Theorem fissuni 9241

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 3923	. . . . 5 ⊢ (𝐴 ⊆ ∪ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ 𝐵)
3		eluni2 4863	. . . . . 6 ⊢ (𝑥 ∈ ∪ 𝐵 ↔ ∃𝑧 ∈ 𝐵 𝑥 ∈ 𝑧)
4	3	ralbii 3078	. . . . 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 2820	. . . 4 ⊢ (𝑧 = (𝑓‘𝑥) → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ (𝑓‘𝑥)))
8	7	ac6sfi 9168	. . 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 6671	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → (𝑓 “ 𝐴) ⊆ 𝐵)
11		vex 3440	. . . . . . . 8 ⊢ 𝑓 ∈ V
12	11	imaex 7844	. . . . . . 7 ⊢ (𝑓 “ 𝐴) ∈ V
13	12	elpw 4554	. . . . . 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 6654	. . . . . 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 9199	. . . . 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 4150	. . 3 ⊢ (((𝐴 ⊆ ∪ 𝐵 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑓‘𝑥))) → (𝑓 “ 𝐴) ∈ (𝒫 𝐵 ∩ Fin))
22		ffn 6651	. . . . . . . . . . 11 ⊢ (𝑓:𝐴⟶𝐵 → 𝑓 Fn 𝐴)
23	22	adantr 480	. . . . . . . . . 10 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑓 Fn 𝐴)
24		ssidd 3958	. . . . . . . . . 10 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ 𝐴)
25		simpr 484	. . . . . . . . . 10 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
26		fnfvima 7167	. . . . . . . . . 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 4889	. . . . . . . . 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 3933	. . . . . . 7 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ (𝑓‘𝑥) → 𝑥 ∈ ∪ (𝑓 “ 𝐴)))
31	30	ralimdva 3144	. . . . . 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 3923	. . . . 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 4870	. . . . 5 ⊢ (𝑐 = (𝑓 “ 𝐴) → ∪ 𝑐 = ∪ (𝑓 “ 𝐴))
37	36	sseq2d 3967	. . . 4 ⊢ (𝑐 = (𝑓 “ 𝐴) → (𝐴 ⊆ ∪ 𝑐 ↔ 𝐴 ⊆ ∪ (𝑓 “ 𝐴)))
38	37	rspcev 3577	. . 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 1936	1 ⊢ ((𝐴 ⊆ ∪ 𝐵 ∧ 𝐴 ∈ Fin) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)𝐴 ⊆ ∪ 𝑐)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056   ∩ cin 3901   ⊆ wss 3902  𝒫 cpw 4550  ∪ cuni 4859   “ cima 5619  Fun wfun 6475   Fn wfn 6476  ⟶wf 6477  ‘cfv 6481  Fincfn 8869

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-om 7797  df-1o 8385  df-en 8870  df-dom 8871  df-fin 8873

This theorem is referenced by:  isacs3lem  18445  isnacs3  42742