Theorem iota2 5582

Description: The unique element such that . (Contributed by Jeff Madsen, 1-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)

Hypothesis

Ref	Expression
iota2.1

Assertion

Ref	Expression
iota2

Distinct variable groups:   ,   ,

Proof of Theorem iota2

Step	Hyp	Ref	Expression
1		elex 3118	. 2
2		simpl 457	. . 3
3		simpr 461	. . 3
4		iota2.1	. . . 4
5	4	adantl 466	. . 3
6		nfv 1707	. . . 4
7		nfeu1 2294	. . . 4
8	6, 7	nfan 1928	. . 3
9		nfvd 1708	. . 3
10		nfcvd 2620	. . 3
11	2, 3, 5, 8, 9, 10	iota2df 5580	. 2
12	1, 11	sylan 471	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  E!weu 2282  cvv 3109  iotacio 5554

This theorem is referenced by:  pczpre  14371  pcdiv  14376  rngurd  27778  unirep  30203  ellimciota  31620  bj-nuliota  34586

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-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

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-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-v 3111  df-sbc 3328  df-un 3480  df-sn 4030  df-pr 4032  df-uni 4250  df-iota 5556