Theorem fissuni 9267

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 3910	. . . . 5 ⊢ (𝐴 ⊆ ∪ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ 𝐵)
3		eluni2 4854	. . . . . 6 ⊢ (𝑥 ∈ ∪ 𝐵 ↔ ∃𝑧 ∈ 𝐵 𝑥 ∈ 𝑧)
4	3	ralbii 3083	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 ∈ 𝑧)
5	2, 4	sylbb 219	. . . 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 2825	. . . 4 ⊢ (𝑧 = (𝑓‘𝑥) → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ (𝑓‘𝑥)))
8	7	ac6sfi 9194	. . 3 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 ∈ 𝑧) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑓‘𝑥)))
9	1, 6, 8	syl2anc 585	. 2 ⊢ ((𝐴 ⊆ ∪ 𝐵 ∧ 𝐴 ∈ Fin) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑓‘𝑥)))
10		fimass 6688	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → (𝑓 “ 𝐴) ⊆ 𝐵)
11		vex 3433	. . . . . . . 8 ⊢ 𝑓 ∈ V
12	11	imaex 7865	. . . . . . 7 ⊢ (𝑓 “ 𝐴) ∈ V
13	12	elpw 4545	. . . . . 6 ⊢ ((𝑓 “ 𝐴) ∈ 𝒫 𝐵 ↔ (𝑓 “ 𝐴) ⊆ 𝐵)
14	10, 13	sylibr 234	. . . . 5 ⊢ (𝑓:𝐴⟶𝐵 → (𝑓 “ 𝐴) ∈ 𝒫 𝐵)
15	14	ad2antrl 729	. . . 4 ⊢ (((𝐴 ⊆ ∪ 𝐵 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑓‘𝑥))) → (𝑓 “ 𝐴) ∈ 𝒫 𝐵)
16		ffun 6671	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → Fun 𝑓)
17	16	ad2antrl 729	. . . . 5 ⊢ (((𝐴 ⊆ ∪ 𝐵 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑓‘𝑥))) → Fun 𝑓)
18		simplr 769	. . . . 5 ⊢ (((𝐴 ⊆ ∪ 𝐵 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑓‘𝑥))) → 𝐴 ∈ Fin)
19		imafi 9225	. . . . 5 ⊢ ((Fun 𝑓 ∧ 𝐴 ∈ Fin) → (𝑓 “ 𝐴) ∈ Fin)
20	17, 18, 19	syl2anc 585	. . . 4 ⊢ (((𝐴 ⊆ ∪ 𝐵 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑓‘𝑥))) → (𝑓 “ 𝐴) ∈ Fin)
21	15, 20	elind 4140	. . 3 ⊢ (((𝐴 ⊆ ∪ 𝐵 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑓‘𝑥))) → (𝑓 “ 𝐴) ∈ (𝒫 𝐵 ∩ Fin))
22		ffn 6668	. . . . . . . . . . 11 ⊢ (𝑓:𝐴⟶𝐵 → 𝑓 Fn 𝐴)
23	22	adantr 480	. . . . . . . . . 10 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑓 Fn 𝐴)
24		ssidd 3945	. . . . . . . . . 10 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ 𝐴)
25		simpr 484	. . . . . . . . . 10 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
26		fnfvima 7188	. . . . . . . . . 10 ⊢ ((𝑓 Fn 𝐴 ∧ 𝐴 ⊆ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ (𝑓 “ 𝐴))
27	23, 24, 25, 26	syl3anc 1374	. . . . . . . . 9 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ (𝑓 “ 𝐴))
28		elssuni 4881	. . . . . . . . 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 3920	. . . . . . 7 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ (𝑓‘𝑥) → 𝑥 ∈ ∪ (𝑓 “ 𝐴)))
31	30	ralimdva 3149	. . . . . 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 3910	. . . . 5 ⊢ (𝐴 ⊆ ∪ (𝑓 “ 𝐴) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑓 “ 𝐴))
34	32, 33	sylibr 234	. . . 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 4861	. . . . 5 ⊢ (𝑐 = (𝑓 “ 𝐴) → ∪ 𝑐 = ∪ (𝑓 “ 𝐴))
37	36	sseq2d 3954	. . . 4 ⊢ (𝑐 = (𝑓 “ 𝐴) → (𝐴 ⊆ ∪ 𝑐 ↔ 𝐴 ⊆ ∪ (𝑓 “ 𝐴)))
38	37	rspcev 3564	. . 3 ⊢ (((𝑓 “ 𝐴) ∈ (𝒫 𝐵 ∩ Fin) ∧ 𝐴 ⊆ ∪ (𝑓 “ 𝐴)) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)𝐴 ⊆ ∪ 𝑐)
39	21, 35, 38	syl2anc 585	. 2 ⊢ (((𝐴 ⊆ ∪ 𝐵 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑓‘𝑥))) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)𝐴 ⊆ ∪ 𝑐)
40	9, 39	exlimddv 1937	1 ⊢ ((𝐴 ⊆ ∪ 𝐵 ∧ 𝐴 ∈ Fin) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)𝐴 ⊆ ∪ 𝑐)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3051  ∃wrex 3061   ∩ cin 3888   ⊆ wss 3889  𝒫 cpw 4541  ∪ cuni 4850   “ cima 5634  Fun wfun 6492   Fn wfn 6493  ⟶wf 6494  ‘cfv 6498  Fincfn 8893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-om 7818  df-1o 8405  df-en 8894  df-dom 8895  df-fin 8897

This theorem is referenced by:  isacs3lem  18508  isnacs3  43142