Theorem fiuni 6977

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 6973	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (fi‘𝐴))
2	1	unissd 3834	. 2 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ⊆ ∪ (fi‘𝐴))
3		eluni 3813	. . . . 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 6971	. . . . . 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 3258	. . . . . . . . . 10 ⊢ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) → 𝑧 ∈ (𝒫 𝐴 ∩ Fin))
12	11	elin1d 3325	. . . . . . . . 9 ⊢ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) → 𝑧 ∈ 𝒫 𝐴)
13	12	elpwid 3587	. . . . . . . 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 3722	. . . . . . . . 9 ⊢ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) → 𝑧 ≠ ∅)
16	11	elin2d 3326	. . . . . . . . . 10 ⊢ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) → 𝑧 ∈ Fin)
17		fin0 6885	. . . . . . . . . 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 3869	. . . . . . 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 3192	. . . . 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 3157	. . . 4 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ∪ (fi‘𝐴)) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (fi‘𝐴))) ∧ (𝑧 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∧ 𝑦 = ∩ 𝑧)) → 𝑥 ∈ ∪ 𝐴)
26	9, 25	rexlimddv 2599	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ∪ (fi‘𝐴)) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (fi‘𝐴))) → 𝑥 ∈ ∪ 𝐴)
27	5, 26	exlimddv 1898	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ∪ (fi‘𝐴)) → 𝑥 ∈ ∪ 𝐴)
28	2, 27	eqelssd 3175	1 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 = ∪ (fi‘𝐴))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353  ∃wex 1492   ∈ wcel 2148   ≠ wne 2347  ∃wrex 2456   ∖ cdif 3127   ∩ cin 3129   ⊆ wss 3130  ∅c0 3423  𝒫 cpw 3576  {csn 3593  ∪ cuni 3810  ∩ cint 3845  ‘cfv 5217  Fincfn 6740  ficfi 6967

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-iinf 4588

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-1o 6417  df-er 6535  df-en 6741  df-fin 6743  df-fi 6968

This theorem is referenced by:  fipwssg  6978