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Theorem predisj 48730
Description: Preimages of disjoint sets are disjoint. (Contributed by Zhi Wang, 9-Sep-2024.)
Hypotheses
Ref Expression
predisj.1 (𝜑 → Fun 𝐹)
predisj.2 (𝜑 → (𝐴𝐵) = ∅)
predisj.3 (𝜑𝑆 ⊆ (𝐹𝐴))
predisj.4 (𝜑𝑇 ⊆ (𝐹𝐵))
Assertion
Ref Expression
predisj (𝜑 → (𝑆𝑇) = ∅)

Proof of Theorem predisj
StepHypRef Expression
1 predisj.4 . 2 (𝜑𝑇 ⊆ (𝐹𝐵))
2 predisj.3 . . 3 (𝜑𝑆 ⊆ (𝐹𝐴))
3 predisj.1 . . . . 5 (𝜑 → Fun 𝐹)
4 inpreima 7084 . . . . 5 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∩ (𝐹𝐵)))
53, 4syl 17 . . . 4 (𝜑 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∩ (𝐹𝐵)))
6 predisj.2 . . . . . 6 (𝜑 → (𝐴𝐵) = ∅)
76imaeq2d 6078 . . . . 5 (𝜑 → (𝐹 “ (𝐴𝐵)) = (𝐹 “ ∅))
8 ima0 6095 . . . . 5 (𝐹 “ ∅) = ∅
97, 8eqtrdi 2793 . . . 4 (𝜑 → (𝐹 “ (𝐴𝐵)) = ∅)
105, 9eqtr3d 2779 . . 3 (𝜑 → ((𝐹𝐴) ∩ (𝐹𝐵)) = ∅)
112, 10ssdisjd 48727 . 2 (𝜑 → (𝑆 ∩ (𝐹𝐵)) = ∅)
121, 11ssdisjdr 48728 1 (𝜑 → (𝑆𝑇) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  cin 3950  wss 3951  c0 4333  ccnv 5684  cima 5688  Fun wfun 6555
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563
This theorem is referenced by:  sepfsepc  48825
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