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Theorem f10d 6082
 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 6081 . . 3 ∅:∅–1-1𝐴
2 dm0 5260 . . . 4 dom ∅ = ∅
3 f1eq2 6010 . . . 4 (dom ∅ = ∅ → (∅:dom ∅–1-1𝐴 ↔ ∅:∅–1-1𝐴))
42, 3ax-mp 5 . . 3 (∅:dom ∅–1-1𝐴 ↔ ∅:∅–1-1𝐴)
51, 4mpbir 220 . 2 ∅:dom ∅–1-1𝐴
6 f10d.f . . 3 (𝜑𝐹 = ∅)
76dmeqd 5248 . . 3 (𝜑 → dom 𝐹 = dom ∅)
8 eqidd 2611 . . 3 (𝜑𝐴 = 𝐴)
96, 7, 8f1eq123d 6044 . 2 (𝜑 → (𝐹:dom 𝐹1-1𝐴 ↔ ∅:dom ∅–1-1𝐴))
105, 9mpbiri 247 1 (𝜑𝐹:dom 𝐹1-1𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475  ∅c0 3874  dom cdm 5038  –1-1→wf1 5801 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809 This theorem is referenced by:  umgr0e  25776  usgra0  25899  usgr0e  40462
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