Theorem fvco2 5673

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 5208	. . . . 5 |- ( ( F o. G ) " { X } ) = ( F " ( G " { X } ) )
2		fnsnfv 5663	. . . . . 6 |- ( ( G Fn A /\ X e. A ) -> { ( G ` X ) } = ( G " { X } ) )
3	2	imaeq2d 5042	. . . . 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 5274	. 2 |- ( ( G Fn A /\ X e. A ) -> ( iota x x e. ( ( F o. G ) " { X } ) ) = ( iota x x e. ( F " { ( G ` X ) } ) ) )
7		dffv3g 5596	. . 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 5617	. . . 4 |- ( ( Fun G /\ X e. dom G ) -> ( G ` X ) e. _V )
10	9	funfni 5396	. . 3 |- ( ( G Fn A /\ X e. A ) -> ( G ` X ) e. _V )
11		dffv3g 5596	. . 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 2777   {csn 3644   "cima 4697   o. ccom 4698   iotacio 5250   Fn wfn 5286   ` cfv 5291

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 4179  ax-pow 4235  ax-pr 4270

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 2779  df-sbc 3007  df-un 3179  df-in 3181  df-ss 3188  df-pw 3629  df-sn 3650  df-pr 3651  df-op 3653  df-uni 3866  df-br 4061  df-opab 4123  df-id 4359  df-xp 4700  df-rel 4701  df-cnv 4702  df-co 4703  df-dm 4704  df-rn 4705  df-res 4706  df-ima 4707  df-iota 5252  df-fun 5293  df-fn 5294  df-fv 5299

This theorem is referenced by:  fvco  5674  fvco3  5675  ofco  6202  updjudhcoinlf  7210  updjudhcoinrg  7211  updjud  7212  caseinl  7221  caseinr  7222  ctm  7239  enomnilem  7268  enmkvlem  7291  enwomnilem  7299  nninfctlemfo  12522  prdsidlem  13440  gsumwmhm  13491  prdsinvlem  13601  ringidvalg  13884  lidlvalg  14394  rspvalg  14395