Theorem fvco2 5671

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 5207	. . . . 5 |- ( ( F o. G ) " { X } ) = ( F " ( G " { X } ) )
2		fnsnfv 5661	. . . . . 6 |- ( ( G Fn A /\ X e. A ) -> { ( G ` X ) } = ( G " { X } ) )
3	2	imaeq2d 5041	. . . . 5 |- ( ( G Fn A /\ X e. A ) -> ( F " { ( G ` X ) } ) = ( F " ( G " { X } ) ) )
4	1, 3	eqtr4id 2259	. . . 4 |- ( ( G Fn A /\ X e. A ) -> ( ( F o. G ) " { X } ) = ( F " { ( G ` X ) } ) )
5	4	eleq2d 2277	. . 3 |- ( ( G Fn A /\ X e. A ) -> ( x e. ( ( F o. G ) " { X } ) <-> x e. ( F " { ( G ` X ) } ) ) )
6	5	iotabidv 5273	. 2 |- ( ( G Fn A /\ X e. A ) -> ( iota x x e. ( ( F o. G ) " { X } ) ) = ( iota x x e. ( F " { ( G ` X ) } ) ) )
7		dffv3g 5595	. . 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 5616	. . . 4 |- ( ( Fun G /\ X e. dom G ) -> ( G ` X ) e. _V )
10	9	funfni 5395	. . 3 |- ( ( G Fn A /\ X e. A ) -> ( G ` X ) e. _V )
11		dffv3g 5595	. . 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 2250	1 |- ( ( G Fn A /\ X e. A ) -> ( ( F o. G ) ` X ) = ( F ` ( G ` X ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   _Vcvv 2776   {csn 3643   "cima 4696   o. ccom 4697   iotacio 5249   Fn wfn 5285   ` cfv 5290

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298

This theorem is referenced by:  fvco  5672  fvco3  5673  ofco  6200  updjudhcoinlf  7208  updjudhcoinrg  7209  updjud  7210  caseinl  7219  caseinr  7220  ctm  7237  enomnilem  7266  enmkvlem  7289  enwomnilem  7297  nninfctlemfo  12476  prdsidlem  13394  gsumwmhm  13445  prdsinvlem  13555  ringidvalg  13838  lidlvalg  14348  rspvalg  14349