Theorem fvdiagfn 6539

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 109	. 2 |- ( ( I e. W /\ X e. B ) -> X e. B )
2		snexg 4066	. . 3 |- ( X e. B -> { X } e. _V )
3		xpexg 4611	. . 3 |- ( ( I e. W /\ { X } e. _V ) -> ( I X. { X } ) e. _V )
4	2, 3	sylan2 282	. 2 |- ( ( I e. W /\ X e. B ) -> ( I X. { X } ) e. _V )
5		sneq 3502	. . . 4 |- ( x = X -> { x } = { X } )
6	5	xpeq2d 4521	. . 3 |- ( x = X -> ( I X. { x } ) = ( I X. { X } ) )
7		fdiagfn.f	. . 3 |- F = ( x e. B |-> ( I X. { x } ) )
8	6, 7	fvmptg 5449	. 2 |- ( ( X e. B /\ ( I X. { X } ) e. _V ) -> ( F ` X ) = ( I X. { X } ) )
9	1, 4, 8	syl2anc 406	1 |- ( ( I e. W /\ X e. B ) -> ( F ` X ) = ( I X. { X } ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1312   e. wcel 1461   _Vcvv 2655   {csn 3491   |-> cmpt 3947   X. cxp 4495   ` cfv 5079

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-sbc 2877  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-iota 5044  df-fun 5081  df-fv 5087

This theorem is referenced by: (None)