Theorem f002 47609

Description: A function with an empty codomain must have empty domain. (Contributed by Zhi Wang, 1-Oct-2024.)

Hypothesis

Ref	Expression
f002.1	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)

Assertion

Ref	Expression
f002	⊢ (𝜑 → (𝐵 = ∅ → 𝐴 = ∅))

Proof of Theorem f002

Step	Hyp	Ref	Expression
1		f002.1	. 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2		feq3 6701	. . 3 ⊢ (𝐵 = ∅ → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐴⟶∅))
3		f00 6774	. . . 4 ⊢ (𝐹:𝐴⟶∅ ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
4	3	simprbi 495	. . 3 ⊢ (𝐹:𝐴⟶∅ → 𝐴 = ∅)
5	2, 4	syl6bi 252	. 2 ⊢ (𝐵 = ∅ → (𝐹:𝐴⟶𝐵 → 𝐴 = ∅))
6	1, 5	syl5com 31	1 ⊢ (𝜑 → (𝐵 = ∅ → 𝐴 = ∅))

Syntax hints:   → wi 4   = wceq 1539  ∅c0 4323  ⟶wf 6540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-fun 6546  df-fn 6547  df-f 6548

This theorem is referenced by:  fullthinc2  47756  thincciso  47758