Theorem fvco2 5490

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		fnsnfv 5480	. . . . . 6 |- ( ( G Fn A /\ X e. A ) -> { ( G ` X ) } = ( G " { X } ) )
2	1	imaeq2d 4881	. . . . 5 |- ( ( G Fn A /\ X e. A ) -> ( F " { ( G ` X ) } ) = ( F " ( G " { X } ) ) )
3		imaco 5044	. . . . 5 |- ( ( F o. G ) " { X } ) = ( F " ( G " { X } ) )
4	2, 3	syl6reqr 2191	. . . 4 |- ( ( G Fn A /\ X e. A ) -> ( ( F o. G ) " { X } ) = ( F " { ( G ` X ) } ) )
5	4	eleq2d 2209	. . 3 |- ( ( G Fn A /\ X e. A ) -> ( x e. ( ( F o. G ) " { X } ) <-> x e. ( F " { ( G ` X ) } ) ) )
6	5	iotabidv 5109	. 2 |- ( ( G Fn A /\ X e. A ) -> ( iota x x e. ( ( F o. G ) " { X } ) ) = ( iota x x e. ( F " { ( G ` X ) } ) ) )
7		dffv3g 5417	. . 3 |- ( X e. A -> ( ( F o. G ) ` X ) = ( iota x x e. ( ( F o. G ) " { X } ) ) )
8	7	adantl 275	. 2 |- ( ( G Fn A /\ X e. A ) -> ( ( F o. G ) ` X ) = ( iota x x e. ( ( F o. G ) " { X } ) ) )
9		funfvex 5438	. . . 4 |- ( ( Fun G /\ X e. dom G ) -> ( G ` X ) e. _V )
10	9	funfni 5223	. . 3 |- ( ( G Fn A /\ X e. A ) -> ( G ` X ) e. _V )
11		dffv3g 5417	. . 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 2182	1 |- ( ( G Fn A /\ X e. A ) -> ( ( F o. G ) ` X ) = ( F ` ( G ` X ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   _Vcvv 2686   {csn 3527   "cima 4542   o. ccom 4543   iotacio 5086   Fn wfn 5118   ` cfv 5123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-fv 5131

This theorem is referenced by:  fvco  5491  fvco3  5492  ofco  6000  updjudhcoinlf  6965  updjudhcoinrg  6966  updjud  6967  caseinl  6976  caseinr  6977  ctm  6994  enomnilem  7010