Theorem strnfvnd 12723

Description: Deduction version of strnfvn 12724. (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 2776	. 2 |- ( ph -> S e. _V )
3		strnfvnd.n	. . 3 |- ( ph -> N e. NN )
4		fvexg 5580	. . 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 5560	. . 3 |- ( x = S -> ( x ` N ) = ( S ` N ) )
7		strnfvnd.c	. . . 4 |- E = Slot N
8		df-slot 12707	. . . 4 |- Slot N = ( x e. _V |-> ( x ` N ) )
9	7, 8	eqtri 2217	. . 3 |- E = ( x e. _V |-> ( x ` N ) )
10	6, 9	fvmptg 5640	. 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 1364   e. wcel 2167   _Vcvv 2763   |-> cmpt 4095   ` cfv 5259   NNcn 9007  Slot cslot 12702

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-iota 5220  df-fun 5261  df-fv 5267  df-slot 12707

This theorem is referenced by:  strnfvn  12724  strfvssn  12725  strndxid  12731  strsetsid  12736  strslfvd  12745  strslfv2d  12746  setsslid  12754  setsslnid  12755  basm  12764