Theorem fipreima 9311

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 3926	. . . . . 6 ⊢ (𝐴 ⊆ ran 𝐹 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ ran 𝐹)
3		fvelrnb 6941	. . . . . . 7 ⊢ (𝐹 Fn 𝐵 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥))
4	3	ralbidv 3188	. . . . . 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 481	. . . 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 6890	. . . 4 ⊢ (𝑦 = (𝑓‘𝑥) → ((𝐹‘𝑦) = 𝑥 ↔ (𝐹‘(𝑓‘𝑥)) = 𝑥))
9	8	ac6sfi 9240	. . 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 595	. 2 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥))
11		fimass 6726	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → (𝑓 “ 𝐴) ⊆ 𝐵)
12		vex 3459	. . . . . . . 8 ⊢ 𝑓 ∈ V
13	12	imaex 7907	. . . . . . 7 ⊢ (𝑓 “ 𝐴) ∈ V
14	13	elpw 4566	. . . . . 6 ⊢ ((𝑓 “ 𝐴) ∈ 𝒫 𝐵 ↔ (𝑓 “ 𝐴) ⊆ 𝐵)
15	11, 14	sylibr 237	. . . . 5 ⊢ (𝑓:𝐴⟶𝐵 → (𝑓 “ 𝐴) ∈ 𝒫 𝐵)
16	15	ad2antrl 740	. . . 4 ⊢ (((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥)) → (𝑓 “ 𝐴) ∈ 𝒫 𝐵)
17		ffun 6708	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → Fun 𝑓)
18	17	ad2antrl 740	. . . . 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 9271	. . . . 5 ⊢ ((Fun 𝑓 ∧ 𝐴 ∈ Fin) → (𝑓 “ 𝐴) ∈ Fin)
21	18, 19, 20	syl2anc 595	. . . 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 4153	. . 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 6981	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘ 𝑓)‘𝑥) = (𝐹‘(𝑓‘𝑥)))
24		fvresi 7171	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
25	24	adantl 486	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (( I ↾ 𝐴)‘𝑥) = 𝑥)
26	23, 25	eqeq12d 2779	. . . . . . . . . 10 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (((𝐹 ∘ 𝑓)‘𝑥) = (( I ↾ 𝐴)‘𝑥) ↔ (𝐹‘(𝑓‘𝑥)) = 𝑥))
27	26	ralbidva 3186	. . . . . . . . 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 486	. . . . . . 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 459	. . . . . 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 6705	. . . . . . . . 9 ⊢ (𝑓:𝐴⟶𝐵 → 𝑓 Fn 𝐴)
33	32	ad2antrl 740	. . . . . . . 8 ⊢ (((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥)) → 𝑓 Fn 𝐴)
34		frn 6713	. . . . . . . . 9 ⊢ (𝑓:𝐴⟶𝐵 → ran 𝑓 ⊆ 𝐵)
35	34	ad2antrl 740	. . . . . . . 8 ⊢ (((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥)) → ran 𝑓 ⊆ 𝐵)
36		fnco 6653	. . . . . . . 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 6664	. . . . . . 7 ⊢ ( I ↾ 𝐴) Fn 𝐴
39		eqfnfv 7025	. . . . . . 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 597	. . . . . 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 6061	. . . 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 3959	. . . . 5 ⊢ 𝐴 ⊆ 𝐴
45		resiima 6078	. . . . 5 ⊢ (𝐴 ⊆ 𝐴 → (( I ↾ 𝐴) “ 𝐴) = 𝐴)
46	44, 45	ax-mp 5	. . . 4 ⊢ (( I ↾ 𝐴) “ 𝐴) = 𝐴
47	42, 43, 46	3eqtr3g 2821	. . 3 ⊢ (((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥)) → (𝐹 “ (𝑓 “ 𝐴)) = 𝐴)
48		imaeq2 6058	. . . . 5 ⊢ (𝑐 = (𝑓 “ 𝐴) → (𝐹 “ 𝑐) = (𝐹 “ (𝑓 “ 𝐴)))
49	48	eqeq1d 2765	. . . 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 595	. 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 1965	1 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)(𝐹 “ 𝑐) = 𝐴)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570  ∃wex 1809   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089   ∩ cin 3904   ⊆ wss 3905  𝒫 cpw 4562   I cid 5555  ran crn 5662   ↾ cres 5663   “ cima 5664   ∘ ccom 5665  Fun wfun 6530   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  Fincfn 8939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-om 7859  df-1o 8449  df-en 8940  df-dom 8941  df-fin 8943

This theorem is referenced by:  fodomfi2  10040  cmpfi  23565  elrfirn  43426  lmhmfgsplit  43813  hbtlem6  43856