Theorem iunopab 4788

Description: Move indexed union inside an ordered-pair abstraction. (Contributed by Stefan O'Rear, 20-Feb-2015.)

Assertion

Ref	Expression
iunopab

Distinct variable groups:   ,   ,   ,   ,

Proof of Theorem iunopab

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elopab 4760	. . . . 5
2	1	rexbii 2959	. . . 4
3		rexcom4 3129	. . . . 5
4		rexcom4 3129	. . . . . . 7
5		r19.42v 3012	. . . . . . . 8
6	5	exbii 1667	. . . . . . 7
7	4, 6	bitri 249	. . . . . 6
8	7	exbii 1667	. . . . 5
9	3, 8	bitri 249	. . . 4
10	2, 9	bitri 249	. . 3
11	10	abbii 2591	. 2
12		df-iun 4332	. 2
13		df-opab 4511	. 2
14	11, 12, 13	3eqtr4i 2496	1

Syntax hints:  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  {cab 2442  E.wrex 2808  <.cop 4035  U_ciun 4330  {copab 4509

This theorem is referenced by:  marypha2lem2  7916

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-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  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-iun 4332  df-opab 4511