Theorem imadisjld 44619

Description: Natural dduction form of one side of imadisj 6039. (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 6039	. 2 ⊢ ((𝐴 “ 𝐵) = ∅ ↔ (dom 𝐴 ∩ 𝐵) = ∅)
3	1, 2	sylibr 236	1 ⊢ (𝜑 → (𝐴 “ 𝐵) = ∅)

Syntax hints:   → wi 4   = wceq 1548   ∩ cin 3884  ∅c0 4264  dom cdm 5621   “ cima 5624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-opab 5138  df-xp 5627  df-cnv 5629  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634

This theorem is referenced by: (None)