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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 8871 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (fi‘𝐴)) | |
2 | 1 | unissd 4854 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ⊆ ∪ (fi‘𝐴)) |
3 | fipwuni 8878 | . . . . 5 ⊢ (fi‘𝐴) ⊆ 𝒫 ∪ 𝐴 | |
4 | 3 | unissi 4853 | . . . 4 ⊢ ∪ (fi‘𝐴) ⊆ ∪ 𝒫 ∪ 𝐴 |
5 | unipw 5333 | . . . 4 ⊢ ∪ 𝒫 ∪ 𝐴 = ∪ 𝐴 | |
6 | 4, 5 | sseqtri 4000 | . . 3 ⊢ ∪ (fi‘𝐴) ⊆ ∪ 𝐴 |
7 | 6 | a1i 11 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∪ (fi‘𝐴) ⊆ ∪ 𝐴) |
8 | 2, 7 | eqssd 3981 | 1 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 = ∪ (fi‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 𝒫 cpw 4535 ∪ cuni 4830 ‘cfv 6348 ficfi 8862 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-fin 8501 df-fi 8863 |
This theorem is referenced by: fipwss 8881 ordttopon 21729 ptbasfi 22117 xkouni 22135 alexsublem 22580 alexsub 22581 alexsubb 22582 alexsubALTlem3 22585 alexsubALTlem4 22586 ptcmplem1 22588 topjoin 33610 |
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