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Theorem fipreima 8313
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 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)(𝐹𝑐) = 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.
StepHypRef Expression
1 simp3 1083 . . 3 ((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) → 𝐴 ∈ Fin)
2 dfss3 3625 . . . . . 6 (𝐴 ⊆ ran 𝐹 ↔ ∀𝑥𝐴 𝑥 ∈ ran 𝐹)
3 fvelrnb 6282 . . . . . . 7 (𝐹 Fn 𝐵 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑦𝐵 (𝐹𝑦) = 𝑥))
43ralbidv 3015 . . . . . 6 (𝐹 Fn 𝐵 → (∀𝑥𝐴 𝑥 ∈ ran 𝐹 ↔ ∀𝑥𝐴𝑦𝐵 (𝐹𝑦) = 𝑥))
52, 4syl5bb 272 . . . . 5 (𝐹 Fn 𝐵 → (𝐴 ⊆ ran 𝐹 ↔ ∀𝑥𝐴𝑦𝐵 (𝐹𝑦) = 𝑥))
65biimpa 500 . . . 4 ((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹) → ∀𝑥𝐴𝑦𝐵 (𝐹𝑦) = 𝑥)
763adant3 1101 . . 3 ((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) → ∀𝑥𝐴𝑦𝐵 (𝐹𝑦) = 𝑥)
8 fveq2 6229 . . . . 5 (𝑦 = (𝑓𝑥) → (𝐹𝑦) = (𝐹‘(𝑓𝑥)))
98eqeq1d 2653 . . . 4 (𝑦 = (𝑓𝑥) → ((𝐹𝑦) = 𝑥 ↔ (𝐹‘(𝑓𝑥)) = 𝑥))
109ac6sfi 8245 . . 3 ((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 (𝐹𝑦) = 𝑥) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥))
111, 7, 10syl2anc 694 . 2 ((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥))
12 fimass 6119 . . . . . 6 (𝑓:𝐴𝐵 → (𝑓𝐴) ⊆ 𝐵)
13 vex 3234 . . . . . . . 8 𝑓 ∈ V
1413imaex 7146 . . . . . . 7 (𝑓𝐴) ∈ V
1514elpw 4197 . . . . . 6 ((𝑓𝐴) ∈ 𝒫 𝐵 ↔ (𝑓𝐴) ⊆ 𝐵)
1612, 15sylibr 224 . . . . 5 (𝑓:𝐴𝐵 → (𝑓𝐴) ∈ 𝒫 𝐵)
1716ad2antrl 764 . . . 4 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → (𝑓𝐴) ∈ 𝒫 𝐵)
18 ffun 6086 . . . . . 6 (𝑓:𝐴𝐵 → Fun 𝑓)
1918ad2antrl 764 . . . . 5 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → Fun 𝑓)
20 simpl3 1086 . . . . 5 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → 𝐴 ∈ Fin)
21 imafi 8300 . . . . 5 ((Fun 𝑓𝐴 ∈ Fin) → (𝑓𝐴) ∈ Fin)
2219, 20, 21syl2anc 694 . . . 4 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → (𝑓𝐴) ∈ Fin)
2317, 22elind 3831 . . 3 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → (𝑓𝐴) ∈ (𝒫 𝐵 ∩ Fin))
24 fvco3 6314 . . . . . . . . . . 11 ((𝑓:𝐴𝐵𝑥𝐴) → ((𝐹𝑓)‘𝑥) = (𝐹‘(𝑓𝑥)))
25 fvresi 6480 . . . . . . . . . . . 12 (𝑥𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
2625adantl 481 . . . . . . . . . . 11 ((𝑓:𝐴𝐵𝑥𝐴) → (( I ↾ 𝐴)‘𝑥) = 𝑥)
2724, 26eqeq12d 2666 . . . . . . . . . 10 ((𝑓:𝐴𝐵𝑥𝐴) → (((𝐹𝑓)‘𝑥) = (( I ↾ 𝐴)‘𝑥) ↔ (𝐹‘(𝑓𝑥)) = 𝑥))
2827ralbidva 3014 . . . . . . . . 9 (𝑓:𝐴𝐵 → (∀𝑥𝐴 ((𝐹𝑓)‘𝑥) = (( I ↾ 𝐴)‘𝑥) ↔ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥))
2928biimprd 238 . . . . . . . 8 (𝑓:𝐴𝐵 → (∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥 → ∀𝑥𝐴 ((𝐹𝑓)‘𝑥) = (( I ↾ 𝐴)‘𝑥)))
3029adantl 481 . . . . . . 7 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ 𝑓:𝐴𝐵) → (∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥 → ∀𝑥𝐴 ((𝐹𝑓)‘𝑥) = (( I ↾ 𝐴)‘𝑥)))
3130impr 648 . . . . . 6 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → ∀𝑥𝐴 ((𝐹𝑓)‘𝑥) = (( I ↾ 𝐴)‘𝑥))
32 simpl1 1084 . . . . . . . 8 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → 𝐹 Fn 𝐵)
33 ffn 6083 . . . . . . . . 9 (𝑓:𝐴𝐵𝑓 Fn 𝐴)
3433ad2antrl 764 . . . . . . . 8 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → 𝑓 Fn 𝐴)
35 frn 6091 . . . . . . . . 9 (𝑓:𝐴𝐵 → ran 𝑓𝐵)
3635ad2antrl 764 . . . . . . . 8 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → ran 𝑓𝐵)
37 fnco 6037 . . . . . . . 8 ((𝐹 Fn 𝐵𝑓 Fn 𝐴 ∧ ran 𝑓𝐵) → (𝐹𝑓) Fn 𝐴)
3832, 34, 36, 37syl3anc 1366 . . . . . . 7 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → (𝐹𝑓) Fn 𝐴)
39 fnresi 6046 . . . . . . 7 ( I ↾ 𝐴) Fn 𝐴
40 eqfnfv 6351 . . . . . . 7 (((𝐹𝑓) Fn 𝐴 ∧ ( I ↾ 𝐴) Fn 𝐴) → ((𝐹𝑓) = ( I ↾ 𝐴) ↔ ∀𝑥𝐴 ((𝐹𝑓)‘𝑥) = (( I ↾ 𝐴)‘𝑥)))
4138, 39, 40sylancl 695 . . . . . 6 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → ((𝐹𝑓) = ( I ↾ 𝐴) ↔ ∀𝑥𝐴 ((𝐹𝑓)‘𝑥) = (( I ↾ 𝐴)‘𝑥)))
4231, 41mpbird 247 . . . . 5 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → (𝐹𝑓) = ( I ↾ 𝐴))
4342imaeq1d 5500 . . . 4 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → ((𝐹𝑓) “ 𝐴) = (( I ↾ 𝐴) “ 𝐴))
44 imaco 5678 . . . 4 ((𝐹𝑓) “ 𝐴) = (𝐹 “ (𝑓𝐴))
45 ssid 3657 . . . . 5 𝐴𝐴
46 resiima 5515 . . . . 5 (𝐴𝐴 → (( I ↾ 𝐴) “ 𝐴) = 𝐴)
4745, 46ax-mp 5 . . . 4 (( I ↾ 𝐴) “ 𝐴) = 𝐴
4843, 44, 473eqtr3g 2708 . . 3 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → (𝐹 “ (𝑓𝐴)) = 𝐴)
49 imaeq2 5497 . . . . 5 (𝑐 = (𝑓𝐴) → (𝐹𝑐) = (𝐹 “ (𝑓𝐴)))
5049eqeq1d 2653 . . . 4 (𝑐 = (𝑓𝐴) → ((𝐹𝑐) = 𝐴 ↔ (𝐹 “ (𝑓𝐴)) = 𝐴))
5150rspcev 3340 . . 3 (((𝑓𝐴) ∈ (𝒫 𝐵 ∩ Fin) ∧ (𝐹 “ (𝑓𝐴)) = 𝐴) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)(𝐹𝑐) = 𝐴)
5223, 48, 51syl2anc 694 . 2 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)(𝐹𝑐) = 𝐴)
5311, 52exlimddv 1903 1 ((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)(𝐹𝑐) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  wral 2941  wrex 2942  cin 3606  wss 3607  𝒫 cpw 4191   I cid 5052  ran crn 5144  cres 5145  cima 5146  ccom 5147  Fun wfun 5920   Fn wfn 5921  wf 5922  cfv 5926  Fincfn 7997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-1o 7605  df-er 7787  df-en 7998  df-dom 7999  df-fin 8001
This theorem is referenced by:  fodomfi2  8921  cmpfi  21259  elrfirn  37575  lmhmfgsplit  37973  hbtlem6  38016
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