Theorem imadisjld 44520

Description: Natural dduction form of one side of imadisj 6047. (Contributed by Stanislas Polu, 9-Mar-2020.)

Hypothesis

Ref	Expression
imadisjld.1	⊢ (𝜑 → (dom 𝐴 ∩ 𝐵) = ∅)

Assertion

Ref	Expression
imadisjld	⊢ (𝜑 → (𝐴 “ 𝐵) = ∅)

Proof of Theorem imadisjld

Step	Hyp	Ref	Expression
1		imadisjld.1	. 2 ⊢ (𝜑 → (dom 𝐴 ∩ 𝐵) = ∅)
2		imadisj 6047	. 2 ⊢ ((𝐴 “ 𝐵) = ∅ ↔ (dom 𝐴 ∩ 𝐵) = ∅)
3	1, 2	sylibr 234	1 ⊢ (𝜑 → (𝐴 “ 𝐵) = ∅)

Syntax hints:   → wi 4   = wceq 1542   ∩ cin 3902  ∅c0 4287  dom cdm 5632   “ cima 5635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5638  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645

This theorem is referenced by: (None)