Theorem cofmpt 5728

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 2194	. 2 |- ( ph -> ( x e. A |-> B ) = ( x e. A |-> B ) )
3		cofmpt.1	. . 3 |- ( ph -> F : C --> D )
4	3	feqmptd 5611	. 2 |- ( ph -> F = ( y e. C |-> ( F ` y ) ) )
5		fveq2 5555	. 2 |- ( y = B -> ( F ` y ) = ( F ` B ) )
6	1, 2, 4, 5	fmptco 5725	1 |- ( ph -> ( F o. ( x e. A |-> B ) ) = ( x e. A |-> ( F ` B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   |-> cmpt 4091   o. ccom 4664   -->wf 5251   ` cfv 5255

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263

This theorem is referenced by:  dvcjbr  14887  dvmptcjx  14903  dvef  14906  lgseisenlem4  15230