Theorem fiuni 6943

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 6939	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (fi‘𝐴))
2	1	unissd 3813	. 2 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ⊆ ∪ (fi‘𝐴))
3		eluni 3792	. . . . 5 ⊢ (𝑥 ∈ ∪ (fi‘𝐴) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (fi‘𝐴)))
4	3	biimpi 119	. . . 4 ⊢ (𝑥 ∈ ∪ (fi‘𝐴) → ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (fi‘𝐴)))
5	4	adantl 275	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ∪ (fi‘𝐴)) → ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (fi‘𝐴)))
6		simprr 522	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ∪ (fi‘𝐴)) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (fi‘𝐴))) → 𝑦 ∈ (fi‘𝐴))
7		elfi2 6937	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ (fi‘𝐴) ↔ ∃𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅})𝑦 = ∩ 𝑧))
8	7	ad2antrr 480	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ∪ (fi‘𝐴)) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (fi‘𝐴))) → (𝑦 ∈ (fi‘𝐴) ↔ ∃𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅})𝑦 = ∩ 𝑧))
9	6, 8	mpbid 146	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ∪ (fi‘𝐴)) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (fi‘𝐴))) → ∃𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅})𝑦 = ∩ 𝑧)
10		simprr 522	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ∪ (fi‘𝐴)) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (fi‘𝐴))) ∧ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∧ 𝑦 = ∩ 𝑧)) → 𝑦 = ∩ 𝑧)
11		eldifi 3244	. . . . . . . . . 10 ⊢ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) → 𝑧 ∈ (𝒫 𝐴 ∩ Fin))
12	11	elin1d 3311	. . . . . . . . 9 ⊢ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) → 𝑧 ∈ 𝒫 𝐴)
13	12	elpwid 3570	. . . . . . . 8 ⊢ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) → 𝑧 ⊆ 𝐴)
14	13	ad2antrl 482	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ∪ (fi‘𝐴)) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (fi‘𝐴))) ∧ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∧ 𝑦 = ∩ 𝑧)) → 𝑧 ⊆ 𝐴)
15		eldifsni 3705	. . . . . . . . 9 ⊢ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) → 𝑧 ≠ ∅)
16	11	elin2d 3312	. . . . . . . . . 10 ⊢ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) → 𝑧 ∈ Fin)
17		fin0 6851	. . . . . . . . . 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 146	. . . . . . . 8 ⊢ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) → ∃𝑤 𝑤 ∈ 𝑧)
20	19	ad2antrl 482	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ∪ (fi‘𝐴)) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (fi‘𝐴))) ∧ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∧ 𝑦 = ∩ 𝑧)) → ∃𝑤 𝑤 ∈ 𝑧)
21		intssuni2m 3848	. . . . . . 7 ⊢ ((𝑧 ⊆ 𝐴 ∧ ∃𝑤 𝑤 ∈ 𝑧) → ∩ 𝑧 ⊆ ∪ 𝐴)
22	14, 20, 21	syl2anc 409	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ∪ (fi‘𝐴)) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (fi‘𝐴))) ∧ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∧ 𝑦 = ∩ 𝑧)) → ∩ 𝑧 ⊆ ∪ 𝐴)
23	10, 22	eqsstrd 3178	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ∪ (fi‘𝐴)) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (fi‘𝐴))) ∧ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∧ 𝑦 = ∩ 𝑧)) → 𝑦 ⊆ ∪ 𝐴)
24		simplrl 525	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ∪ (fi‘𝐴)) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (fi‘𝐴))) ∧ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∧ 𝑦 = ∩ 𝑧)) → 𝑥 ∈ 𝑦)
25	23, 24	sseldd 3143	. . . 4 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ∪ (fi‘𝐴)) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (fi‘𝐴))) ∧ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∧ 𝑦 = ∩ 𝑧)) → 𝑥 ∈ ∪ 𝐴)
26	9, 25	rexlimddv 2588	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ∪ (fi‘𝐴)) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (fi‘𝐴))) → 𝑥 ∈ ∪ 𝐴)
27	5, 26	exlimddv 1886	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ∪ (fi‘𝐴)) → 𝑥 ∈ ∪ 𝐴)
28	2, 27	eqelssd 3161	1 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 = ∪ (fi‘𝐴))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1343  ∃wex 1480   ∈ wcel 2136   ≠ wne 2336  ∃wrex 2445   ∖ cdif 3113   ∩ cin 3115   ⊆ wss 3116  ∅c0 3409  𝒫 cpw 3559  {csn 3576  ∪ cuni 3789  ∩ cint 3824  ‘cfv 5188  Fincfn 6706  ficfi 6933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-1o 6384  df-er 6501  df-en 6707  df-fin 6709  df-fi 6934

This theorem is referenced by:  fipwssg  6944