Theorem f10d 6802

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–1-1→𝐴)

Proof of Theorem f10d

Step	Hyp	Ref	Expression
1		f10 6801	. . 3 ⊢ ∅:∅–1-1→𝐴
2		dm0 5864	. . . 4 ⊢ dom ∅ = ∅
3		f1eq2 6720	. . . 4 ⊢ (dom ∅ = ∅ → (∅:dom ∅–1-1→𝐴 ↔ ∅:∅–1-1→𝐴))
4	2, 3	ax-mp 5	. . 3 ⊢ (∅:dom ∅–1-1→𝐴 ↔ ∅:∅–1-1→𝐴)
5	1, 4	mpbir 231	. 2 ⊢ ∅:dom ∅–1-1→𝐴
6		f10d.f	. . 3 ⊢ (𝜑 → 𝐹 = ∅)
7	6	dmeqd 5849	. . 3 ⊢ (𝜑 → dom 𝐹 = dom ∅)
8		eqidd 2734	. . 3 ⊢ (𝜑 → 𝐴 = 𝐴)
9	6, 7, 8	f1eq123d 6760	. 2 ⊢ (𝜑 → (𝐹:dom 𝐹–1-1→𝐴 ↔ ∅:dom ∅–1-1→𝐴))
10	5, 9	mpbiri 258	1 ⊢ (𝜑 → 𝐹:dom 𝐹–1-1→𝐴)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541  ∅c0 4282  dom cdm 5619  –1-1→wf1 6483

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2537  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-br 5094  df-opab 5156  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491

This theorem is referenced by:  umgr0e  29090  usgr0e  29216