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Theorem 1st0 7928
Description: The value of the first-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
Assertion
Ref Expression
1st0 (1st ‘∅) = ∅

Proof of Theorem 1st0
StepHypRef Expression
1 1stval 7924 . 2 (1st ‘∅) = dom {∅}
2 dmsn0 6162 . . 3 dom {∅} = ∅
32unieqi 4879 . 2 dom {∅} =
4 uni0 4897 . 2 ∅ = ∅
51, 3, 43eqtri 2769 1 (1st ‘∅) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  c0 4283  {csn 4587   cuni 4866  dom cdm 5634  cfv 6497  1st c1st 7920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fv 6505  df-1st 7922
This theorem is referenced by:  vafval  29548
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