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Theorem mpt2eq123 5592
 Description: An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) (Revised by Mario Carneiro, 19-Mar-2015.)
Assertion
Ref Expression
mpt2eq123
Distinct variable groups:   ,,   ,   ,,   ,
Allowed substitution hints:   ()   (,)   ()   (,)

Proof of Theorem mpt2eq123
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfv 1437 . . . 4
2 nfra1 2372 . . . 4
31, 2nfan 1473 . . 3
4 nfv 1437 . . . 4
5 nfcv 2194 . . . . 5
6 nfv 1437 . . . . . 6
7 nfra1 2372 . . . . . 6
86, 7nfan 1473 . . . . 5
95, 8nfralxy 2377 . . . 4
104, 9nfan 1473 . . 3
11 nfv 1437 . . 3
12 rsp 2386 . . . . . . 7
13 rsp 2386 . . . . . . . . . 10
14 eqeq2 2065 . . . . . . . . . 10
1513, 14syl6 33 . . . . . . . . 9
1615pm5.32d 431 . . . . . . . 8
17 eleq2 2117 . . . . . . . . 9
1817anbi1d 446 . . . . . . . 8
1916, 18sylan9bbr 444 . . . . . . 7
2012, 19syl6 33 . . . . . 6
2120pm5.32d 431 . . . . 5
22 eleq2 2117 . . . . . 6
2322anbi1d 446 . . . . 5
2421, 23sylan9bbr 444 . . . 4
25 anass 387 . . . 4
26 anass 387 . . . 4
2724, 25, 263bitr4g 216 . . 3
283, 10, 11, 27oprabbid 5586 . 2
29 df-mpt2 5545 . 2
30 df-mpt2 5545 . 2
3128, 29, 303eqtr4g 2113 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 101   wb 102   wceq 1259   wcel 1409  wral 2323  coprab 5541   cmpt2 5542 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-oprab 5544  df-mpt2 5545 This theorem is referenced by:  mpt2eq12  5593
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