Theorem fiuni 7106

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.)

Assertion

Ref	Expression
fiuni	⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 = ∪ (fi‘𝐴))

Proof of Theorem fiuni

Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssfii 7102	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (fi‘𝐴))
2	1	unissd 3888	. 2 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ⊆ ∪ (fi‘𝐴))
3		eluni 3867	. . . . 5 ⊢ (𝑥 ∈ ∪ (fi‘𝐴) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (fi‘𝐴)))
4	3	biimpi 120	. . . 4 ⊢ (𝑥 ∈ ∪ (fi‘𝐴) → ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (fi‘𝐴)))
5	4	adantl 277	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ∪ (fi‘𝐴)) → ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (fi‘𝐴)))
6		simprr 531	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ∪ (fi‘𝐴)) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (fi‘𝐴))) → 𝑦 ∈ (fi‘𝐴))
7		elfi2 7100	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ (fi‘𝐴) ↔ ∃𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅})𝑦 = ∩ 𝑧))
8	7	ad2antrr 488	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ∪ (fi‘𝐴)) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (fi‘𝐴))) → (𝑦 ∈ (fi‘𝐴) ↔ ∃𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅})𝑦 = ∩ 𝑧))
9	6, 8	mpbid 147	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ∪ (fi‘𝐴)) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (fi‘𝐴))) → ∃𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅})𝑦 = ∩ 𝑧)
10		simprr 531	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ∪ (fi‘𝐴)) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (fi‘𝐴))) ∧ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∧ 𝑦 = ∩ 𝑧)) → 𝑦 = ∩ 𝑧)
11		eldifi 3303	. . . . . . . . . 10 ⊢ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) → 𝑧 ∈ (𝒫 𝐴 ∩ Fin))
12	11	elin1d 3370	. . . . . . . . 9 ⊢ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) → 𝑧 ∈ 𝒫 𝐴)
13	12	elpwid 3637	. . . . . . . 8 ⊢ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) → 𝑧 ⊆ 𝐴)
14	13	ad2antrl 490	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ∪ (fi‘𝐴)) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (fi‘𝐴))) ∧ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∧ 𝑦 = ∩ 𝑧)) → 𝑧 ⊆ 𝐴)
15		eldifsni 3773	. . . . . . . . 9 ⊢ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) → 𝑧 ≠ ∅)
16	11	elin2d 3371	. . . . . . . . . 10 ⊢ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) → 𝑧 ∈ Fin)
17		fin0 7008	. . . . . . . . . 10 ⊢ (𝑧 ∈ Fin → (𝑧 ≠ ∅ ↔ ∃𝑤 𝑤 ∈ 𝑧))
18	16, 17	syl 14	. . . . . . . . 9 ⊢ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) → (𝑧 ≠ ∅ ↔ ∃𝑤 𝑤 ∈ 𝑧))
19	15, 18	mpbid 147	. . . . . . . 8 ⊢ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) → ∃𝑤 𝑤 ∈ 𝑧)
20	19	ad2antrl 490	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ∪ (fi‘𝐴)) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (fi‘𝐴))) ∧ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∧ 𝑦 = ∩ 𝑧)) → ∃𝑤 𝑤 ∈ 𝑧)
21		intssuni2m 3923	. . . . . . 7 ⊢ ((𝑧 ⊆ 𝐴 ∧ ∃𝑤 𝑤 ∈ 𝑧) → ∩ 𝑧 ⊆ ∪ 𝐴)
22	14, 20, 21	syl2anc 411	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ∪ (fi‘𝐴)) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (fi‘𝐴))) ∧ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∧ 𝑦 = ∩ 𝑧)) → ∩ 𝑧 ⊆ ∪ 𝐴)
23	10, 22	eqsstrd 3237	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ∪ (fi‘𝐴)) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (fi‘𝐴))) ∧ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∧ 𝑦 = ∩ 𝑧)) → 𝑦 ⊆ ∪ 𝐴)
24		simplrl 535	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ∪ (fi‘𝐴)) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (fi‘𝐴))) ∧ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∧ 𝑦 = ∩ 𝑧)) → 𝑥 ∈ 𝑦)
25	23, 24	sseldd 3202	. . . 4 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ∪ (fi‘𝐴)) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (fi‘𝐴))) ∧ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∧ 𝑦 = ∩ 𝑧)) → 𝑥 ∈ ∪ 𝐴)
26	9, 25	rexlimddv 2630	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ∪ (fi‘𝐴)) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (fi‘𝐴))) → 𝑥 ∈ ∪ 𝐴)
27	5, 26	exlimddv 1923	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ∪ (fi‘𝐴)) → 𝑥 ∈ ∪ 𝐴)
28	2, 27	eqelssd 3220	1 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 = ∪ (fi‘𝐴))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1373  ∃wex 1516   ∈ wcel 2178   ≠ wne 2378  ∃wrex 2487   ∖ cdif 3171   ∩ cin 3173   ⊆ wss 3174  ∅c0 3468  𝒫 cpw 3626  {csn 3643  ∪ cuni 3864  ∩ cint 3899  ‘cfv 5290  Fincfn 6850  ficfi 7096

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-iinf 4654

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-1o 6525  df-er 6643  df-en 6851  df-fin 6853  df-fi 7097

This theorem is referenced by:  fipwssg  7107