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Theorem f10d 6857
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 6856 . . 3 ∅:∅–1-1𝐴
2 dm0 5910 . . . 4 dom ∅ = ∅
3 f1eq2 6773 . . . 4 (dom ∅ = ∅ → (∅:dom ∅–1-1𝐴 ↔ ∅:∅–1-1𝐴))
42, 3ax-mp 5 . . 3 (∅:dom ∅–1-1𝐴 ↔ ∅:∅–1-1𝐴)
51, 4mpbir 230 . 2 ∅:dom ∅–1-1𝐴
6 f10d.f . . 3 (𝜑𝐹 = ∅)
76dmeqd 5895 . . 3 (𝜑 → dom 𝐹 = dom ∅)
8 eqidd 2725 . . 3 (𝜑𝐴 = 𝐴)
96, 7, 8f1eq123d 6815 . 2 (𝜑 → (𝐹:dom 𝐹1-1𝐴 ↔ ∅:dom ∅–1-1𝐴))
105, 9mpbiri 258 1 (𝜑𝐹:dom 𝐹1-1𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  c0 4314  dom cdm 5666  1-1wf1 6530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538
This theorem is referenced by:  umgr0e  28805  usgr0e  28928
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