Theorem strnfvnd 13232

Description: Deduction version of strnfvn 13233. (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 2827	. 2 |- ( ph -> S e. _V )
3		strnfvnd.n	. . 3 |- ( ph -> N e. NN )
4		fvexg 5689	. . 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 5669	. . 3 |- ( x = S -> ( x ` N ) = ( S ` N ) )
7		strnfvnd.c	. . . 4 |- E = Slot N
8		df-slot 13216	. . . 4 |- Slot N = ( x e. _V |-> ( x ` N ) )
9	7, 8	eqtri 2253	. . 3 |- E = ( x e. _V |-> ( x ` N ) )
10	6, 9	fvmptg 5753	. 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 1398   e. wcel 2203   _Vcvv 2813   |-> cmpt 4171   ` cfv 5352   NNcn 9237  Slot cslot 13211

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-iota 5312  df-fun 5354  df-fv 5360  df-slot 13216

This theorem is referenced by:  strnfvn  13233  strfvssn  13234  strndxid  13240  strsetsid  13245  strslfvd  13254  strslfv2d  13255  setsslid  13263  setsslnid  13264  basm  13274  edgfndxid  16004