Theorem fipreima 9302

Description: Given a finite subset 𝐴 of the range of a function, there exists a finite subset of the domain whose image is 𝐴. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 22-Feb-2015.)

Assertion

Ref	Expression
fipreima	⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)(𝐹 “ 𝑐) = 𝐴)

Distinct variable groups:   𝐴,𝑐   𝐵,𝑐   𝐹,𝑐

Proof of Theorem fipreima

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

Step	Hyp	Ref	Expression
1		simp3 1138	. . 3 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) → 𝐴 ∈ Fin)
2		dfss3 3932	. . . . . 6 ⊢ (𝐴 ⊆ ran 𝐹 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ ran 𝐹)
3		fvelrnb 6903	. . . . . . 7 ⊢ (𝐹 Fn 𝐵 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥))
4	3	ralbidv 3174	. . . . . 6 ⊢ (𝐹 Fn 𝐵 → (∀𝑥 ∈ 𝐴 𝑥 ∈ ran 𝐹 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥))
5	2, 4	bitrid 282	. . . . 5 ⊢ (𝐹 Fn 𝐵 → (𝐴 ⊆ ran 𝐹 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥))
6	5	biimpa 477	. . . 4 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥)
7	6	3adant3 1132	. . 3 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥)
8		fveqeq2 6851	. . . 4 ⊢ (𝑦 = (𝑓‘𝑥) → ((𝐹‘𝑦) = 𝑥 ↔ (𝐹‘(𝑓‘𝑥)) = 𝑥))
9	8	ac6sfi 9231	. . 3 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥))
10	1, 7, 9	syl2anc 584	. 2 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥))
11		fimass 6689	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → (𝑓 “ 𝐴) ⊆ 𝐵)
12		vex 3449	. . . . . . . 8 ⊢ 𝑓 ∈ V
13	12	imaex 7853	. . . . . . 7 ⊢ (𝑓 “ 𝐴) ∈ V
14	13	elpw 4564	. . . . . 6 ⊢ ((𝑓 “ 𝐴) ∈ 𝒫 𝐵 ↔ (𝑓 “ 𝐴) ⊆ 𝐵)
15	11, 14	sylibr 233	. . . . 5 ⊢ (𝑓:𝐴⟶𝐵 → (𝑓 “ 𝐴) ∈ 𝒫 𝐵)
16	15	ad2antrl 726	. . . 4 ⊢ (((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥)) → (𝑓 “ 𝐴) ∈ 𝒫 𝐵)
17		ffun 6671	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → Fun 𝑓)
18	17	ad2antrl 726	. . . . 5 ⊢ (((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥)) → Fun 𝑓)
19		simpl3 1193	. . . . 5 ⊢ (((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥)) → 𝐴 ∈ Fin)
20		imafi 9119	. . . . 5 ⊢ ((Fun 𝑓 ∧ 𝐴 ∈ Fin) → (𝑓 “ 𝐴) ∈ Fin)
21	18, 19, 20	syl2anc 584	. . . 4 ⊢ (((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥)) → (𝑓 “ 𝐴) ∈ Fin)
22	16, 21	elind 4154	. . 3 ⊢ (((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥)) → (𝑓 “ 𝐴) ∈ (𝒫 𝐵 ∩ Fin))
23		fvco3 6940	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘ 𝑓)‘𝑥) = (𝐹‘(𝑓‘𝑥)))
24		fvresi 7119	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
25	24	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (( I ↾ 𝐴)‘𝑥) = 𝑥)
26	23, 25	eqeq12d 2752	. . . . . . . . . 10 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (((𝐹 ∘ 𝑓)‘𝑥) = (( I ↾ 𝐴)‘𝑥) ↔ (𝐹‘(𝑓‘𝑥)) = 𝑥))
27	26	ralbidva 3172	. . . . . . . . 9 ⊢ (𝑓:𝐴⟶𝐵 → (∀𝑥 ∈ 𝐴 ((𝐹 ∘ 𝑓)‘𝑥) = (( I ↾ 𝐴)‘𝑥) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥))
28	27	biimprd 247	. . . . . . . 8 ⊢ (𝑓:𝐴⟶𝐵 → (∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥 → ∀𝑥 ∈ 𝐴 ((𝐹 ∘ 𝑓)‘𝑥) = (( I ↾ 𝐴)‘𝑥)))
29	28	adantl 482	. . . . . . 7 ⊢ (((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) ∧ 𝑓:𝐴⟶𝐵) → (∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥 → ∀𝑥 ∈ 𝐴 ((𝐹 ∘ 𝑓)‘𝑥) = (( I ↾ 𝐴)‘𝑥)))
30	29	impr 455	. . . . . 6 ⊢ (((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥)) → ∀𝑥 ∈ 𝐴 ((𝐹 ∘ 𝑓)‘𝑥) = (( I ↾ 𝐴)‘𝑥))
31		simpl1 1191	. . . . . . . 8 ⊢ (((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥)) → 𝐹 Fn 𝐵)
32		ffn 6668	. . . . . . . . 9 ⊢ (𝑓:𝐴⟶𝐵 → 𝑓 Fn 𝐴)
33	32	ad2antrl 726	. . . . . . . 8 ⊢ (((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥)) → 𝑓 Fn 𝐴)
34		frn 6675	. . . . . . . . 9 ⊢ (𝑓:𝐴⟶𝐵 → ran 𝑓 ⊆ 𝐵)
35	34	ad2antrl 726	. . . . . . . 8 ⊢ (((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥)) → ran 𝑓 ⊆ 𝐵)
36		fnco 6618	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐵 ∧ 𝑓 Fn 𝐴 ∧ ran 𝑓 ⊆ 𝐵) → (𝐹 ∘ 𝑓) Fn 𝐴)
37	31, 33, 35, 36	syl3anc 1371	. . . . . . 7 ⊢ (((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥)) → (𝐹 ∘ 𝑓) Fn 𝐴)
38		fnresi 6630	. . . . . . 7 ⊢ ( I ↾ 𝐴) Fn 𝐴
39		eqfnfv 6982	. . . . . . 7 ⊢ (((𝐹 ∘ 𝑓) Fn 𝐴 ∧ ( I ↾ 𝐴) Fn 𝐴) → ((𝐹 ∘ 𝑓) = ( I ↾ 𝐴) ↔ ∀𝑥 ∈ 𝐴 ((𝐹 ∘ 𝑓)‘𝑥) = (( I ↾ 𝐴)‘𝑥)))
40	37, 38, 39	sylancl 586	. . . . . 6 ⊢ (((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥)) → ((𝐹 ∘ 𝑓) = ( I ↾ 𝐴) ↔ ∀𝑥 ∈ 𝐴 ((𝐹 ∘ 𝑓)‘𝑥) = (( I ↾ 𝐴)‘𝑥)))
41	30, 40	mpbird 256	. . . . 5 ⊢ (((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥)) → (𝐹 ∘ 𝑓) = ( I ↾ 𝐴))
42	41	imaeq1d 6012	. . . 4 ⊢ (((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥)) → ((𝐹 ∘ 𝑓) “ 𝐴) = (( I ↾ 𝐴) “ 𝐴))
43		imaco 6203	. . . 4 ⊢ ((𝐹 ∘ 𝑓) “ 𝐴) = (𝐹 “ (𝑓 “ 𝐴))
44		ssid 3966	. . . . 5 ⊢ 𝐴 ⊆ 𝐴
45		resiima 6028	. . . . 5 ⊢ (𝐴 ⊆ 𝐴 → (( I ↾ 𝐴) “ 𝐴) = 𝐴)
46	44, 45	ax-mp 5	. . . 4 ⊢ (( I ↾ 𝐴) “ 𝐴) = 𝐴
47	42, 43, 46	3eqtr3g 2799	. . 3 ⊢ (((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥)) → (𝐹 “ (𝑓 “ 𝐴)) = 𝐴)
48		imaeq2 6009	. . . . 5 ⊢ (𝑐 = (𝑓 “ 𝐴) → (𝐹 “ 𝑐) = (𝐹 “ (𝑓 “ 𝐴)))
49	48	eqeq1d 2738	. . . 4 ⊢ (𝑐 = (𝑓 “ 𝐴) → ((𝐹 “ 𝑐) = 𝐴 ↔ (𝐹 “ (𝑓 “ 𝐴)) = 𝐴))
50	49	rspcev 3581	. . 3 ⊢ (((𝑓 “ 𝐴) ∈ (𝒫 𝐵 ∩ Fin) ∧ (𝐹 “ (𝑓 “ 𝐴)) = 𝐴) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)(𝐹 “ 𝑐) = 𝐴)
51	22, 47, 50	syl2anc 584	. 2 ⊢ (((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥)) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)(𝐹 “ 𝑐) = 𝐴)
52	10, 51	exlimddv 1938	1 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)(𝐹 “ 𝑐) = 𝐴)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541  ∃wex 1781   ∈ wcel 2106  ∀wral 3064  ∃wrex 3073   ∩ cin 3909   ⊆ wss 3910  𝒫 cpw 4560   I cid 5530  ran crn 5634   ↾ cres 5635   “ cima 5636   ∘ ccom 5637  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-csb 3856  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-mpt 5189  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:  fodomfi2  9996  cmpfi  22759  elrfirn  41004  lmhmfgsplit  41399  hbtlem6  41442