ( x e. A |-> B ) = ( x e. A |-> B ) )
3		cofmpt.1	. . 3 |- ( ph -> F : C --> D )
4	3	feqmptd 5708	. 2 |- ( ph -> F = ( y e. C |-> ( F ` y ) ) )
5		fveq2 5648	. 2 |- ( y = B -> ( F ` y ) = ( F ` B ) )
6	1, 2, 4, 5	fmptco 5821	1 |- ( ph -> ( F o. ( x e. A |-> B ) ) = ( x e. A |-> ( F ` B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   |-> cmpt 4155   o. ccom 4735   -->wf 5329   ` cfv 5333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341

This theorem is referenced by:  dvcjbr  15519  dvmptcjx  15535  dvef  15538  lgseisenlem4  15892  gfsumsn  16814