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Theorem fipreima 8155
 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 1056 . . 3 ((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) → 𝐴 ∈ Fin)
2 dfss3 3558 . . . . . 6 (𝐴 ⊆ ran 𝐹 ↔ ∀𝑥𝐴 𝑥 ∈ ran 𝐹)
3 fvelrnb 6153 . . . . . . 7 (𝐹 Fn 𝐵 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑦𝐵 (𝐹𝑦) = 𝑥))
43ralbidv 2969 . . . . . 6 (𝐹 Fn 𝐵 → (∀𝑥𝐴 𝑥 ∈ ran 𝐹 ↔ ∀𝑥𝐴𝑦𝐵 (𝐹𝑦) = 𝑥))
52, 4syl5bb 271 . . . . 5 (𝐹 Fn 𝐵 → (𝐴 ⊆ ran 𝐹 ↔ ∀𝑥𝐴𝑦𝐵 (𝐹𝑦) = 𝑥))
65biimpa 500 . . . 4 ((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹) → ∀𝑥𝐴𝑦𝐵 (𝐹𝑦) = 𝑥)
763adant3 1074 . . 3 ((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) → ∀𝑥𝐴𝑦𝐵 (𝐹𝑦) = 𝑥)
8 fveq2 6103 . . . . 5 (𝑦 = (𝑓𝑥) → (𝐹𝑦) = (𝐹‘(𝑓𝑥)))
98eqeq1d 2612 . . . 4 (𝑦 = (𝑓𝑥) → ((𝐹𝑦) = 𝑥 ↔ (𝐹‘(𝑓𝑥)) = 𝑥))
109ac6sfi 8089 . . 3 ((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 (𝐹𝑦) = 𝑥) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥))
111, 7, 10syl2anc 691 . 2 ((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥))
12 fimass 5994 . . . . . 6 (𝑓:𝐴𝐵 → (𝑓𝐴) ⊆ 𝐵)
13 vex 3176 . . . . . . . 8 𝑓 ∈ V
1413imaex 6996 . . . . . . 7 (𝑓𝐴) ∈ V
1514elpw 4114 . . . . . 6 ((𝑓𝐴) ∈ 𝒫 𝐵 ↔ (𝑓𝐴) ⊆ 𝐵)
1612, 15sylibr 223 . . . . 5 (𝑓:𝐴𝐵 → (𝑓𝐴) ∈ 𝒫 𝐵)
1716ad2antrl 760 . . . 4 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → (𝑓𝐴) ∈ 𝒫 𝐵)
18 ffun 5961 . . . . . 6 (𝑓:𝐴𝐵 → Fun 𝑓)
1918ad2antrl 760 . . . . 5 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → Fun 𝑓)
20 simpl3 1059 . . . . 5 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → 𝐴 ∈ Fin)
21 imafi 8142 . . . . 5 ((Fun 𝑓𝐴 ∈ Fin) → (𝑓𝐴) ∈ Fin)
2219, 20, 21syl2anc 691 . . . 4 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → (𝑓𝐴) ∈ Fin)
2317, 22elind 3760 . . 3 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → (𝑓𝐴) ∈ (𝒫 𝐵 ∩ Fin))
24 fvco3 6185 . . . . . . . . . . 11 ((𝑓:𝐴𝐵𝑥𝐴) → ((𝐹𝑓)‘𝑥) = (𝐹‘(𝑓𝑥)))
25 fvresi 6344 . . . . . . . . . . . 12 (𝑥𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
2625adantl 481 . . . . . . . . . . 11 ((𝑓:𝐴𝐵𝑥𝐴) → (( I ↾ 𝐴)‘𝑥) = 𝑥)
2724, 26eqeq12d 2625 . . . . . . . . . 10 ((𝑓:𝐴𝐵𝑥𝐴) → (((𝐹𝑓)‘𝑥) = (( I ↾ 𝐴)‘𝑥) ↔ (𝐹‘(𝑓𝑥)) = 𝑥))
2827ralbidva 2968 . . . . . . . . 9 (𝑓:𝐴𝐵 → (∀𝑥𝐴 ((𝐹𝑓)‘𝑥) = (( I ↾ 𝐴)‘𝑥) ↔ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥))
2928biimprd 237 . . . . . . . 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 647 . . . . . 6 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → ∀𝑥𝐴 ((𝐹𝑓)‘𝑥) = (( I ↾ 𝐴)‘𝑥))
32 simpl1 1057 . . . . . . . 8 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → 𝐹 Fn 𝐵)
33 ffn 5958 . . . . . . . . 9 (𝑓:𝐴𝐵𝑓 Fn 𝐴)
3433ad2antrl 760 . . . . . . . 8 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → 𝑓 Fn 𝐴)
35 frn 5966 . . . . . . . . 9 (𝑓:𝐴𝐵 → ran 𝑓𝐵)
3635ad2antrl 760 . . . . . . . 8 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → ran 𝑓𝐵)
37 fnco 5913 . . . . . . . 8 ((𝐹 Fn 𝐵𝑓 Fn 𝐴 ∧ ran 𝑓𝐵) → (𝐹𝑓) Fn 𝐴)
3832, 34, 36, 37syl3anc 1318 . . . . . . 7 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → (𝐹𝑓) Fn 𝐴)
39 fnresi 5922 . . . . . . 7 ( I ↾ 𝐴) Fn 𝐴
40 eqfnfv 6219 . . . . . . 7 (((𝐹𝑓) Fn 𝐴 ∧ ( I ↾ 𝐴) Fn 𝐴) → ((𝐹𝑓) = ( I ↾ 𝐴) ↔ ∀𝑥𝐴 ((𝐹𝑓)‘𝑥) = (( I ↾ 𝐴)‘𝑥)))
4138, 39, 40sylancl 693 . . . . . 6 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → ((𝐹𝑓) = ( I ↾ 𝐴) ↔ ∀𝑥𝐴 ((𝐹𝑓)‘𝑥) = (( I ↾ 𝐴)‘𝑥)))
4231, 41mpbird 246 . . . . 5 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → (𝐹𝑓) = ( I ↾ 𝐴))
4342imaeq1d 5384 . . . 4 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → ((𝐹𝑓) “ 𝐴) = (( I ↾ 𝐴) “ 𝐴))
44 imaco 5557 . . . 4 ((𝐹𝑓) “ 𝐴) = (𝐹 “ (𝑓𝐴))
45 ssid 3587 . . . . 5 𝐴𝐴
46 resiima 5399 . . . . 5 (𝐴𝐴 → (( I ↾ 𝐴) “ 𝐴) = 𝐴)
4745, 46ax-mp 5 . . . 4 (( I ↾ 𝐴) “ 𝐴) = 𝐴
4843, 44, 473eqtr3g 2667 . . 3 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → (𝐹 “ (𝑓𝐴)) = 𝐴)
49 imaeq2 5381 . . . . 5 (𝑐 = (𝑓𝐴) → (𝐹𝑐) = (𝐹 “ (𝑓𝐴)))
5049eqeq1d 2612 . . . 4 (𝑐 = (𝑓𝐴) → ((𝐹𝑐) = 𝐴 ↔ (𝐹 “ (𝑓𝐴)) = 𝐴))
5150rspcev 3282 . . 3 (((𝑓𝐴) ∈ (𝒫 𝐵 ∩ Fin) ∧ (𝐹 “ (𝑓𝐴)) = 𝐴) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)(𝐹𝑐) = 𝐴)
5223, 48, 51syl2anc 691 . 2 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)(𝐹𝑐) = 𝐴)
5311, 52exlimddv 1850 1 ((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)(𝐹𝑐) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108   I cid 4948  ran crn 5039   ↾ cres 5040   “ cima 5041   ∘ ccom 5042  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  Fincfn 7841 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-1o 7447  df-er 7629  df-en 7842  df-dom 7843  df-fin 7845 This theorem is referenced by:  fodomfi2  8766  cmpfi  21021  elrfirn  36276  lmhmfgsplit  36674  hbtlem6  36718
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