Theorem imadisjld 41236

Description: Natural dduction form of one side of imadisj 5920. (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 5920	. 2 ⊢ ((𝐴 “ 𝐵) = ∅ ↔ (dom 𝐴 ∩ 𝐵) = ∅)
3	1, 2	sylibr 237	1 ⊢ (𝜑 → (𝐴 “ 𝐵) = ∅)

Syntax hints:   → wi 4   = wceq 1538   ∩ cin 3857  ∅c0 4225  dom cdm 5524   “ cima 5527

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-dif 3861  df-un 3863  df-in 3865  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-br 5033  df-opab 5095  df-xp 5530  df-cnv 5532  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537

This theorem is referenced by: (None)