Theorem fissuni 9301

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 485	. . 3 ⊢ ((𝐴 ⊆ ∪ 𝐵 ∧ 𝐴 ∈ Fin) → 𝐴 ∈ Fin)
2		dfss3 3932	. . . . 5 ⊢ (𝐴 ⊆ ∪ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ 𝐵)
3		eluni2 4869	. . . . . 6 ⊢ (𝑥 ∈ ∪ 𝐵 ↔ ∃𝑧 ∈ 𝐵 𝑥 ∈ 𝑧)
4	3	ralbii 3096	. . . . 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 481	. . 3 ⊢ ((𝐴 ⊆ ∪ 𝐵 ∧ 𝐴 ∈ Fin) → ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 ∈ 𝑧)
7		eleq2 2826	. . . 4 ⊢ (𝑧 = (𝑓‘𝑥) → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ (𝑓‘𝑥)))
8	7	ac6sfi 9231	. . 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 6689	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → (𝑓 “ 𝐴) ⊆ 𝐵)
11		vex 3449	. . . . . . . 8 ⊢ 𝑓 ∈ V
12	11	imaex 7853	. . . . . . 7 ⊢ (𝑓 “ 𝐴) ∈ V
13	12	elpw 4564	. . . . . 6 ⊢ ((𝑓 “ 𝐴) ∈ 𝒫 𝐵 ↔ (𝑓 “ 𝐴) ⊆ 𝐵)
14	10, 13	sylibr 233	. . . . 5 ⊢ (𝑓:𝐴⟶𝐵 → (𝑓 “ 𝐴) ∈ 𝒫 𝐵)
15	14	ad2antrl 726	. . . 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 726	. . . . 5 ⊢ (((𝐴 ⊆ ∪ 𝐵 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑓‘𝑥))) → Fun 𝑓)
18		simplr 767	. . . . 5 ⊢ (((𝐴 ⊆ ∪ 𝐵 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑓‘𝑥))) → 𝐴 ∈ Fin)
19		imafi 9119	. . . . 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 4154	. . 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 481	. . . . . . . . . 10 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑓 Fn 𝐴)
24		ssidd 3967	. . . . . . . . . 10 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ 𝐴)
25		simpr 485	. . . . . . . . . 10 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
26		fnfvima 7183	. . . . . . . . . 10 ⊢ ((𝑓 Fn 𝐴 ∧ 𝐴 ⊆ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ (𝑓 “ 𝐴))
27	23, 24, 25, 26	syl3anc 1371	. . . . . . . . 9 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ (𝑓 “ 𝐴))
28		elssuni 4898	. . . . . . . . 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 3943	. . . . . . 7 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ (𝑓‘𝑥) → 𝑥 ∈ ∪ (𝑓 “ 𝐴)))
31	30	ralimdva 3164	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → (∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑓‘𝑥) → ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑓 “ 𝐴)))
32	31	imp 407	. . . . 5 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑓‘𝑥)) → ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑓 “ 𝐴))
33		dfss3 3932	. . . . 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 482	. . 3 ⊢ (((𝐴 ⊆ ∪ 𝐵 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑓‘𝑥))) → 𝐴 ⊆ ∪ (𝑓 “ 𝐴))
36		unieq 4876	. . . . 5 ⊢ (𝑐 = (𝑓 “ 𝐴) → ∪ 𝑐 = ∪ (𝑓 “ 𝐴))
37	36	sseq2d 3976	. . . 4 ⊢ (𝑐 = (𝑓 “ 𝐴) → (𝐴 ⊆ ∪ 𝑐 ↔ 𝐴 ⊆ ∪ (𝑓 “ 𝐴)))
38	37	rspcev 3581	. . 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 1938	1 ⊢ ((𝐴 ⊆ ∪ 𝐵 ∧ 𝐴 ∈ Fin) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)𝐴 ⊆ ∪ 𝑐)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  ∀wral 3064  ∃wrex 3073   ∩ cin 3909   ⊆ wss 3910  𝒫 cpw 4560  ∪ cuni 4865   “ cima 5636  Fun wfun 6490   Fn wfn 6491  ⟶wf 6492  ‘cfv 6496  Fincfn 8883

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-om 7803  df-1o 8412  df-en 8884  df-fin 8887

This theorem is referenced by:  isacs3lem  18431  isnacs3  41019