Theorem bnj97 30936

Description: Technical lemma for bnj150 30946. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj96.1	|- F = { <. (/) , pred ( x , A , R ) >. }

Assertion

Ref	Expression
bnj97	|- ( ( R FrSe A /\ x e. A ) -> ( F ` (/) ) = pred ( x , A , R ) )

Distinct variable groups:   x, A   x, R

Allowed substitution hint:   F( x)

Proof of Theorem bnj97

Step	Hyp	Ref	Expression
1		bnj93 30933	. . 3 |- ( ( R FrSe A /\ x e. A ) -> pred ( x , A , R ) e. _V )
2		0ex 4790	. . . . 5 |- (/) e. _V
3	2	bnj519 30804	. . . 4 |- ( pred ( x , A , R ) e. _V -> Fun { <. (/) , pred ( x , A , R ) >. } )
4		bnj96.1	. . . . 5 |- F = { <. (/) , pred ( x , A , R ) >. }
5	4	funeqi 5909	. . . 4 |- ( Fun F <-> Fun { <. (/) , pred ( x , A , R ) >. } )
6	3, 5	sylibr 224	. . 3 |- ( pred ( x , A , R ) e. _V -> Fun F )
7	1, 6	syl 17	. 2 |- ( ( R FrSe A /\ x e. A ) -> Fun F )
8		opex 4932	. . . 4 |- <. (/) , pred ( x , A , R ) >. e. _V
9	8	snid 4208	. . 3 |- <. (/) , pred ( x , A , R ) >. e. { <. (/) , pred ( x , A , R ) >. }
10	9, 4	eleqtrri 2700	. 2 |- <. (/) , pred ( x , A , R ) >. e. F
11		funopfv 6235	. 2 |- ( Fun F -> ( <. (/) , pred ( x , A , R ) >. e. F -> ( F ` (/) ) = pred ( x , A , R ) ) )
12	7, 10, 11	mpisyl 21	1 |- ( ( R FrSe A /\ x e. A ) -> ( F ` (/) ) = pred ( x , A , R ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   {csn 4177   <.cop 4183   Fun wfun 5882   ` cfv 5888   predc-bnj14 30754   FrSe w-bnj15 30758

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-bnj13 30757  df-bnj15 30759

This theorem is referenced by:  bnj150  30946