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Mirrors > Home > MPE Home > Th. List > iinfi | Structured version Visualization version GIF version |
Description: An indexed intersection of elements of 𝐶 is an element of the finite intersections of 𝐶. (Contributed by Mario Carneiro, 30-Aug-2015.) |
Ref | Expression |
---|---|
iinfi | ⊢ ((𝐶 ∈ 𝑉 ∧ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ (fi‘𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr1 1190 | . . . 4 ⊢ ((𝐶 ∈ 𝑉 ∧ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) | |
2 | dfiin2g 4957 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) |
4 | eqid 2821 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
5 | 4 | rnmpt 5827 | . . . 4 ⊢ ran (𝑥 ∈ 𝐴 ↦ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} |
6 | 5 | inteqi 4880 | . . 3 ⊢ ∩ ran (𝑥 ∈ 𝐴 ↦ 𝐵) = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} |
7 | 3, 6 | syl6eqr 2874 | . 2 ⊢ ((𝐶 ∈ 𝑉 ∧ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) |
8 | 4 | fmpt 6874 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶) |
9 | 8 | 3anbi1i 1153 | . . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin) ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) |
10 | intrnfi 8880 | . . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ ((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ∩ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (fi‘𝐶)) | |
11 | 9, 10 | sylan2b 595 | . 2 ⊢ ((𝐶 ∈ 𝑉 ∧ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ∩ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (fi‘𝐶)) |
12 | 7, 11 | eqeltrd 2913 | 1 ⊢ ((𝐶 ∈ 𝑉 ∧ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ (fi‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 {cab 2799 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 ∅c0 4291 ∩ cint 4876 ∩ ciin 4920 ↦ cmpt 5146 ran crn 5556 ⟶wf 6351 ‘cfv 6355 Fincfn 8509 ficfi 8874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-om 7581 df-1o 8102 df-er 8289 df-en 8510 df-dom 8511 df-fin 8513 df-fi 8875 |
This theorem is referenced by: firest 16706 iscmet3 23896 sigapildsyslem 31420 |
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