Theorem imaiinfv 37256

Description: Indexed intersection of an image. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Assertion

Ref	Expression
imaiinfv	|- ( ( F Fn A /\ B C_ A ) -> |^|_ x e. B ( F ` x ) = |^| ( F " B ) )

Distinct variable groups:   x, B   x, F

Allowed substitution hint:   A( x)

Proof of Theorem imaiinfv

Step	Hyp	Ref	Expression
1		fnssres 6004	. . 3 |- ( ( F Fn A /\ B C_ A ) -> ( F |` B ) Fn B )
2		fniinfv 6257	. . 3 |- ( ( F |` B ) Fn B -> |^|_ x e. B ( ( F |` B ) ` x ) = |^| ran ( F |` B ) )
3	1, 2	syl 17	. 2 |- ( ( F Fn A /\ B C_ A ) -> |^|_ x e. B ( ( F |` B ) ` x ) = |^| ran ( F |` B ) )
4		fvres 6207	. . . 4 |- ( x e. B -> ( ( F |` B ) ` x ) = ( F ` x ) )
5	4	iineq2i 4540	. . 3 |- |^|_ x e. B ( ( F |` B ) ` x ) = |^|_ x e. B ( F ` x )
6	5	eqcomi 2631	. 2 |- |^|_ x e. B ( F ` x ) = |^|_ x e. B ( ( F |` B ) ` x )
7		df-ima 5127	. . 3 |- ( F " B ) = ran ( F |` B )
8	7	inteqi 4479	. 2 |- |^| ( F " B ) = |^| ran ( F |` B )
9	3, 6, 8	3eqtr4g 2681	1 |- ( ( F Fn A /\ B C_ A ) -> |^|_ x e. B ( F ` x ) = |^| ( F " B ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   C_ wss 3574   |^|cint 4475   |^|_ciin 4521   ran crn 5115   |` cres 5116   "cima 5117   Fn wfn 5883   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  elrfirn  37258