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Theorem f10d 6137
 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 6136 . . 3 ∅:∅–1-1𝐴
2 dm0 5309 . . . 4 dom ∅ = ∅
3 f1eq2 6064 . . . 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 5296 . . 3 (𝜑 → dom 𝐹 = dom ∅)
8 eqidd 2622 . . 3 (𝜑𝐴 = 𝐴)
96, 7, 8f1eq123d 6098 . 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 1480  ∅c0 3897  dom cdm 5084  –1-1→wf1 5854 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862 This theorem is referenced by:  umgr0e  25934  usgr0e  26055
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