Theorem f002 49512

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 6683	. . 3 ⊢ (𝐵 = ∅ → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐴⟶∅))
3		f00 6758	. . . 4 ⊢ (𝐹:𝐴⟶∅ ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
4	3	simprbi 502	. . 3 ⊢ (𝐹:𝐴⟶∅ → 𝐴 = ∅)
5	2, 4	biimtrdi 256	. 2 ⊢ (𝐵 = ∅ → (𝐹:𝐴⟶𝐵 → 𝐴 = ∅))
6	1, 5	syl5com 32	1 ⊢ (𝜑 → (𝐵 = ∅ → 𝐴 = ∅))

Syntax hints:   → wi 4   = wceq 1567  ∅c0 4294  ⟶wf 6530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-mo 2573  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5175  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-fun 6536  df-fn 6537  df-f 6538

This theorem is referenced by:  func0g  49747  functhincfun  50107  fullthinc2  50109  thincciso  50111