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Theorem f10d 6627
Description: The empty set maps one-to-one into any class, deduction version. (Contributed by AV, 25-Nov-2020.)
Hypothesis
Ref Expression
f10d.f (𝜑𝐹 = ∅)
Assertion
Ref Expression
f10d (𝜑𝐹:dom 𝐹1-1𝐴)

Proof of Theorem f10d
StepHypRef Expression
1 f10 6626 . . 3 ∅:∅–1-1𝐴
2 dm0 5758 . . . 4 dom ∅ = ∅
3 f1eq2 6549 . . . 4 (dom ∅ = ∅ → (∅:dom ∅–1-1𝐴 ↔ ∅:∅–1-1𝐴))
42, 3ax-mp 5 . . 3 (∅:dom ∅–1-1𝐴 ↔ ∅:∅–1-1𝐴)
51, 4mpbir 234 . 2 ∅:dom ∅–1-1𝐴
6 f10d.f . . 3 (𝜑𝐹 = ∅)
76dmeqd 5742 . . 3 (𝜑 → dom 𝐹 = dom ∅)
8 eqidd 2802 . . 3 (𝜑𝐴 = 𝐴)
96, 7, 8f1eq123d 6587 . 2 (𝜑 → (𝐹:dom 𝐹1-1𝐴 ↔ ∅:dom ∅–1-1𝐴))
105, 9mpbiri 261 1 (𝜑𝐹:dom 𝐹1-1𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  c0 4246  dom cdm 5523  1-1wf1 6325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333
This theorem is referenced by:  umgr0e  26907  usgr0e  27030
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