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Theorem f10d 6312
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 6311 . . 3 ∅:∅–1-1𝐴
2 dm0 5476 . . . 4 dom ∅ = ∅
3 f1eq2 6238 . . . 4 (dom ∅ = ∅ → (∅:dom ∅–1-1𝐴 ↔ ∅:∅–1-1𝐴))
42, 3ax-mp 5 . . 3 (∅:dom ∅–1-1𝐴 ↔ ∅:∅–1-1𝐴)
51, 4mpbir 221 . 2 ∅:dom ∅–1-1𝐴
6 f10d.f . . 3 (𝜑𝐹 = ∅)
76dmeqd 5463 . . 3 (𝜑 → dom 𝐹 = dom ∅)
8 eqidd 2772 . . 3 (𝜑𝐴 = 𝐴)
96, 7, 8f1eq123d 6273 . 2 (𝜑 → (𝐹:dom 𝐹1-1𝐴 ↔ ∅:dom ∅–1-1𝐴))
105, 9mpbiri 248 1 (𝜑𝐹:dom 𝐹1-1𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1631  c0 4063  dom cdm 5250  1-1wf1 6027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035
This theorem is referenced by:  umgr0e  26226  usgr0e  26351
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