Theorem f002 49632

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 6685	. . 3 ⊢ (𝐵 = ∅ → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐴⟶∅))
3		f00 6760	. . . 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 1570  ∅c0 4286  ⟶wf 6532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-fun 6538  df-fn 6539  df-f 6540

This theorem is referenced by:  func0g  49867  functhincfun  50227  fullthinc2  50229  thincciso  50231