Theorem strnfvnd 12414

Description: Deduction version of strnfvn 12415. (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 2739	. 2 |- ( ph -> S e. _V )
3		strnfvnd.n	. . 3 |- ( ph -> N e. NN )
4		fvexg 5505	. . 3 |- ( ( S e. V /\ N e. NN ) -> ( S ` N ) e. _V )
5	1, 3, 4	syl2anc 409	. 2 |- ( ph -> ( S ` N ) e. _V )
6		fveq1 5485	. . 3 |- ( x = S -> ( x ` N ) = ( S ` N ) )
7		strnfvnd.c	. . . 4 |- E = Slot N
8		df-slot 12398	. . . 4 |- Slot N = ( x e. _V |-> ( x ` N ) )
9	7, 8	eqtri 2186	. . 3 |- E = ( x e. _V |-> ( x ` N ) )
10	6, 9	fvmptg 5562	. 2 |- ( ( S e. _V /\ ( S ` N ) e. _V ) -> ( E ` S ) = ( S ` N ) )
11	2, 5, 10	syl2anc 409	1 |- ( ph -> ( E ` S ) = ( S ` N ) )

Syntax hints:   -> wi 4   = wceq 1343   e. wcel 2136   _Vcvv 2726   |-> cmpt 4043   ` cfv 5188   NNcn 8857  Slot cslot 12393

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fv 5196  df-slot 12398

This theorem is referenced by:  strnfvn  12415  strfvssn  12416  strndxid  12422  strsetsid  12427  strslfvd  12435  strslfv2d  12436  setsslid  12444  setsslnid  12445  ressid  12456