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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 3274 | . . . . 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 218 | . . . 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 38782 | . . . 4 ⊢ (𝑦 ∈ (∩ 𝐴 “ 𝐵) ↔ ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 〈𝑏, 𝑦〉 ∈ 𝑎) | |
5 | elintima 38787 | . . . 4 ⊢ (𝑦 ∈ ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)} ↔ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 〈𝑏, 𝑦〉 ∈ 𝑎) | |
6 | 3, 4, 5 | 3bitr4g 306 | . . 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 1912 | . 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 2820 | . 2 ⊢ ((∩ 𝐴 “ 𝐵) = ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)} ↔ ∀𝑦(𝑦 ∈ (∩ 𝐴 “ 𝐵) ↔ 𝑦 ∈ ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)})) | |
9 | 7, 8 | sylibr 226 | 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 198 ∀wal 1656 = wceq 1658 ∈ wcel 2166 {cab 2812 ∀wral 3118 ∃wrex 3119 〈cop 4404 ∩ cint 4698 “ cima 5346 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pr 5128 ax-un 7210 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-int 4699 df-br 4875 df-opab 4937 df-xp 5349 df-cnv 5351 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 |
This theorem is referenced by: intimasn 38791 |
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