Theorem strslssd 12879

Description: Deduction version of strslss 12880. (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 3194	. . 3 |- ( ph -> <. ( E ` ndx ) , C >. e. T )
7	1, 2, 3, 6	strslfvd 12874	. 2 |- ( ph -> C = ( E ` T ) )
8	2, 4	ssexd 4184	. . 3 |- ( ph -> S e. _V )
9		funss 5290	. . . 4 |- ( S C_ T -> ( Fun T -> Fun S ) )
10	4, 3, 9	sylc 62	. . 3 |- ( ph -> Fun S )
11	1, 8, 10, 5	strslfvd 12874	. 2 |- ( ph -> C = ( E ` S ) )
12	7, 11	eqtr3d 2240	1 |- ( ph -> ( E ` T ) = ( E ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   _Vcvv 2772   C_ wss 3166   <.cop 3636   Fun wfun 5265   ` cfv 5271   NNcn 9036   ndxcnx 12829  Slot cslot 12831

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-iota 5232  df-fun 5273  df-fv 5279  df-slot 12836

This theorem is referenced by:  strslss  12880