Theorem fvmpt2i 6290

Description: Value of a function given by the "maps to" notation. (Contributed by Mario Carneiro, 23-Apr-2014.)

Hypothesis

Ref	Expression
mptrcl.1	|- F = ( x e. A |-> B )

Assertion

Ref	Expression
fvmpt2i	|- ( x e. A -> ( F ` x ) = ( _I ` B ) )

Distinct variable group:   x, A

Allowed substitution hints:   B( x)   F( x)

Proof of Theorem fvmpt2i

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3536	. . 3 |- ( y = x -> [_ y / x ]_ B = [_ x / x ]_ B )
2		csbid 3541	. . 3 |- [_ x / x ]_ B = B
3	1, 2	syl6eq 2672	. 2 |- ( y = x -> [_ y / x ]_ B = B )
4		mptrcl.1	. . 3 |- F = ( x e. A |-> B )
5		nfcv 2764	. . . 4 |- F/_ y B
6		nfcsb1v 3549	. . . 4 |- F/_ x [_ y / x ]_ B
7		csbeq1a 3542	. . . 4 |- ( x = y -> B = [_ y / x ]_ B )
8	5, 6, 7	cbvmpt 4749	. . 3 |- ( x e. A |-> B ) = ( y e. A |-> [_ y / x ]_ B )
9	4, 8	eqtri 2644	. 2 |- F = ( y e. A |-> [_ y / x ]_ B )
10	3, 9	fvmpti 6281	1 |- ( x e. A -> ( F ` x ) = ( _I ` B ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   [_csb 3533   |-> cmpt 4729   _I cid 5023   ` 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-8 1992  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-pow 4843  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-csb 3534  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-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-fv 5896

This theorem is referenced by:  fvmpt2  6291  sumfc  14440  fsumf1o  14454  sumss  14455  isumshft  14571  prodfc  14675  fprodf1o  14676  mbfsup  23431  itg2splitlem  23515  dgrle  23999