Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > fipwss | Structured version Visualization version GIF version |
Description: If a set is a family of subsets of some base set, then so is its finite intersection. (Contributed by Stefan O'Rear, 2-Aug-2015.) |
Ref | Expression |
---|---|
fipwss | ⊢ (𝐴 ⊆ 𝒫 𝑋 → (fi‘𝐴) ⊆ 𝒫 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fiuni 8892 | . . . . 5 ⊢ (𝐴 ∈ V → ∪ 𝐴 = ∪ (fi‘𝐴)) | |
2 | 1 | sseq1d 3998 | . . . 4 ⊢ (𝐴 ∈ V → (∪ 𝐴 ⊆ 𝑋 ↔ ∪ (fi‘𝐴) ⊆ 𝑋)) |
3 | sspwuni 5022 | . . . 4 ⊢ (𝐴 ⊆ 𝒫 𝑋 ↔ ∪ 𝐴 ⊆ 𝑋) | |
4 | sspwuni 5022 | . . . 4 ⊢ ((fi‘𝐴) ⊆ 𝒫 𝑋 ↔ ∪ (fi‘𝐴) ⊆ 𝑋) | |
5 | 2, 3, 4 | 3bitr4g 316 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ⊆ 𝒫 𝑋 ↔ (fi‘𝐴) ⊆ 𝒫 𝑋)) |
6 | 5 | biimpa 479 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐴 ⊆ 𝒫 𝑋) → (fi‘𝐴) ⊆ 𝒫 𝑋) |
7 | fvprc 6663 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (fi‘𝐴) = ∅) | |
8 | 0ss 4350 | . . . 4 ⊢ ∅ ⊆ 𝒫 𝑋 | |
9 | 7, 8 | eqsstrdi 4021 | . . 3 ⊢ (¬ 𝐴 ∈ V → (fi‘𝐴) ⊆ 𝒫 𝑋) |
10 | 9 | adantr 483 | . 2 ⊢ ((¬ 𝐴 ∈ V ∧ 𝐴 ⊆ 𝒫 𝑋) → (fi‘𝐴) ⊆ 𝒫 𝑋) |
11 | 6, 10 | pm2.61ian 810 | 1 ⊢ (𝐴 ⊆ 𝒫 𝑋 → (fi‘𝐴) ⊆ 𝒫 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2114 Vcvv 3494 ⊆ wss 3936 ∅c0 4291 𝒫 cpw 4539 ∪ cuni 4838 ‘cfv 6355 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-csb 3884 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-iun 4921 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-pred 6148 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-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-fin 8513 df-fi 8875 |
This theorem is referenced by: fsubbas 22475 |
Copyright terms: Public domain | W3C validator |