Theorem strnfvnd 12967

Description: Deduction version of strnfvn 12968. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Jim Kingdon, 19-Jan-2023.)

Hypotheses

Ref	Expression
strnfvnd.c	|- E = Slot N
strnfvnd.f	|- ( ph -> S e. V )
strnfvnd.n	|- ( ph -> N e. NN )

Assertion

Ref	Expression
strnfvnd	|- ( ph -> ( E ` S ) = ( S ` N ) )

Proof of Theorem strnfvnd

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		strnfvnd.f	. . 3 |- ( ph -> S e. V )
2	1	elexd 2790	. 2 |- ( ph -> S e. _V )
3		strnfvnd.n	. . 3 |- ( ph -> N e. NN )
4		fvexg 5618	. . 3 |- ( ( S e. V /\ N e. NN ) -> ( S ` N ) e. _V )
5	1, 3, 4	syl2anc 411	. 2 |- ( ph -> ( S ` N ) e. _V )
6		fveq1 5598	. . 3 |- ( x = S -> ( x ` N ) = ( S ` N ) )
7		strnfvnd.c	. . . 4 |- E = Slot N
8		df-slot 12951	. . . 4 |- Slot N = ( x e. _V |-> ( x ` N ) )
9	7, 8	eqtri 2228	. . 3 |- E = ( x e. _V |-> ( x ` N ) )
10	6, 9	fvmptg 5678	. 2 |- ( ( S e. _V /\ ( S ` N ) e. _V ) -> ( E ` S ) = ( S ` N ) )
11	2, 5, 10	syl2anc 411	1 |- ( ph -> ( E ` S ) = ( S ` N ) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2178   _Vcvv 2776   |-> cmpt 4121   ` cfv 5290   NNcn 9071  Slot cslot 12946

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-iota 5251  df-fun 5292  df-fv 5298  df-slot 12951

This theorem is referenced by:  strnfvn  12968  strfvssn  12969  strndxid  12975  strsetsid  12980  strslfvd  12989  strslfv2d  12990  setsslid  12998  setsslnid  12999  basm  13008  edgfndxid  15723