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Theorem mpt2eq123 6135
 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 1630 . . . 4
2 nfra1 2758 . . . 4
31, 2nfan 1847 . . 3
4 nfv 1630 . . . 4
5 nfcv 2574 . . . . 5
6 nfv 1630 . . . . . 6
7 nfra1 2758 . . . . . 6
86, 7nfan 1847 . . . . 5
95, 8nfral 2761 . . . 4
104, 9nfan 1847 . . 3
11 nfv 1630 . . 3
12 rsp 2768 . . . . . . 7
13 rsp 2768 . . . . . . . . . 10
14 eqeq2 2447 . . . . . . . . . 10
1513, 14syl6 32 . . . . . . . . 9
1615pm5.32d 622 . . . . . . . 8
17 eleq2 2499 . . . . . . . . 9
1817anbi1d 687 . . . . . . . 8
1916, 18sylan9bbr 683 . . . . . . 7
2012, 19syl6 32 . . . . . 6
2120pm5.32d 622 . . . . 5
22 eleq2 2499 . . . . . 6
2322anbi1d 687 . . . . 5
2421, 23sylan9bbr 683 . . . 4
25 anass 632 . . . 4
26 anass 632 . . . 4
2724, 25, 263bitr4g 281 . . 3
283, 10, 11, 27oprabbid 6129 . 2
29 df-mpt2 6088 . 2
30 df-mpt2 6088 . 2
3128, 29, 303eqtr4g 2495 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360   wceq 1653   wcel 1726  wral 2707  coprab 6084   cmpt2 6085 This theorem is referenced by:  mpt2eq12  6136  mapxpen  7275  xkoptsub  17688  xkocnv  17848 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-oprab 6087  df-mpt2 6088
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