Theorem imai 5354

Description: Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38. (Contributed by NM, 30-Apr-1998.)

Assertion

Ref	Expression
imai

Proof of Theorem imai

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

Step	Hyp	Ref	Expression
1		dfima3 5345	. 2
2		df-br 4453	. . . . . . . 8
3		vex 3112	. . . . . . . . 9
4	3	ideq 5160	. . . . . . . 8
5	2, 4	bitr3i 251	. . . . . . 7
6	5	anbi2i 694	. . . . . 6
7		ancom 450	. . . . . 6
8	6, 7	bitri 249	. . . . 5
9	8	exbii 1667	. . . 4
10		eleq1 2529	. . . . 5
11	3, 10	ceqsexv 3146	. . . 4
12	9, 11	bitri 249	. . 3
13	12	abbii 2591	. 2
14		abid2 2597	. 2
15	1, 13, 14	3eqtri 2490	1

Syntax hints:  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  {cab 2442  <.cop 4035   class class class wbr 4452  cid 4795  "cima 5007

This theorem is referenced by:  rnresi  5355  cnvresid  5663  ecidsn  7379  mbfid  22043

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-rab 2816  df-v 3111  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-br 4453  df-opab 4511  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017