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Theorem mpt22eqb 6393
 Description: Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnov2 6391. (Contributed by Mario Carneiro, 4-Jan-2017.)
Assertion
Ref Expression
mpt22eqb
Distinct variable groups:   ,,   ,
Allowed substitution hints:   ()   (,)   (,)   (,)

Proof of Theorem mpt22eqb
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 pm13.183 3244 . . . . . 6
21ralimi 2857 . . . . 5
3 ralbi 2993 . . . . 5
42, 3syl 16 . . . 4
54ralimi 2857 . . 3
6 ralbi 2993 . . 3
75, 6syl 16 . 2
8 df-mpt2 6287 . . . 4
9 df-mpt2 6287 . . . 4
108, 9eqeq12i 2487 . . 3
11 eqoprab2b 6337 . . 3
12 pm5.32 636 . . . . . . 7
1312albii 1620 . . . . . 6
14 19.21v 1930 . . . . . 6
1513, 14bitr3i 251 . . . . 5
16152albii 1621 . . . 4
17 r2al 2842 . . . 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  wal 1377   wceq 1379   wcel 1767  wral 2814  coprab 6283   cmpt2 6284 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-oprab 6286  df-mpt2 6287 This theorem is referenced by:  homfeq  14943  comfeq  14955
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