Theorem fipreima 9358

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 1139	. . 3 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) → 𝐴 ∈ Fin)
2		dfss3 3971	. . . . . 6 ⊢ (𝐴 ⊆ ran 𝐹 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ ran 𝐹)
3		fvelrnb 6953	. . . . . . 7 ⊢ (𝐹 Fn 𝐵 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥))
4	3	ralbidv 3178	. . . . . 6 ⊢ (𝐹 Fn 𝐵 → (∀𝑥 ∈ 𝐴 𝑥 ∈ ran 𝐹 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥))
5	2, 4	bitrid 283	. . . . 5 ⊢ (𝐹 Fn 𝐵 → (𝐴 ⊆ ran 𝐹 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥))
6	5	biimpa 478	. . . 4 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥)
7	6	3adant3 1133	. . 3 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥)
8		fveqeq2 6901	. . . 4 ⊢ (𝑦 = (𝑓‘𝑥) → ((𝐹‘𝑦) = 𝑥 ↔ (𝐹‘(𝑓‘𝑥)) = 𝑥))
9	8	ac6sfi 9287	. . 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 585	. 2 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥))
11		fimass 6739	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → (𝑓 “ 𝐴) ⊆ 𝐵)
12		vex 3479	. . . . . . . 8 ⊢ 𝑓 ∈ V
13	12	imaex 7907	. . . . . . 7 ⊢ (𝑓 “ 𝐴) ∈ V
14	13	elpw 4607	. . . . . 6 ⊢ ((𝑓 “ 𝐴) ∈ 𝒫 𝐵 ↔ (𝑓 “ 𝐴) ⊆ 𝐵)
15	11, 14	sylibr 233	. . . . 5 ⊢ (𝑓:𝐴⟶𝐵 → (𝑓 “ 𝐴) ∈ 𝒫 𝐵)
16	15	ad2antrl 727	. . . 4 ⊢ (((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥)) → (𝑓 “ 𝐴) ∈ 𝒫 𝐵)
17		ffun 6721	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → Fun 𝑓)
18	17	ad2antrl 727	. . . . 5 ⊢ (((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥)) → Fun 𝑓)
19		simpl3 1194	. . . . 5 ⊢ (((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥)) → 𝐴 ∈ Fin)
20		imafi 9175	. . . . 5 ⊢ ((Fun 𝑓 ∧ 𝐴 ∈ Fin) → (𝑓 “ 𝐴) ∈ Fin)
21	18, 19, 20	syl2anc 585	. . . 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 4195	. . 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 6991	. . . . . . . . . . 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 483	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (( I ↾ 𝐴)‘𝑥) = 𝑥)
26	23, 25	eqeq12d 2749	. . . . . . . . . 10 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (((𝐹 ∘ 𝑓)‘𝑥) = (( I ↾ 𝐴)‘𝑥) ↔ (𝐹‘(𝑓‘𝑥)) = 𝑥))
27	26	ralbidva 3176	. . . . . . . . 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 483	. . . . . . 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 456	. . . . . 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 1192	. . . . . . . 8 ⊢ (((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥)) → 𝐹 Fn 𝐵)
32		ffn 6718	. . . . . . . . 9 ⊢ (𝑓:𝐴⟶𝐵 → 𝑓 Fn 𝐴)
33	32	ad2antrl 727	. . . . . . . 8 ⊢ (((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥)) → 𝑓 Fn 𝐴)
34		frn 6725	. . . . . . . . 9 ⊢ (𝑓:𝐴⟶𝐵 → ran 𝑓 ⊆ 𝐵)
35	34	ad2antrl 727	. . . . . . . 8 ⊢ (((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥)) → ran 𝑓 ⊆ 𝐵)
36		fnco 6668	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐵 ∧ 𝑓 Fn 𝐴 ∧ ran 𝑓 ⊆ 𝐵) → (𝐹 ∘ 𝑓) Fn 𝐴)
37	31, 33, 35, 36	syl3anc 1372	. . . . . . 7 ⊢ (((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥)) → (𝐹 ∘ 𝑓) Fn 𝐴)
38		fnresi 6680	. . . . . . 7 ⊢ ( I ↾ 𝐴) Fn 𝐴
39		eqfnfv 7033	. . . . . . 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 587	. . . . . 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 257	. . . . 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 6059	. . . 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 6251	. . . 4 ⊢ ((𝐹 ∘ 𝑓) “ 𝐴) = (𝐹 “ (𝑓 “ 𝐴))
44		ssid 4005	. . . . 5 ⊢ 𝐴 ⊆ 𝐴
45		resiima 6076	. . . . 5 ⊢ (𝐴 ⊆ 𝐴 → (( I ↾ 𝐴) “ 𝐴) = 𝐴)
46	44, 45	ax-mp 5	. . . 4 ⊢ (( I ↾ 𝐴) “ 𝐴) = 𝐴
47	42, 43, 46	3eqtr3g 2796	. . 3 ⊢ (((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝑓‘𝑥)) = 𝑥)) → (𝐹 “ (𝑓 “ 𝐴)) = 𝐴)
48		imaeq2 6056	. . . . 5 ⊢ (𝑐 = (𝑓 “ 𝐴) → (𝐹 “ 𝑐) = (𝐹 “ (𝑓 “ 𝐴)))
49	48	eqeq1d 2735	. . . 4 ⊢ (𝑐 = (𝑓 “ 𝐴) → ((𝐹 “ 𝑐) = 𝐴 ↔ (𝐹 “ (𝑓 “ 𝐴)) = 𝐴))
50	49	rspcev 3613	. . 3 ⊢ (((𝑓 “ 𝐴) ∈ (𝒫 𝐵 ∩ Fin) ∧ (𝐹 “ (𝑓 “ 𝐴)) = 𝐴) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)(𝐹 “ 𝑐) = 𝐴)
51	22, 47, 50	syl2anc 585	. 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 1939	1 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)(𝐹 “ 𝑐) = 𝐴)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  ∃wex 1782   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071   ∩ cin 3948   ⊆ wss 3949  𝒫 cpw 4603   I cid 5574  ran crn 5678   ↾ cres 5679   “ cima 5680   ∘ ccom 5681  Fun wfun 6538   Fn wfn 6539  ⟶wf 6540  ‘cfv 6544  Fincfn 8939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-om 7856  df-1o 8466  df-en 8940  df-fin 8943

This theorem is referenced by:  fodomfi2  10055  cmpfi  22912  elrfirn  41433  lmhmfgsplit  41828  hbtlem6  41871