Theorem idref 6153

Description: TODO: This is the same as issref 5385 (which has a much longer proof). Should we replace issref 5385 with this one? - NM 9-May-2016.

Two ways to state a relation is reflexive. (Adapted from Tarski.) (Contributed by FL, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 3-Nov-2015.) (Proof modification is discouraged.)

Assertion

Ref	Expression
idref

Distinct variable groups:   ,   ,

Proof of Theorem idref

Step	Hyp	Ref	Expression
1		eqid 2457	. . . 4
2	1	fmpt 6052	. . 3
3		opex 4716	. . . . 5
4	3, 1	fnmpti 5714	. . . 4
5		df-f 5597	. . . 4
6	4, 5	mpbiran 918	. . 3
7	2, 6	bitri 249	. 2
8		df-br 4453	. . 3
9	8	ralbii 2888	. 2
10		mptresid 5333	. . . 4
11		vex 3112	. . . . 5
12	11	fnasrn 6077	. . . 4
13	10, 12	eqtr3i 2488	. . 3
14	13	sseq1i 3527	. 2
15	7, 9, 14	3bitr4ri 278	1

Syntax hints:  <->wb 184  e.wcel 1818  A.wral 2807  C_wss 3475  <.cop 4035   class class class wbr 4452  e.cmpt 4510  cid 4795  rancrn 5005  |`cres 5006  Fnwfn 5588  -->wf 5589

This theorem is referenced by:  retos  18654  filnetlem2  30197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601