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Theorem fipreima 9396
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 1137 . . 3 ((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) → 𝐴 ∈ Fin)
2 dfss3 3984 . . . . . 6 (𝐴 ⊆ ran 𝐹 ↔ ∀𝑥𝐴 𝑥 ∈ ran 𝐹)
3 fvelrnb 6969 . . . . . . 7 (𝐹 Fn 𝐵 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑦𝐵 (𝐹𝑦) = 𝑥))
43ralbidv 3176 . . . . . 6 (𝐹 Fn 𝐵 → (∀𝑥𝐴 𝑥 ∈ ran 𝐹 ↔ ∀𝑥𝐴𝑦𝐵 (𝐹𝑦) = 𝑥))
52, 4bitrid 283 . . . . 5 (𝐹 Fn 𝐵 → (𝐴 ⊆ ran 𝐹 ↔ ∀𝑥𝐴𝑦𝐵 (𝐹𝑦) = 𝑥))
65biimpa 476 . . . 4 ((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹) → ∀𝑥𝐴𝑦𝐵 (𝐹𝑦) = 𝑥)
763adant3 1131 . . 3 ((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) → ∀𝑥𝐴𝑦𝐵 (𝐹𝑦) = 𝑥)
8 fveqeq2 6916 . . . 4 (𝑦 = (𝑓𝑥) → ((𝐹𝑦) = 𝑥 ↔ (𝐹‘(𝑓𝑥)) = 𝑥))
98ac6sfi 9318 . . 3 ((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 (𝐹𝑦) = 𝑥) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥))
101, 7, 9syl2anc 584 . 2 ((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥))
11 fimass 6757 . . . . . 6 (𝑓:𝐴𝐵 → (𝑓𝐴) ⊆ 𝐵)
12 vex 3482 . . . . . . . 8 𝑓 ∈ V
1312imaex 7937 . . . . . . 7 (𝑓𝐴) ∈ V
1413elpw 4609 . . . . . 6 ((𝑓𝐴) ∈ 𝒫 𝐵 ↔ (𝑓𝐴) ⊆ 𝐵)
1511, 14sylibr 234 . . . . 5 (𝑓:𝐴𝐵 → (𝑓𝐴) ∈ 𝒫 𝐵)
1615ad2antrl 728 . . . 4 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → (𝑓𝐴) ∈ 𝒫 𝐵)
17 ffun 6740 . . . . . 6 (𝑓:𝐴𝐵 → Fun 𝑓)
1817ad2antrl 728 . . . . 5 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → Fun 𝑓)
19 simpl3 1192 . . . . 5 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → 𝐴 ∈ Fin)
20 imafi 9351 . . . . 5 ((Fun 𝑓𝐴 ∈ Fin) → (𝑓𝐴) ∈ Fin)
2118, 19, 20syl2anc 584 . . . 4 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → (𝑓𝐴) ∈ Fin)
2216, 21elind 4210 . . 3 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → (𝑓𝐴) ∈ (𝒫 𝐵 ∩ Fin))
23 fvco3 7008 . . . . . . . . . . 11 ((𝑓:𝐴𝐵𝑥𝐴) → ((𝐹𝑓)‘𝑥) = (𝐹‘(𝑓𝑥)))
24 fvresi 7193 . . . . . . . . . . . 12 (𝑥𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
2524adantl 481 . . . . . . . . . . 11 ((𝑓:𝐴𝐵𝑥𝐴) → (( I ↾ 𝐴)‘𝑥) = 𝑥)
2623, 25eqeq12d 2751 . . . . . . . . . 10 ((𝑓:𝐴𝐵𝑥𝐴) → (((𝐹𝑓)‘𝑥) = (( I ↾ 𝐴)‘𝑥) ↔ (𝐹‘(𝑓𝑥)) = 𝑥))
2726ralbidva 3174 . . . . . . . . 9 (𝑓:𝐴𝐵 → (∀𝑥𝐴 ((𝐹𝑓)‘𝑥) = (( I ↾ 𝐴)‘𝑥) ↔ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥))
2827biimprd 248 . . . . . . . 8 (𝑓:𝐴𝐵 → (∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥 → ∀𝑥𝐴 ((𝐹𝑓)‘𝑥) = (( I ↾ 𝐴)‘𝑥)))
2928adantl 481 . . . . . . 7 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ 𝑓:𝐴𝐵) → (∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥 → ∀𝑥𝐴 ((𝐹𝑓)‘𝑥) = (( I ↾ 𝐴)‘𝑥)))
3029impr 454 . . . . . 6 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → ∀𝑥𝐴 ((𝐹𝑓)‘𝑥) = (( I ↾ 𝐴)‘𝑥))
31 simpl1 1190 . . . . . . . 8 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → 𝐹 Fn 𝐵)
32 ffn 6737 . . . . . . . . 9 (𝑓:𝐴𝐵𝑓 Fn 𝐴)
3332ad2antrl 728 . . . . . . . 8 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → 𝑓 Fn 𝐴)
34 frn 6744 . . . . . . . . 9 (𝑓:𝐴𝐵 → ran 𝑓𝐵)
3534ad2antrl 728 . . . . . . . 8 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → ran 𝑓𝐵)
36 fnco 6687 . . . . . . . 8 ((𝐹 Fn 𝐵𝑓 Fn 𝐴 ∧ ran 𝑓𝐵) → (𝐹𝑓) Fn 𝐴)
3731, 33, 35, 36syl3anc 1370 . . . . . . 7 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → (𝐹𝑓) Fn 𝐴)
38 fnresi 6698 . . . . . . 7 ( I ↾ 𝐴) Fn 𝐴
39 eqfnfv 7051 . . . . . . 7 (((𝐹𝑓) Fn 𝐴 ∧ ( I ↾ 𝐴) Fn 𝐴) → ((𝐹𝑓) = ( I ↾ 𝐴) ↔ ∀𝑥𝐴 ((𝐹𝑓)‘𝑥) = (( I ↾ 𝐴)‘𝑥)))
4037, 38, 39sylancl 586 . . . . . 6 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → ((𝐹𝑓) = ( I ↾ 𝐴) ↔ ∀𝑥𝐴 ((𝐹𝑓)‘𝑥) = (( I ↾ 𝐴)‘𝑥)))
4130, 40mpbird 257 . . . . 5 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → (𝐹𝑓) = ( I ↾ 𝐴))
4241imaeq1d 6079 . . . 4 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → ((𝐹𝑓) “ 𝐴) = (( I ↾ 𝐴) “ 𝐴))
43 imaco 6273 . . . 4 ((𝐹𝑓) “ 𝐴) = (𝐹 “ (𝑓𝐴))
44 ssid 4018 . . . . 5 𝐴𝐴
45 resiima 6096 . . . . 5 (𝐴𝐴 → (( I ↾ 𝐴) “ 𝐴) = 𝐴)
4644, 45ax-mp 5 . . . 4 (( I ↾ 𝐴) “ 𝐴) = 𝐴
4742, 43, 463eqtr3g 2798 . . 3 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → (𝐹 “ (𝑓𝐴)) = 𝐴)
48 imaeq2 6076 . . . . 5 (𝑐 = (𝑓𝐴) → (𝐹𝑐) = (𝐹 “ (𝑓𝐴)))
4948eqeq1d 2737 . . . 4 (𝑐 = (𝑓𝐴) → ((𝐹𝑐) = 𝐴 ↔ (𝐹 “ (𝑓𝐴)) = 𝐴))
5049rspcev 3622 . . 3 (((𝑓𝐴) ∈ (𝒫 𝐵 ∩ Fin) ∧ (𝐹 “ (𝑓𝐴)) = 𝐴) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)(𝐹𝑐) = 𝐴)
5122, 47, 50syl2anc 584 . 2 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)(𝐹𝑐) = 𝐴)
5210, 51exlimddv 1933 1 ((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)(𝐹𝑐) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  wral 3059  wrex 3068  cin 3962  wss 3963  𝒫 cpw 4605   I cid 5582  ran crn 5690  cres 5691  cima 5692  ccom 5693  Fun wfun 6557   Fn wfn 6558  wf 6559  cfv 6563  Fincfn 8984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-1o 8505  df-en 8985  df-dom 8986  df-fin 8988
This theorem is referenced by:  fodomfi2  10098  cmpfi  23432  elrfirn  42683  lmhmfgsplit  43075  hbtlem6  43118
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