Theorem fvdiagfn 6840

Description: Functionality of the diagonal map. (Contributed by Stefan O'Rear, 24-Jan-2015.)

Hypothesis

Ref	Expression
fdiagfn.f	|- F = ( x e. B |-> ( I X. { x } ) )

Assertion

Ref	Expression
fvdiagfn	|- ( ( I e. W /\ X e. B ) -> ( F ` X ) = ( I X. { X } ) )

Distinct variable groups:   x, B   x, I   x, W   x, X

Allowed substitution hint:   F( x)

Proof of Theorem fvdiagfn

Step	Hyp	Ref	Expression
1		simpr 110	. 2 |- ( ( I e. W /\ X e. B ) -> X e. B )
2		snexg 4268	. . 3 |- ( X e. B -> { X } e. _V )
3		xpexg 4833	. . 3 |- ( ( I e. W /\ { X } e. _V ) -> ( I X. { X } ) e. _V )
4	2, 3	sylan2 286	. 2 |- ( ( I e. W /\ X e. B ) -> ( I X. { X } ) e. _V )
5		sneq 3677	. . . 4 |- ( x = X -> { x } = { X } )
6	5	xpeq2d 4743	. . 3 |- ( x = X -> ( I X. { x } ) = ( I X. { X } ) )
7		fdiagfn.f	. . 3 |- F = ( x e. B |-> ( I X. { x } ) )
8	6, 7	fvmptg 5710	. 2 |- ( ( X e. B /\ ( I X. { X } ) e. _V ) -> ( F ` X ) = ( I X. { X } ) )
9	1, 4, 8	syl2anc 411	1 |- ( ( I e. W /\ X e. B ) -> ( F ` X ) = ( I X. { X } ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   _Vcvv 2799   {csn 3666   |-> cmpt 4145   X. cxp 4717   ` cfv 5318

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326

This theorem is referenced by: (None)