Theorem strslssd 13276

Description: Deduction version of strslss 13277. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 31-Jan-2023.)

Hypotheses

Ref	Expression
strslssd.e	|- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
strssd.t	|- ( ph -> T e. V )
strssd.f	|- ( ph -> Fun T )
strssd.s	|- ( ph -> S C_ T )
strssd.n	|- ( ph -> <. ( E ` ndx ) , C >. e. S )

Assertion

Ref	Expression
strslssd	|- ( ph -> ( E ` T ) = ( E ` S ) )

Proof of Theorem strslssd

Step	Hyp	Ref	Expression
1		strslssd.e	. . 3 |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
2		strssd.t	. . 3 |- ( ph -> T e. V )
3		strssd.f	. . 3 |- ( ph -> Fun T )
4		strssd.s	. . . 4 |- ( ph -> S C_ T )
5		strssd.n	. . . 4 |- ( ph -> <. ( E ` ndx ) , C >. e. S )
6	4, 5	sseldd 3241	. . 3 |- ( ph -> <. ( E ` ndx ) , C >. e. T )
7	1, 2, 3, 6	strslfvd 13271	. 2 |- ( ph -> C = ( E ` T ) )
8	2, 4	ssexd 4252	. . 3 |- ( ph -> S e. _V )
9		funss 5373	. . . 4 |- ( S C_ T -> ( Fun T -> Fun S ) )
10	4, 3, 9	sylc 62	. . 3 |- ( ph -> Fun S )
11	1, 8, 10, 5	strslfvd 13271	. 2 |- ( ph -> C = ( E ` S ) )
12	7, 11	eqtr3d 2269	1 |- ( ph -> ( E ` T ) = ( E ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   _Vcvv 2815   C_ wss 3213   <.cop 3694   Fun wfun 5348   ` cfv 5354   NNcn 9239   ndxcnx 13226  Slot cslot 13228

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-iota 5314  df-fun 5356  df-fv 5362  df-slot 13233

This theorem is referenced by:  strslss  13277