Theorem f10d 6885

Description: The empty set maps one-to-one into any class, deduction version. (Contributed by AV, 25-Nov-2020.)

Hypothesis

Ref	Expression
f10d.f	⊢ (𝜑 → 𝐹 = ∅)

Assertion

Ref	Expression
f10d	⊢ (𝜑 → 𝐹:dom 𝐹⟶1-1𝐴)

Proof of Theorem f10d

Step	Hyp	Ref	Expression
1		f10 6884	. . 3 ⊢ ∅:∅⟶1-1𝐴
2		dm0 5910	. . . 4 ⊢ dom ∅ = ∅
3		f1eq2 6799	. . . 4 ⊢ (dom ∅ = ∅ → (∅:dom ∅⟶1-1𝐴 ↔ ∅:∅⟶1-1𝐴))
4	2, 3	ax-mp 5	. . 3 ⊢ (∅:dom ∅⟶1-1𝐴 ↔ ∅:∅⟶1-1𝐴)
5	1, 4	mpbir 234	. 2 ⊢ ∅:dom ∅⟶1-1𝐴
6		f10d.f	. . 3 ⊢ (𝜑 → 𝐹 = ∅)
7	6	dmeqd 5895	. . 3 ⊢ (𝜑 → dom 𝐹 = dom ∅)
8		eqidd 2764	. . 3 ⊢ (𝜑 → 𝐴 = 𝐴)
9	6, 7, 8	f1eq123d 6841	. 2 ⊢ (𝜑 → (𝐹:dom 𝐹⟶1-1𝐴 ↔ ∅:dom ∅⟶1-1𝐴))
10	5, 9	mpbiri 261	1 ⊢ (𝜑 → 𝐹:dom 𝐹⟶1-1𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  ∅c0 4281  dom cdm 5656  ⟶1-1wf1 6558

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5172  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566

This theorem is used by:  umgr0e  29721  usgr0e  29850