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| Mirrors > Home > MPE Home > Th. List > Mathboxes > intimag | Structured version Visualization version GIF version | ||
| Description: Requirement for the image under the intersection of relations to equal the intersection of the images of those relations. (Contributed by RP, 13-Apr-2020.) |
| Ref | Expression |
|---|---|
| intimag | ⊢ (∀𝑦(∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 〈𝑏, 𝑦〉 ∈ 𝑎 → ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 〈𝑏, 𝑦〉 ∈ 𝑎) → (∩ 𝐴 “ 𝐵) = ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | r19.12 3314 | . . . . 5 ⊢ (∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 〈𝑏, 𝑦〉 ∈ 𝑎 → ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 〈𝑏, 𝑦〉 ∈ 𝑎) | |
| 2 | id 22 | . . . . 5 ⊢ ((∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 〈𝑏, 𝑦〉 ∈ 𝑎 → ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 〈𝑏, 𝑦〉 ∈ 𝑎) → (∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 〈𝑏, 𝑦〉 ∈ 𝑎 → ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 〈𝑏, 𝑦〉 ∈ 𝑎)) | |
| 3 | 1, 2 | impbid2 226 | . . . 4 ⊢ ((∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 〈𝑏, 𝑦〉 ∈ 𝑎 → ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 〈𝑏, 𝑦〉 ∈ 𝑎) → (∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 〈𝑏, 𝑦〉 ∈ 𝑎 ↔ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 〈𝑏, 𝑦〉 ∈ 𝑎)) |
| 4 | elimaint 43662 | . . . 4 ⊢ (𝑦 ∈ (∩ 𝐴 “ 𝐵) ↔ ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 〈𝑏, 𝑦〉 ∈ 𝑎) | |
| 5 | elintima 43666 | . . . 4 ⊢ (𝑦 ∈ ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)} ↔ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 〈𝑏, 𝑦〉 ∈ 𝑎) | |
| 6 | 3, 4, 5 | 3bitr4g 314 | . . 3 ⊢ ((∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 〈𝑏, 𝑦〉 ∈ 𝑎 → ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 〈𝑏, 𝑦〉 ∈ 𝑎) → (𝑦 ∈ (∩ 𝐴 “ 𝐵) ↔ 𝑦 ∈ ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)})) |
| 7 | 6 | alimi 1811 | . 2 ⊢ (∀𝑦(∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 〈𝑏, 𝑦〉 ∈ 𝑎 → ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 〈𝑏, 𝑦〉 ∈ 𝑎) → ∀𝑦(𝑦 ∈ (∩ 𝐴 “ 𝐵) ↔ 𝑦 ∈ ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)})) |
| 8 | dfcleq 2730 | . 2 ⊢ ((∩ 𝐴 “ 𝐵) = ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)} ↔ ∀𝑦(𝑦 ∈ (∩ 𝐴 “ 𝐵) ↔ 𝑦 ∈ ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)})) | |
| 9 | 7, 8 | sylibr 234 | 1 ⊢ (∀𝑦(∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 〈𝑏, 𝑦〉 ∈ 𝑎 → ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 〈𝑏, 𝑦〉 ∈ 𝑎) → (∩ 𝐴 “ 𝐵) = ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 = wceq 1540 ∈ wcel 2108 {cab 2714 ∀wral 3061 ∃wrex 3070 〈cop 4632 ∩ cint 4946 “ cima 5688 |
| 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 ax-un 7755 |
| 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-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-uni 4908 df-int 4947 df-br 5144 df-opab 5206 df-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 |
| This theorem is referenced by: intimasn 43670 |
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