Theorem fvco2 5650

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 5189	. . . . 5 |- ( ( F o. G ) " { X } ) = ( F " ( G " { X } ) )
2		fnsnfv 5640	. . . . . 6 |- ( ( G Fn A /\ X e. A ) -> { ( G ` X ) } = ( G " { X } ) )
3	2	imaeq2d 5023	. . . . 5 |- ( ( G Fn A /\ X e. A ) -> ( F " { ( G ` X ) } ) = ( F " ( G " { X } ) ) )
4	1, 3	eqtr4id 2257	. . . 4 |- ( ( G Fn A /\ X e. A ) -> ( ( F o. G ) " { X } ) = ( F " { ( G ` X ) } ) )
5	4	eleq2d 2275	. . 3 |- ( ( G Fn A /\ X e. A ) -> ( x e. ( ( F o. G ) " { X } ) <-> x e. ( F " { ( G ` X ) } ) ) )
6	5	iotabidv 5255	. 2 |- ( ( G Fn A /\ X e. A ) -> ( iota x x e. ( ( F o. G ) " { X } ) ) = ( iota x x e. ( F " { ( G ` X ) } ) ) )
7		dffv3g 5574	. . 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 5595	. . . 4 |- ( ( Fun G /\ X e. dom G ) -> ( G ` X ) e. _V )
10	9	funfni 5377	. . 3 |- ( ( G Fn A /\ X e. A ) -> ( G ` X ) e. _V )
11		dffv3g 5574	. . 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 2248	1 |- ( ( G Fn A /\ X e. A ) -> ( ( F o. G ) ` X ) = ( F ` ( G ` X ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   _Vcvv 2772   {csn 3633   "cima 4679   o. ccom 4680   iotacio 5231   Fn wfn 5267   ` cfv 5272

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-opab 4107  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-fv 5280

This theorem is referenced by:  fvco  5651  fvco3  5652  ofco  6179  updjudhcoinlf  7184  updjudhcoinrg  7185  updjud  7186  caseinl  7195  caseinr  7196  ctm  7213  enomnilem  7242  enmkvlem  7265  enwomnilem  7273  nninfctlemfo  12394  prdsidlem  13312  gsumwmhm  13363  prdsinvlem  13473  ringidvalg  13756  lidlvalg  14266  rspvalg  14267