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Theorem fipwuni 8866
 Description: The set of finite intersections of a set is contained in the powerset of the union of the elements of 𝐴. (Contributed by Mario Carneiro, 24-Nov-2013.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
fipwuni (fi‘𝐴) ⊆ 𝒫 𝐴

Proof of Theorem fipwuni
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uniexg 7441 . . . . 5 (𝐴 ∈ V → 𝐴 ∈ V)
21pwexd 5253 . . . 4 (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
3 pwuni 4848 . . . 4 𝐴 ⊆ 𝒫 𝐴
4 fiss 8864 . . . 4 ((𝒫 𝐴 ∈ V ∧ 𝐴 ⊆ 𝒫 𝐴) → (fi‘𝐴) ⊆ (fi‘𝒫 𝐴))
52, 3, 4sylancl 589 . . 3 (𝐴 ∈ V → (fi‘𝐴) ⊆ (fi‘𝒫 𝐴))
6 ssinss1 4189 . . . . . . 7 (𝑥 𝐴 → (𝑥𝑦) ⊆ 𝐴)
7 vex 3474 . . . . . . . 8 𝑥 ∈ V
87elpw 4516 . . . . . . 7 (𝑥 ∈ 𝒫 𝐴𝑥 𝐴)
97inex1 5194 . . . . . . . 8 (𝑥𝑦) ∈ V
109elpw 4516 . . . . . . 7 ((𝑥𝑦) ∈ 𝒫 𝐴 ↔ (𝑥𝑦) ⊆ 𝐴)
116, 8, 103imtr4i 295 . . . . . 6 (𝑥 ∈ 𝒫 𝐴 → (𝑥𝑦) ∈ 𝒫 𝐴)
1211adantr 484 . . . . 5 ((𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴) → (𝑥𝑦) ∈ 𝒫 𝐴)
1312rgen2 3191 . . . 4 𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴(𝑥𝑦) ∈ 𝒫 𝐴
14 inficl 8865 . . . . 5 (𝒫 𝐴 ∈ V → (∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴(𝑥𝑦) ∈ 𝒫 𝐴 ↔ (fi‘𝒫 𝐴) = 𝒫 𝐴))
152, 14syl 17 . . . 4 (𝐴 ∈ V → (∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴(𝑥𝑦) ∈ 𝒫 𝐴 ↔ (fi‘𝒫 𝐴) = 𝒫 𝐴))
1613, 15mpbii 236 . . 3 (𝐴 ∈ V → (fi‘𝒫 𝐴) = 𝒫 𝐴)
175, 16sseqtrd 3983 . 2 (𝐴 ∈ V → (fi‘𝐴) ⊆ 𝒫 𝐴)
18 fvprc 6636 . . 3 𝐴 ∈ V → (fi‘𝐴) = ∅)
19 0ss 4323 . . 3 ∅ ⊆ 𝒫 𝐴
2018, 19eqsstrdi 3997 . 2 𝐴 ∈ V → (fi‘𝐴) ⊆ 𝒫 𝐴)
2117, 20pm2.61i 185 1 (fi‘𝐴) ⊆ 𝒫 𝐴
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209   = wceq 1538   ∈ wcel 2115  ∀wral 3126  Vcvv 3471   ∩ cin 3909   ⊆ wss 3910  ∅c0 4266  𝒫 cpw 4512  ∪ cuni 4811  ‘cfv 6328  ficfi 8850 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436 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 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-1o 8077  df-oadd 8081  df-er 8264  df-en 8485  df-fin 8488  df-fi 8851 This theorem is referenced by:  fiuni  8868  ordtbas  21776
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