Theorem fvdiagfn 6969

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 4319	. . 3 |- ( X e. B -> { X } e. _V )
3		xpexg 4887	. . 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 3719	. . . 4 |- ( x = X -> { x } = { X } )
6	5	xpeq2d 4796	. . 3 |- ( x = X -> ( I X. { x } ) = ( I X. { X } ) )
7		fdiagfn.f	. . 3 |- F = ( x e. B |-> ( I X. { x } ) )
8	6, 7	fvmptg 5778	. 2 |- ( ( X e. B /\ ( I X. { X } ) e. _V ) -> ( F ` X ) = ( I X. { X } ) )
9	1, 4, 8	syl2anc 415	1 |- ( ( I e. W /\ X e. B ) -> ( F ` X ) = ( I X. { X } ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   {csn 3708   |-> cmpt 4190   X. cxp 4770   ` cfv 5375

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-iota 5335  df-fun 5377  df-fv 5383

This theorem is used by: (None)