Theorem f002 48815

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 6650	. . 3 ⊢ (𝐵 = ∅ → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐴⟶∅))
3		f00 6724	. . . 4 ⊢ (𝐹:𝐴⟶∅ ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
4	3	simprbi 496	. . 3 ⊢ (𝐹:𝐴⟶∅ → 𝐴 = ∅)
5	2, 4	biimtrdi 253	. 2 ⊢ (𝐵 = ∅ → (𝐹:𝐴⟶𝐵 → 𝐴 = ∅))
6	1, 5	syl5com 31	1 ⊢ (𝜑 → (𝐵 = ∅ → 𝐴 = ∅))

Syntax hints:   → wi 4   = wceq 1540  ∅c0 4292  ⟶wf 6495

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-fun 6501  df-fn 6502  df-f 6503

This theorem is referenced by:  func0g  49051  functhincfun  49411  fullthinc2  49413  thincciso  49415