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Theorem mpt2eq123 5661
 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 1619 . . . 4
2 nfra1 2664 . . . 4
31, 2nfan 1824 . . 3
4 nfv 1619 . . . 4
5 nfcv 2489 . . . . 5
6 nfv 1619 . . . . . 6
7 nfra1 2664 . . . . . 6
86, 7nfan 1824 . . . . 5
95, 8nfral 2667 . . . 4
104, 9nfan 1824 . . 3
11 nfv 1619 . . 3
12 rsp 2674 . . . . . . 7
13 rsp 2674 . . . . . . . . . 10
14 eqeq2 2362 . . . . . . . . . 10
1513, 14syl6 29 . . . . . . . . 9
1615pm5.32d 620 . . . . . . . 8
17 eleq2 2414 . . . . . . . . 9
1817anbi1d 685 . . . . . . . 8
1916, 18sylan9bbr 681 . . . . . . 7
2012, 19syl6 29 . . . . . 6
2120pm5.32d 620 . . . . 5
22 eleq2 2414 . . . . . 6
2322anbi1d 685 . . . . 5
2421, 23sylan9bbr 681 . . . 4
25 anass 630 . . . 4
26 anass 630 . . . 4
2724, 25, 263bitr4g 279 . . 3
283, 10, 11, 27oprabbid 5563 . 2
29 df-mpt2 5654 . 2
30 df-mpt2 5654 . 2
3128, 29, 303eqtr4g 2410 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176   wa 358   wceq 1642   wcel 1710  wral 2614  coprab 5527   cmpt2 5653 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-oprab 5528  df-mpt2 5654 This theorem is referenced by:  mpt2eq12  5662
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