Theorem resiexg 5005

Description: The existence of a restricted identity function, proved without using the Axiom of Replacement. (Contributed by NM, 13-Jan-2007.)

Assertion

Ref	Expression
resiexg	|- ( A e. V -> ( _I |` A ) e. _V )

Proof of Theorem resiexg

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relres 4988	. . 3 |- Rel ( _I |` A )
2		simpr 110	. . . . 5 |- ( ( x = y /\ x e. A ) -> x e. A )
3		eleq1 2268	. . . . . 6 |- ( x = y -> ( x e. A <-> y e. A ) )
4	3	biimpa 296	. . . . 5 |- ( ( x = y /\ x e. A ) -> y e. A )
5	2, 4	jca 306	. . . 4 |- ( ( x = y /\ x e. A ) -> ( x e. A /\ y e. A ) )
6		vex 2775	. . . . . 6 |- y e. _V
7	6	opelres 4965	. . . . 5 |- ( <. x , y >. e. ( _I |` A ) <-> ( <. x , y >. e. _I /\ x e. A ) )
8		df-br 4046	. . . . . . 7 |- ( x _I y <-> <. x , y >. e. _I )
9	6	ideq 4831	. . . . . . 7 |- ( x _I y <-> x = y )
10	8, 9	bitr3i 186	. . . . . 6 |- ( <. x , y >. e. _I <-> x = y )
11	10	anbi1i 458	. . . . 5 |- ( ( <. x , y >. e. _I /\ x e. A ) <-> ( x = y /\ x e. A ) )
12	7, 11	bitri 184	. . . 4 |- ( <. x , y >. e. ( _I |` A ) <-> ( x = y /\ x e. A ) )
13		opelxp 4706	. . . 4 |- ( <. x , y >. e. ( A X. A ) <-> ( x e. A /\ y e. A ) )
14	5, 12, 13	3imtr4i 201	. . 3 |- ( <. x , y >. e. ( _I |` A ) -> <. x , y >. e. ( A X. A ) )
15	1, 14	relssi 4767	. 2 |- ( _I |` A ) C_ ( A X. A )
16		xpexg 4790	. . 3 |- ( ( A e. V /\ A e. V ) -> ( A X. A ) e. _V )
17	16	anidms 397	. 2 |- ( A e. V -> ( A X. A ) e. _V )
18		ssexg 4184	. 2 |- ( ( ( _I |` A ) C_ ( A X. A ) /\ ( A X. A ) e. _V ) -> ( _I |` A ) e. _V )
19	15, 17, 18	sylancr 414	1 |- ( A e. V -> ( _I |` A ) e. _V )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2176   _Vcvv 2772   C_ wss 3166   <.cop 3636   class class class wbr 4045   _I cid 4336   X. cxp 4674   |` cres 4678

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 4163  ax-pow 4219  ax-pr 4254  ax-un 4481

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-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 4046  df-opab 4107  df-id 4341  df-xp 4682  df-rel 4683  df-res 4688

This theorem is referenced by:  ordiso  7140  omct  7221  ctssexmid  7254  ssomct  12849  ndxarg  12888  subctctexmid  15974