Theorem fipreima 9351

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 1156	. . 3 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) → 𝐴 ∈ Fin)
2		dfss3 3897	. . . . . 6 ⊢ (𝐴 ⊆ ran 𝐹 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ ran 𝐹)
3		fvelrnb 6946	. . . . . . 7 ⊢ (𝐹 Fn 𝐵 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥))
4	3	ralbidv 3185	. . . . . 6 ⊢ (𝐹 Fn 𝐵 → (∀𝑥 ∈ 𝐴 𝑥 ∈ ran 𝐹 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥))
5	2, 4	bitrid 286	. . . . 5 ⊢ (𝐹 Fn 𝐵 → (𝐴 ⊆ ran 𝐹 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥))
6	5	biimpa 482	. . . 4 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥)
7	6	3adant3 1150	. . 3 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥)
8		fveqeq2 6895	. . . 4 ⊢ (𝑦 = (𝑓‘𝑥) → ((𝐹‘𝑦) = 𝑥 ↔ (𝐹‘(𝑓‘𝑥)) = 𝑥))
9	8	ac6sfi 9279	. . 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 596	. 2 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥))
11		fimass 6731	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → (𝑓 “ 𝐴) ⊆ 𝐵)
12		vex 3454	. . . . . . . 8 ⊢ 𝑓 ∈ V
13	12	imaex 7927	. . . . . . 7 ⊢ (𝑓 “ 𝐴) ∈ V
14	13	elpw 4560	. . . . . 6 ⊢ ((𝑓 “ 𝐴) ∈ 𝒫 𝐵 ↔ (𝑓 “ 𝐴) ⊆ 𝐵)
15	11, 14	sylibr 237	. . . . 5 ⊢ (𝑓:𝐴⟶𝐵 → (𝑓 “ 𝐴) ∈ 𝒫 𝐵)
16	15	ad2antrl 741	. . . 4 ⊢ (((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥)) → (𝑓 “ 𝐴) ∈ 𝒫 𝐵)
17		ffun 6713	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → Fun 𝑓)
18	17	ad2antrl 741	. . . . 5 ⊢ (((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥)) → Fun 𝑓)
19		simpl3 1212	. . . . 5 ⊢ (((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥)) → 𝐴 ∈ Fin)
20		imafi 9311	. . . . 5 ⊢ ((Fun 𝑓 ∧ 𝐴 ∈ Fin) → (𝑓 “ 𝐴) ∈ Fin)
21	18, 19, 20	syl2anc 596	. . . 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 4097	. . 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 6986	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘ 𝑓)‘𝑥) = (𝐹‘(𝑓‘𝑥)))
24		fvresi 7179	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
25	24	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (( I ↾ 𝐴)‘𝑥) = 𝑥)
26	23, 25	eqeq12d 2776	. . . . . . . . . 10 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (((𝐹 ∘ 𝑓)‘𝑥) = (( I ↾ 𝐴)‘𝑥) ↔ (𝐹‘(𝑓‘𝑥)) = 𝑥))
27	26	ralbidva 3183	. . . . . . . . 9 ⊢ (𝑓:𝐴⟶𝐵 → (∀𝑥 ∈ 𝐴 ((𝐹 ∘ 𝑓)‘𝑥) = (( I ↾ 𝐴)‘𝑥) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥))
28	27	biimprd 251	. . . . . . . 8 ⊢ (𝑓:𝐴⟶𝐵 → (∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥 → ∀𝑥 ∈ 𝐴 ((𝐹 ∘ 𝑓)‘𝑥) = (( I ↾ 𝐴)‘𝑥)))
29	28	adantl 487	. . . . . . 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 460	. . . . . 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 1210	. . . . . . . 8 ⊢ (((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥)) → 𝐹 Fn 𝐵)
32		ffn 6710	. . . . . . . . 9 ⊢ (𝑓:𝐴⟶𝐵 → 𝑓 Fn 𝐴)
33	32	ad2antrl 741	. . . . . . . 8 ⊢ (((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥)) → 𝑓 Fn 𝐴)
34		frn 6718	. . . . . . . . 9 ⊢ (𝑓:𝐴⟶𝐵 → ran 𝑓 ⊆ 𝐵)
35	34	ad2antrl 741	. . . . . . . 8 ⊢ (((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥)) → ran 𝑓 ⊆ 𝐵)
36		fnco 6658	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐵 ∧ 𝑓 Fn 𝐴 ∧ ran 𝑓 ⊆ 𝐵) → (𝐹 ∘ 𝑓) Fn 𝐴)
37	31, 33, 35, 36	syl3anc 1398	. . . . . . 7 ⊢ (((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥)) → (𝐹 ∘ 𝑓) Fn 𝐴)
38		fnresi 6669	. . . . . . 7 ⊢ ( I ↾ 𝐴) Fn 𝐴
39		eqfnfv 7030	. . . . . . 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 598	. . . . . 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 260	. . . . 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 6050	. . . 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 6252	. . . 4 ⊢ ((𝐹 ∘ 𝑓) “ 𝐴) = (𝐹 “ (𝑓 “ 𝐴))
44		ssid 3930	. . . . 5 ⊢ 𝐴 ⊆ 𝐴
45		resiima 6073	. . . . 5 ⊢ (𝐴 ⊆ 𝐴 → (( I ↾ 𝐴) “ 𝐴) = 𝐴)
46	44, 45	ax-mp 5	. . . 4 ⊢ (( I ↾ 𝐴) “ 𝐴) = 𝐴
47	42, 43, 46	3eqtr3g 2818	. . 3 ⊢ (((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥)) → (𝐹 “ (𝑓 “ 𝐴)) = 𝐴)
48		imaeq2 6047	. . . . 5 ⊢ (𝑐 = (𝑓 “ 𝐴) → (𝐹 “ 𝑐) = (𝐹 “ (𝑓 “ 𝐴)))
49	48	eqeq1d 2762	. . . 4 ⊢ (𝑐 = (𝑓 “ 𝐴) → ((𝐹 “ 𝑐) = 𝐴 ↔ (𝐹 “ (𝑓 “ 𝐴)) = 𝐴))
50	49	rspcev 3576	. . 3 ⊢ (((𝑓 “ 𝐴) ∈ (𝒫 𝐵 ∩ Fin) ∧ (𝐹 “ (𝑓 “ 𝐴)) = 𝐴) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)(𝐹 “ 𝑐) = 𝐴)
51	22, 47, 50	syl2anc 596	. 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 1967	1 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)(𝐹 “ 𝑐) = 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086   ⊆ wss 3895   ∩ cin 4089  𝒫 cpw 4556   I cid 5544  ran crn 5651   ↾ cres 5652   “ cima 5653   ∘ ccom 5654  Fun wfun 6533   Fn wfn 6534  ⟶wf 6535  ‘cfv 6539  Fincfn 8977

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-om 7879  df-1o 8480  df-en 8978  df-dom 8979  df-fin 8981

This theorem is used by:  fodomfi2  10143  cmpfi  23732  elrfirn  43800  lmhmfgsplit  44182  hbtlem6  44225