Theorem fvco2 5587

Description: Value of a function composition. Similar to second part of Theorem 3H of [Enderton] p. 47. (Contributed by NM, 9-Oct-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by Stefan O'Rear, 16-Oct-2014.)

Assertion

Ref	Expression
fvco2	|- ( ( G Fn A /\ X e. A ) -> ( ( F o. G ) ` X ) = ( F ` ( G ` X ) ) )

Proof of Theorem fvco2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		imaco 5136	. . . . 5 |- ( ( F o. G ) " { X } ) = ( F " ( G " { X } ) )
2		fnsnfv 5577	. . . . . 6 |- ( ( G Fn A /\ X e. A ) -> { ( G ` X ) } = ( G " { X } ) )
3	2	imaeq2d 4972	. . . . 5 |- ( ( G Fn A /\ X e. A ) -> ( F " { ( G ` X ) } ) = ( F " ( G " { X } ) ) )
4	1, 3	eqtr4id 2229	. . . 4 |- ( ( G Fn A /\ X e. A ) -> ( ( F o. G ) " { X } ) = ( F " { ( G ` X ) } ) )
5	4	eleq2d 2247	. . 3 |- ( ( G Fn A /\ X e. A ) -> ( x e. ( ( F o. G ) " { X } ) <-> x e. ( F " { ( G ` X ) } ) ) )
6	5	iotabidv 5201	. 2 |- ( ( G Fn A /\ X e. A ) -> ( iota x x e. ( ( F o. G ) " { X } ) ) = ( iota x x e. ( F " { ( G ` X ) } ) ) )
7		dffv3g 5513	. . 3 |- ( X e. A -> ( ( F o. G ) ` X ) = ( iota x x e. ( ( F o. G ) " { X } ) ) )
8	7	adantl 277	. 2 |- ( ( G Fn A /\ X e. A ) -> ( ( F o. G ) ` X ) = ( iota x x e. ( ( F o. G ) " { X } ) ) )
9		funfvex 5534	. . . 4 |- ( ( Fun G /\ X e. dom G ) -> ( G ` X ) e. _V )
10	9	funfni 5318	. . 3 |- ( ( G Fn A /\ X e. A ) -> ( G ` X ) e. _V )
11		dffv3g 5513	. . 3 |- ( ( G ` X ) e. _V -> ( F ` ( G ` X ) ) = ( iota x x e. ( F " { ( G ` X ) } ) ) )
12	10, 11	syl 14	. 2 |- ( ( G Fn A /\ X e. A ) -> ( F ` ( G ` X ) ) = ( iota x x e. ( F " { ( G ` X ) } ) ) )
13	6, 8, 12	3eqtr4d 2220	1 |- ( ( G Fn A /\ X e. A ) -> ( ( F o. G ) ` X ) = ( F ` ( G ` X ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   _Vcvv 2739   {csn 3594   "cima 4631   o. ccom 4632   iotacio 5178   Fn wfn 5213   ` cfv 5218

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226

This theorem is referenced by:  fvco  5588  fvco3  5589  ofco  6103  updjudhcoinlf  7081  updjudhcoinrg  7082  updjud  7083  caseinl  7092  caseinr  7093  ctm  7110  enomnilem  7138  enmkvlem  7161  enwomnilem  7169  ringidvalg  13149