Theorem fiuni 6953

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 6949	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (fi‘𝐴))
2	1	unissd 3818	. 2 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ⊆ ∪ (fi‘𝐴))
3		eluni 3797	. . . . 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 527	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ∪ (fi‘𝐴)) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (fi‘𝐴))) → 𝑦 ∈ (fi‘𝐴))
7		elfi2 6947	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ (fi‘𝐴) ↔ ∃𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅})𝑦 = ∩ 𝑧))
8	7	ad2antrr 485	. . . . 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 527	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ∪ (fi‘𝐴)) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (fi‘𝐴))) ∧ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∧ 𝑦 = ∩ 𝑧)) → 𝑦 = ∩ 𝑧)
11		eldifi 3249	. . . . . . . . . 10 ⊢ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) → 𝑧 ∈ (𝒫 𝐴 ∩ Fin))
12	11	elin1d 3316	. . . . . . . . 9 ⊢ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) → 𝑧 ∈ 𝒫 𝐴)
13	12	elpwid 3575	. . . . . . . 8 ⊢ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) → 𝑧 ⊆ 𝐴)
14	13	ad2antrl 487	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ∪ (fi‘𝐴)) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (fi‘𝐴))) ∧ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∧ 𝑦 = ∩ 𝑧)) → 𝑧 ⊆ 𝐴)
15		eldifsni 3710	. . . . . . . . 9 ⊢ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) → 𝑧 ≠ ∅)
16	11	elin2d 3317	. . . . . . . . . 10 ⊢ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) → 𝑧 ∈ Fin)
17		fin0 6861	. . . . . . . . . 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 487	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ∪ (fi‘𝐴)) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (fi‘𝐴))) ∧ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∧ 𝑦 = ∩ 𝑧)) → ∃𝑤 𝑤 ∈ 𝑧)
21		intssuni2m 3853	. . . . . . 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 3183	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ∪ (fi‘𝐴)) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (fi‘𝐴))) ∧ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∧ 𝑦 = ∩ 𝑧)) → 𝑦 ⊆ ∪ 𝐴)
24		simplrl 530	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ∪ (fi‘𝐴)) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (fi‘𝐴))) ∧ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∧ 𝑦 = ∩ 𝑧)) → 𝑥 ∈ 𝑦)
25	23, 24	sseldd 3148	. . . 4 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ∪ (fi‘𝐴)) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (fi‘𝐴))) ∧ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∧ 𝑦 = ∩ 𝑧)) → 𝑥 ∈ ∪ 𝐴)
26	9, 25	rexlimddv 2592	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ∪ (fi‘𝐴)) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (fi‘𝐴))) → 𝑥 ∈ ∪ 𝐴)
27	5, 26	exlimddv 1891	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ∪ (fi‘𝐴)) → 𝑥 ∈ ∪ 𝐴)
28	2, 27	eqelssd 3166	1 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 = ∪ (fi‘𝐴))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1348  ∃wex 1485   ∈ wcel 2141   ≠ wne 2340  ∃wrex 2449   ∖ cdif 3118   ∩ cin 3120   ⊆ wss 3121  ∅c0 3414  𝒫 cpw 3564  {csn 3581  ∪ cuni 3794  ∩ cint 3829  ‘cfv 5196  Fincfn 6716  ficfi 6943

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-iinf 4570

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-1o 6393  df-er 6511  df-en 6717  df-fin 6719  df-fi 6944

This theorem is referenced by:  fipwssg  6954