Theorem strnfvnd 12436

Description: Deduction version of strnfvn 12437. (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 2743	. 2 |- ( ph -> S e. _V )
3		strnfvnd.n	. . 3 |- ( ph -> N e. NN )
4		fvexg 5515	. . 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 5495	. . 3 |- ( x = S -> ( x ` N ) = ( S ` N ) )
7		strnfvnd.c	. . . 4 |- E = Slot N
8		df-slot 12420	. . . 4 |- Slot N = ( x e. _V |-> ( x ` N ) )
9	7, 8	eqtri 2191	. . 3 |- E = ( x e. _V |-> ( x ` N ) )
10	6, 9	fvmptg 5572	. 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 1348   e. wcel 2141   _Vcvv 2730   |-> cmpt 4050   ` cfv 5198   NNcn 8878  Slot cslot 12415

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fv 5206  df-slot 12420

This theorem is referenced by:  strnfvn  12437  strfvssn  12438  strndxid  12444  strsetsid  12449  strslfvd  12457  strslfv2d  12458  setsslid  12466  setsslnid  12467  ressid  12479