![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > elfir | Structured version Visualization version GIF version |
Description: Sufficient condition for an element of (fi‘𝐵). (Contributed by Mario Carneiro, 24-Nov-2013.) |
Ref | Expression |
---|---|
elfir | ⊢ ((𝐵 ∈ 𝑉 ∧ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ∩ 𝐴 ∈ (fi‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1133 | . . . . . 6 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin) → 𝐴 ⊆ 𝐵) | |
2 | elpw2g 5211 | . . . . . 6 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) | |
3 | 1, 2 | syl5ibr 249 | . . . . 5 ⊢ (𝐵 ∈ 𝑉 → ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin) → 𝐴 ∈ 𝒫 𝐵)) |
4 | 3 | imp 410 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → 𝐴 ∈ 𝒫 𝐵) |
5 | simpr3 1193 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → 𝐴 ∈ Fin) | |
6 | 4, 5 | elind 4121 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → 𝐴 ∈ (𝒫 𝐵 ∩ Fin)) |
7 | eqid 2798 | . . 3 ⊢ ∩ 𝐴 = ∩ 𝐴 | |
8 | inteq 4841 | . . . 4 ⊢ (𝑥 = 𝐴 → ∩ 𝑥 = ∩ 𝐴) | |
9 | 8 | rspceeqv 3586 | . . 3 ⊢ ((𝐴 ∈ (𝒫 𝐵 ∩ Fin) ∧ ∩ 𝐴 = ∩ 𝐴) → ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)∩ 𝐴 = ∩ 𝑥) |
10 | 6, 7, 9 | sylancl 589 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)∩ 𝐴 = ∩ 𝑥) |
11 | simp2 1134 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin) → 𝐴 ≠ ∅) | |
12 | intex 5204 | . . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∩ 𝐴 ∈ V) | |
13 | 11, 12 | sylib 221 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin) → ∩ 𝐴 ∈ V) |
14 | id 22 | . . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ 𝑉) | |
15 | elfi 8861 | . . 3 ⊢ ((∩ 𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (∩ 𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)∩ 𝐴 = ∩ 𝑥)) | |
16 | 13, 14, 15 | syl2anr 599 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → (∩ 𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)∩ 𝐴 = ∩ 𝑥)) |
17 | 10, 16 | mpbird 260 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ∩ 𝐴 ∈ (fi‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∃wrex 3107 Vcvv 3441 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 𝒫 cpw 4497 ∩ cint 4838 ‘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-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-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-int 4839 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-fi 8859 |
This theorem is referenced by: intrnfi 8864 ssfii 8867 elfiun 8878 ptbasfi 22186 fbssint 22443 filintn0 22466 alexsublem 22649 ispisys2 31522 |
Copyright terms: Public domain | W3C validator |