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Mirrors > Home > MPE Home > Th. List > intrnfi | Structured version Visualization version GIF version |
Description: Sufficient condition for the intersection of the range of a function to be in the set of finite intersections. (Contributed by Mario Carneiro, 30-Aug-2015.) |
Ref | Expression |
---|---|
intrnfi | ⊢ ((𝐵 ∈ 𝑉 ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ∩ ran 𝐹 ∈ (fi‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr1 1191 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → 𝐹:𝐴⟶𝐵) | |
2 | 1 | frnd 6494 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ran 𝐹 ⊆ 𝐵) |
3 | 1 | fdmd 6497 | . . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → dom 𝐹 = 𝐴) |
4 | simpr2 1192 | . . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → 𝐴 ≠ ∅) | |
5 | 3, 4 | eqnetrd 3054 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → dom 𝐹 ≠ ∅) |
6 | dm0rn0 5759 | . . . . 5 ⊢ (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅) | |
7 | 6 | necon3bii 3039 | . . . 4 ⊢ (dom 𝐹 ≠ ∅ ↔ ran 𝐹 ≠ ∅) |
8 | 5, 7 | sylib 221 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ran 𝐹 ≠ ∅) |
9 | simpr3 1193 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → 𝐴 ∈ Fin) | |
10 | 1 | ffnd 6488 | . . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → 𝐹 Fn 𝐴) |
11 | dffn4 6571 | . . . . 5 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴–onto→ran 𝐹) | |
12 | 10, 11 | sylib 221 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → 𝐹:𝐴–onto→ran 𝐹) |
13 | fofi 8794 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–onto→ran 𝐹) → ran 𝐹 ∈ Fin) | |
14 | 9, 12, 13 | syl2anc 587 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ran 𝐹 ∈ Fin) |
15 | 2, 8, 14 | 3jca 1125 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → (ran 𝐹 ⊆ 𝐵 ∧ ran 𝐹 ≠ ∅ ∧ ran 𝐹 ∈ Fin)) |
16 | elfir 8863 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ (ran 𝐹 ⊆ 𝐵 ∧ ran 𝐹 ≠ ∅ ∧ ran 𝐹 ∈ Fin)) → ∩ ran 𝐹 ∈ (fi‘𝐵)) | |
17 | 15, 16 | syldan 594 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ∩ ran 𝐹 ∈ (fi‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 ∈ wcel 2111 ≠ wne 2987 ⊆ wss 3881 ∅c0 4243 ∩ cint 4838 dom cdm 5519 ran crn 5520 Fn wfn 6319 ⟶wf 6320 –onto→wfo 6322 ‘cfv 6324 Fincfn 8492 ficfi 8858 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-om 7561 df-1o 8085 df-er 8272 df-en 8493 df-dom 8494 df-fin 8496 df-fi 8859 |
This theorem is referenced by: iinfi 8865 firest 16698 |
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