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Theorem mpt0 4948
Description: A mapping operation with empty domain. (Contributed by Mario Carneiro, 28-Dec-2014.)
Assertion
Ref Expression
mpt0 (x ∅ ↦ A) = ∅

Proof of Theorem mpt0
StepHypRef Expression
1 ral0 3297 . . 3 x A V
2 eqid 2018 . . . 4 (x ∅ ↦ A) = (x ∅ ↦ A)
32fnmpt 4947 . . 3 (x A V → (x ∅ ↦ A) Fn ∅)
41, 3ax-mp 7 . 2 (x ∅ ↦ A) Fn ∅
5 fn0 4940 . 2 ((x ∅ ↦ A) Fn ∅ ↔ (x ∅ ↦ A) = ∅)
64, 5mpbi 133 1 (x ∅ ↦ A) = ∅
Colors of variables: wff set class
Syntax hints:   = wceq 1226   wcel 1370  wral 2280  Vcvv 2531  c0 3197  cmpt 3788   Fn wfn 4820
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-nul 3853  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904  df-nul 3198  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-mpt 3790  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-fun 4827  df-fn 4828
This theorem is referenced by:  fmptpr  5276
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