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Mirrors > Home > MPE Home > Th. List > fiuni | Structured version Visualization version GIF version |
Description: The union of the finite intersections of a set is simply the union of the set itself. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Mario Carneiro, 24-Nov-2013.) |
Ref | Expression |
---|---|
fiuni | ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 = ∪ (fi‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssfii 8867 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (fi‘𝐴)) | |
2 | 1 | unissd 4810 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ⊆ ∪ (fi‘𝐴)) |
3 | fipwuni 8874 | . . . . 5 ⊢ (fi‘𝐴) ⊆ 𝒫 ∪ 𝐴 | |
4 | 3 | unissi 4809 | . . . 4 ⊢ ∪ (fi‘𝐴) ⊆ ∪ 𝒫 ∪ 𝐴 |
5 | unipw 5308 | . . . 4 ⊢ ∪ 𝒫 ∪ 𝐴 = ∪ 𝐴 | |
6 | 4, 5 | sseqtri 3951 | . . 3 ⊢ ∪ (fi‘𝐴) ⊆ ∪ 𝐴 |
7 | 6 | a1i 11 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∪ (fi‘𝐴) ⊆ ∪ 𝐴) |
8 | 2, 7 | eqssd 3932 | 1 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 = ∪ (fi‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 𝒫 cpw 4497 ∪ cuni 4800 ‘cfv 6324 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-csb 3829 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-iun 4883 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-pred 6116 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-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-fin 8496 df-fi 8859 |
This theorem is referenced by: fipwss 8877 ordttopon 21798 ptbasfi 22186 xkouni 22204 alexsublem 22649 alexsub 22650 alexsubb 22651 alexsubALTlem3 22654 alexsubALTlem4 22655 ptcmplem1 22657 topjoin 33826 |
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