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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.
StepHypRef Expression
1 pm13.183 3240 . . . . . 6
21ralimi 2850 . . . . 5
3 ralbi 2988 . . . . 5
42, 3syl 16 . . . 4
54ralimi 2850 . . 3
6 ralbi 2988 . . 3
75, 6syl 16 . 2
8 df-mpt2 6301 . . . 4
9 df-mpt2 6301 . . . 4
108, 9eqeq12i 2477 . . 3
11 eqoprab2b 6355 . . 3
12 pm5.32 636 . . . . . . 7
1312albii 1640 . . . . . 6
14 19.21v 1729 . . . . . 6
1513, 14bitr3i 251 . . . . 5
16152albii 1641 . . . 4
17 r2al 2835 . . . 4
1816, 17bitr4i 252 . . 3
1910, 11, 183bitri 271 . 2
207, 19syl6rbbr 264 1
 Colors of variables: wff setvar class 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
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