Theorem mpt22eqb 6411

Description: Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnov2 6409. (Contributed by Mario Carneiro, 4-Jan-2017.)

Assertion

Ref	Expression
mpt22eqb

Distinct variable groups:   ,,   ,

Proof of Theorem mpt22eqb

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pm13.183 3240	. . . . . 6
2	1	ralimi 2850	. . . . 5
3		ralbi 2988	. . . . 5
4	2, 3	syl 16	. . . 4
5	4	ralimi 2850	. . 3
6		ralbi 2988	. . 3
7	5, 6	syl 16	. 2
8		df-mpt2 6301	. . . 4
9		df-mpt2 6301	. . . 4
10	8, 9	eqeq12i 2477	. . 3
11		eqoprab2b 6355	. . 3
12		pm5.32 636	. . . . . . 7
13	12	albii 1640	. . . . . 6
14		19.21v 1729	. . . . . 6
15	13, 14	bitr3i 251	. . . . 5
16	15	2albii 1641	. . . 4
17		r2al 2835	. . . 4
18	16, 17	bitr4i 252	. . 3
19	10, 11, 18	3bitri 271	. 2
20	7, 19	syl6rbbr 264	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  =wceq 1395  e.wcel 1818  A.wral 2807  {coprab 6297  e.cmpt2 6298

This theorem is referenced by:  homfeq  15089  comfeq  15101

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-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-oprab 6300  df-mpt2 6301