Theorem fvdiagfn 6752

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 4217	. . 3 |- ( X e. B -> { X } e. _V )
3		xpexg 4777	. . 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 3633	. . . 4 |- ( x = X -> { x } = { X } )
6	5	xpeq2d 4687	. . 3 |- ( x = X -> ( I X. { x } ) = ( I X. { X } ) )
7		fdiagfn.f	. . 3 |- F = ( x e. B |-> ( I X. { x } ) )
8	6, 7	fvmptg 5637	. 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 1364   e. wcel 2167   _Vcvv 2763   {csn 3622   |-> cmpt 4094   X. cxp 4661   ` cfv 5258

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fv 5266

This theorem is referenced by: (None)