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Theorem mpt2eq123 6356
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:   , ,   ,   , ,   ,

Proof of Theorem mpt2eq123
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfv 1707 . . . 4
2 nfra1 2838 . . . 4
31, 2nfan 1928 . . 3
4 nfv 1707 . . . 4
5 nfcv 2619 . . . . 5
6 nfv 1707 . . . . . 6
7 nfra1 2838 . . . . . 6
86, 7nfan 1928 . . . . 5
95, 8nfral 2843 . . . 4
104, 9nfan 1928 . . 3
11 nfv 1707 . . 3
12 rsp 2823 . . . . . . 7
13 rsp 2823 . . . . . . . . . 10
14 eqeq2 2472 . . . . . . . . . 10
1513, 14syl6 33 . . . . . . . . 9
1615pm5.32d 639 . . . . . . . 8
17 eleq2 2530 . . . . . . . . 9
1817anbi1d 704 . . . . . . . 8
1916, 18sylan9bbr 700 . . . . . . 7
2012, 19syl6 33 . . . . . 6
2120pm5.32d 639 . . . . 5
22 eleq2 2530 . . . . . 6
2322anbi1d 704 . . . . 5
2421, 23sylan9bbr 700 . . . 4
25 anass 649 . . . 4
26 anass 649 . . . 4
2724, 25, 263bitr4g 288 . . 3
283, 10, 11, 27oprabbid 6350 . 2
29 df-mpt2 6301 . 2
30 df-mpt2 6301 . 2
3128, 29, 303eqtr4g 2523 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  A.wral 2807  {coprab 6297  e.cmpt2 6298
This theorem is referenced by:  mpt2eq12  6357  mapxpen  7703  pmatcollpw2lem  19278  xkoptsub  20155  xkocnv  20315
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-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-oprab 6300  df-mpt2 6301
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