![]() |
Mathbox for Zhi Wang |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > predisj | Structured version Visualization version GIF version |
Description: Preimages of disjoint sets are disjoint. (Contributed by Zhi Wang, 9-Sep-2024.) |
Ref | Expression |
---|---|
predisj.1 | ⊢ (𝜑 → Fun 𝐹) |
predisj.2 | ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) |
predisj.3 | ⊢ (𝜑 → 𝑆 ⊆ (◡𝐹 “ 𝐴)) |
predisj.4 | ⊢ (𝜑 → 𝑇 ⊆ (◡𝐹 “ 𝐵)) |
Ref | Expression |
---|---|
predisj | ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | predisj.4 | . 2 ⊢ (𝜑 → 𝑇 ⊆ (◡𝐹 “ 𝐵)) | |
2 | predisj.3 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ (◡𝐹 “ 𝐴)) | |
3 | predisj.1 | . . . . 5 ⊢ (𝜑 → Fun 𝐹) | |
4 | inpreima 7084 | . . . . 5 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∩ 𝐵)) = ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ 𝐵))) | |
5 | 3, 4 | syl 17 | . . . 4 ⊢ (𝜑 → (◡𝐹 “ (𝐴 ∩ 𝐵)) = ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ 𝐵))) |
6 | predisj.2 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) | |
7 | 6 | imaeq2d 6080 | . . . . 5 ⊢ (𝜑 → (◡𝐹 “ (𝐴 ∩ 𝐵)) = (◡𝐹 “ ∅)) |
8 | ima0 6097 | . . . . 5 ⊢ (◡𝐹 “ ∅) = ∅ | |
9 | 7, 8 | eqtrdi 2791 | . . . 4 ⊢ (𝜑 → (◡𝐹 “ (𝐴 ∩ 𝐵)) = ∅) |
10 | 5, 9 | eqtr3d 2777 | . . 3 ⊢ (𝜑 → ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ 𝐵)) = ∅) |
11 | 2, 10 | ssdisjd 48656 | . 2 ⊢ (𝜑 → (𝑆 ∩ (◡𝐹 “ 𝐵)) = ∅) |
12 | 1, 11 | ssdisjdr 48657 | 1 ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∩ cin 3962 ⊆ wss 3963 ∅c0 4339 ◡ccnv 5688 “ cima 5692 Fun wfun 6557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-fun 6565 |
This theorem is referenced by: sepfsepc 48724 |
Copyright terms: Public domain | W3C validator |