Theorem fissuni 9054

Description: A finite subset of a union is covered by finitely many elements. (Contributed by Stefan O'Rear, 2-Apr-2015.)

Assertion

Ref	Expression
fissuni	⊢ ((𝐴 ⊆ ∪ 𝐵 ∧ 𝐴 ∈ Fin) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)𝐴 ⊆ ∪ 𝑐)

Distinct variable groups:   𝐴,𝑐   𝐵,𝑐

Proof of Theorem fissuni

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

Step	Hyp	Ref	Expression
1		simpr 484	. . 3 ⊢ ((𝐴 ⊆ ∪ 𝐵 ∧ 𝐴 ∈ Fin) → 𝐴 ∈ Fin)
2		dfss3 3905	. . . . 5 ⊢ (𝐴 ⊆ ∪ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ 𝐵)
3		eluni2 4840	. . . . . 6 ⊢ (𝑥 ∈ ∪ 𝐵 ↔ ∃𝑧 ∈ 𝐵 𝑥 ∈ 𝑧)
4	3	ralbii 3090	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 ∈ 𝑧)
5	2, 4	sylbb 218	. . . 4 ⊢ (𝐴 ⊆ ∪ 𝐵 → ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 ∈ 𝑧)
6	5	adantr 480	. . 3 ⊢ ((𝐴 ⊆ ∪ 𝐵 ∧ 𝐴 ∈ Fin) → ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 ∈ 𝑧)
7		eleq2 2827	. . . 4 ⊢ (𝑧 = (𝑓‘𝑥) → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ (𝑓‘𝑥)))
8	7	ac6sfi 8988	. . 3 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 ∈ 𝑧) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑓‘𝑥)))
9	1, 6, 8	syl2anc 583	. 2 ⊢ ((𝐴 ⊆ ∪ 𝐵 ∧ 𝐴 ∈ Fin) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑓‘𝑥)))
10		fimass 6605	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → (𝑓 “ 𝐴) ⊆ 𝐵)
11		vex 3426	. . . . . . . 8 ⊢ 𝑓 ∈ V
12	11	imaex 7737	. . . . . . 7 ⊢ (𝑓 “ 𝐴) ∈ V
13	12	elpw 4534	. . . . . 6 ⊢ ((𝑓 “ 𝐴) ∈ 𝒫 𝐵 ↔ (𝑓 “ 𝐴) ⊆ 𝐵)
14	10, 13	sylibr 233	. . . . 5 ⊢ (𝑓:𝐴⟶𝐵 → (𝑓 “ 𝐴) ∈ 𝒫 𝐵)
15	14	ad2antrl 724	. . . 4 ⊢ (((𝐴 ⊆ ∪ 𝐵 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑓‘𝑥))) → (𝑓 “ 𝐴) ∈ 𝒫 𝐵)
16		ffun 6587	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → Fun 𝑓)
17	16	ad2antrl 724	. . . . 5 ⊢ (((𝐴 ⊆ ∪ 𝐵 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑓‘𝑥))) → Fun 𝑓)
18		simplr 765	. . . . 5 ⊢ (((𝐴 ⊆ ∪ 𝐵 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑓‘𝑥))) → 𝐴 ∈ Fin)
19		imafi 8920	. . . . 5 ⊢ ((Fun 𝑓 ∧ 𝐴 ∈ Fin) → (𝑓 “ 𝐴) ∈ Fin)
20	17, 18, 19	syl2anc 583	. . . 4 ⊢ (((𝐴 ⊆ ∪ 𝐵 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑓‘𝑥))) → (𝑓 “ 𝐴) ∈ Fin)
21	15, 20	elind 4124	. . 3 ⊢ (((𝐴 ⊆ ∪ 𝐵 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑓‘𝑥))) → (𝑓 “ 𝐴) ∈ (𝒫 𝐵 ∩ Fin))
22		ffn 6584	. . . . . . . . . . 11 ⊢ (𝑓:𝐴⟶𝐵 → 𝑓 Fn 𝐴)
23	22	adantr 480	. . . . . . . . . 10 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑓 Fn 𝐴)
24		ssidd 3940	. . . . . . . . . 10 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ 𝐴)
25		simpr 484	. . . . . . . . . 10 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
26		fnfvima 7091	. . . . . . . . . 10 ⊢ ((𝑓 Fn 𝐴 ∧ 𝐴 ⊆ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ (𝑓 “ 𝐴))
27	23, 24, 25, 26	syl3anc 1369	. . . . . . . . 9 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ (𝑓 “ 𝐴))
28		elssuni 4868	. . . . . . . . 9 ⊢ ((𝑓‘𝑥) ∈ (𝑓 “ 𝐴) → (𝑓‘𝑥) ⊆ ∪ (𝑓 “ 𝐴))
29	27, 28	syl 17	. . . . . . . 8 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ⊆ ∪ (𝑓 “ 𝐴))
30	29	sseld 3916	. . . . . . 7 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ (𝑓‘𝑥) → 𝑥 ∈ ∪ (𝑓 “ 𝐴)))
31	30	ralimdva 3102	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → (∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑓‘𝑥) → ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑓 “ 𝐴)))
32	31	imp 406	. . . . 5 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑓‘𝑥)) → ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑓 “ 𝐴))
33		dfss3 3905	. . . . 5 ⊢ (𝐴 ⊆ ∪ (𝑓 “ 𝐴) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑓 “ 𝐴))
34	32, 33	sylibr 233	. . . 4 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑓‘𝑥)) → 𝐴 ⊆ ∪ (𝑓 “ 𝐴))
35	34	adantl 481	. . 3 ⊢ (((𝐴 ⊆ ∪ 𝐵 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑓‘𝑥))) → 𝐴 ⊆ ∪ (𝑓 “ 𝐴))
36		unieq 4847	. . . . 5 ⊢ (𝑐 = (𝑓 “ 𝐴) → ∪ 𝑐 = ∪ (𝑓 “ 𝐴))
37	36	sseq2d 3949	. . . 4 ⊢ (𝑐 = (𝑓 “ 𝐴) → (𝐴 ⊆ ∪ 𝑐 ↔ 𝐴 ⊆ ∪ (𝑓 “ 𝐴)))
38	37	rspcev 3552	. . 3 ⊢ (((𝑓 “ 𝐴) ∈ (𝒫 𝐵 ∩ Fin) ∧ 𝐴 ⊆ ∪ (𝑓 “ 𝐴)) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)𝐴 ⊆ ∪ 𝑐)
39	21, 35, 38	syl2anc 583	. 2 ⊢ (((𝐴 ⊆ ∪ 𝐵 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑓‘𝑥))) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)𝐴 ⊆ ∪ 𝑐)
40	9, 39	exlimddv 1939	1 ⊢ ((𝐴 ⊆ ∪ 𝐵 ∧ 𝐴 ∈ Fin) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)𝐴 ⊆ ∪ 𝑐)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064   ∩ cin 3882   ⊆ wss 3883  𝒫 cpw 4530  ∪ cuni 4836   “ cima 5583  Fun wfun 6412   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  Fincfn 8691

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-om 7688  df-1o 8267  df-en 8692  df-fin 8695

This theorem is referenced by:  isacs3lem  18175  isnacs3  40448