Theorem mptun 5717

Description: Union of mappings which are mutually compatible. (Contributed by Mario Carneiro, 31-Aug-2015.)

Assertion

Ref	Expression
mptun

Proof of Theorem mptun

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mpt 4512	. 2
2		df-mpt 4512	. . . 4
3		df-mpt 4512	. . . 4
4	2, 3	uneq12i 3655	. . 3
5		elun 3644	. . . . . . 7
6	5	anbi1i 695	. . . . . 6
7		andir 868	. . . . . 6
8	6, 7	bitri 249	. . . . 5
9	8	opabbii 4516	. . . 4
10		unopab 4527	. . . 4
11	9, 10	eqtr4i 2489	. . 3
12	4, 11	eqtr4i 2489	. 2
13	1, 12	eqtr4i 2489	1

Syntax hints:  \/wo 368  /\wa 369  =wceq 1395  e.wcel 1818  u.cun 3473  {copab 4509  e.cmpt 4510

This theorem is referenced by:  fmptap  6094  fmptapd  6095  partfun  27516  ptrest  30048

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-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-un 3480  df-opab 4511  df-mpt 4512