Theorem fissuni 9266

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 3926	. . . . 5 ⊢ (𝐴 ⊆ ∪ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ 𝐵)
3		eluni2 4865	. . . . . 6 ⊢ (𝑥 ∈ ∪ 𝐵 ↔ ∃𝑧 ∈ 𝐵 𝑥 ∈ 𝑧)
4	3	ralbii 3075	. . . . 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 2817	. . . 4 ⊢ (𝑧 = (𝑓‘𝑥) → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ (𝑓‘𝑥)))
8	7	ac6sfi 9189	. . 3 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 ∈ 𝑧) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑓‘𝑥)))
9	1, 6, 8	syl2anc 584	. 2 ⊢ ((𝐴 ⊆ ∪ 𝐵 ∧ 𝐴 ∈ Fin) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑓‘𝑥)))
10		fimass 6676	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → (𝑓 “ 𝐴) ⊆ 𝐵)
11		vex 3442	. . . . . . . 8 ⊢ 𝑓 ∈ V
12	11	imaex 7854	. . . . . . 7 ⊢ (𝑓 “ 𝐴) ∈ V
13	12	elpw 4557	. . . . . 6 ⊢ ((𝑓 “ 𝐴) ∈ 𝒫 𝐵 ↔ (𝑓 “ 𝐴) ⊆ 𝐵)
14	10, 13	sylibr 234	. . . . 5 ⊢ (𝑓:𝐴⟶𝐵 → (𝑓 “ 𝐴) ∈ 𝒫 𝐵)
15	14	ad2antrl 728	. . . 4 ⊢ (((𝐴 ⊆ ∪ 𝐵 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑓‘𝑥))) → (𝑓 “ 𝐴) ∈ 𝒫 𝐵)
16		ffun 6659	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → Fun 𝑓)
17	16	ad2antrl 728	. . . . 5 ⊢ (((𝐴 ⊆ ∪ 𝐵 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑓‘𝑥))) → Fun 𝑓)
18		simplr 768	. . . . 5 ⊢ (((𝐴 ⊆ ∪ 𝐵 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑓‘𝑥))) → 𝐴 ∈ Fin)
19		imafi 9222	. . . . 5 ⊢ ((Fun 𝑓 ∧ 𝐴 ∈ Fin) → (𝑓 “ 𝐴) ∈ Fin)
20	17, 18, 19	syl2anc 584	. . . 4 ⊢ (((𝐴 ⊆ ∪ 𝐵 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑓‘𝑥))) → (𝑓 “ 𝐴) ∈ Fin)
21	15, 20	elind 4153	. . 3 ⊢ (((𝐴 ⊆ ∪ 𝐵 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑓‘𝑥))) → (𝑓 “ 𝐴) ∈ (𝒫 𝐵 ∩ Fin))
22		ffn 6656	. . . . . . . . . . 11 ⊢ (𝑓:𝐴⟶𝐵 → 𝑓 Fn 𝐴)
23	22	adantr 480	. . . . . . . . . 10 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑓 Fn 𝐴)
24		ssidd 3961	. . . . . . . . . 10 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ 𝐴)
25		simpr 484	. . . . . . . . . 10 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
26		fnfvima 7173	. . . . . . . . . 10 ⊢ ((𝑓 Fn 𝐴 ∧ 𝐴 ⊆ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ (𝑓 “ 𝐴))
27	23, 24, 25, 26	syl3anc 1373	. . . . . . . . 9 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ (𝑓 “ 𝐴))
28		elssuni 4891	. . . . . . . . 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 3936	. . . . . . 7 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ (𝑓‘𝑥) → 𝑥 ∈ ∪ (𝑓 “ 𝐴)))
31	30	ralimdva 3141	. . . . . 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 3926	. . . . 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 4872	. . . . 5 ⊢ (𝑐 = (𝑓 “ 𝐴) → ∪ 𝑐 = ∪ (𝑓 “ 𝐴))
37	36	sseq2d 3970	. . . 4 ⊢ (𝑐 = (𝑓 “ 𝐴) → (𝐴 ⊆ ∪ 𝑐 ↔ 𝐴 ⊆ ∪ (𝑓 “ 𝐴)))
38	37	rspcev 3579	. . 3 ⊢ (((𝑓 “ 𝐴) ∈ (𝒫 𝐵 ∩ Fin) ∧ 𝐴 ⊆ ∪ (𝑓 “ 𝐴)) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)𝐴 ⊆ ∪ 𝑐)
39	21, 35, 38	syl2anc 584	. 2 ⊢ (((𝐴 ⊆ ∪ 𝐵 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑓‘𝑥))) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)𝐴 ⊆ ∪ 𝑐)
40	9, 39	exlimddv 1935	1 ⊢ ((𝐴 ⊆ ∪ 𝐵 ∧ 𝐴 ∈ Fin) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)𝐴 ⊆ ∪ 𝑐)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ∩ cin 3904   ⊆ wss 3905  𝒫 cpw 4553  ∪ cuni 4861   “ cima 5626  Fun wfun 6480   Fn wfn 6481  ⟶wf 6482  ‘cfv 6486  Fincfn 8879

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-om 7807  df-1o 8395  df-en 8880  df-dom 8881  df-fin 8883

This theorem is referenced by:  isacs3lem  18466  isnacs3  42683