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Theorem fipreima 8818
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 1135 . . 3 ((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) → 𝐴 ∈ Fin)
2 dfss3 3906 . . . . . 6 (𝐴 ⊆ ran 𝐹 ↔ ∀𝑥𝐴 𝑥 ∈ ran 𝐹)
3 fvelrnb 6705 . . . . . . 7 (𝐹 Fn 𝐵 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑦𝐵 (𝐹𝑦) = 𝑥))
43ralbidv 3165 . . . . . 6 (𝐹 Fn 𝐵 → (∀𝑥𝐴 𝑥 ∈ ran 𝐹 ↔ ∀𝑥𝐴𝑦𝐵 (𝐹𝑦) = 𝑥))
52, 4syl5bb 286 . . . . 5 (𝐹 Fn 𝐵 → (𝐴 ⊆ ran 𝐹 ↔ ∀𝑥𝐴𝑦𝐵 (𝐹𝑦) = 𝑥))
65biimpa 480 . . . 4 ((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹) → ∀𝑥𝐴𝑦𝐵 (𝐹𝑦) = 𝑥)
763adant3 1129 . . 3 ((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) → ∀𝑥𝐴𝑦𝐵 (𝐹𝑦) = 𝑥)
8 fveqeq2 6658 . . . 4 (𝑦 = (𝑓𝑥) → ((𝐹𝑦) = 𝑥 ↔ (𝐹‘(𝑓𝑥)) = 𝑥))
98ac6sfi 8750 . . 3 ((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 (𝐹𝑦) = 𝑥) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥))
101, 7, 9syl2anc 587 . 2 ((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥))
11 fimass 6533 . . . . . 6 (𝑓:𝐴𝐵 → (𝑓𝐴) ⊆ 𝐵)
12 vex 3447 . . . . . . . 8 𝑓 ∈ V
1312imaex 7607 . . . . . . 7 (𝑓𝐴) ∈ V
1413elpw 4504 . . . . . 6 ((𝑓𝐴) ∈ 𝒫 𝐵 ↔ (𝑓𝐴) ⊆ 𝐵)
1511, 14sylibr 237 . . . . 5 (𝑓:𝐴𝐵 → (𝑓𝐴) ∈ 𝒫 𝐵)
1615ad2antrl 727 . . . 4 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → (𝑓𝐴) ∈ 𝒫 𝐵)
17 ffun 6494 . . . . . 6 (𝑓:𝐴𝐵 → Fun 𝑓)
1817ad2antrl 727 . . . . 5 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → Fun 𝑓)
19 simpl3 1190 . . . . 5 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → 𝐴 ∈ Fin)
20 imafi 8805 . . . . 5 ((Fun 𝑓𝐴 ∈ Fin) → (𝑓𝐴) ∈ Fin)
2118, 19, 20syl2anc 587 . . . 4 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → (𝑓𝐴) ∈ Fin)
2216, 21elind 4124 . . 3 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → (𝑓𝐴) ∈ (𝒫 𝐵 ∩ Fin))
23 fvco3 6741 . . . . . . . . . . 11 ((𝑓:𝐴𝐵𝑥𝐴) → ((𝐹𝑓)‘𝑥) = (𝐹‘(𝑓𝑥)))
24 fvresi 6916 . . . . . . . . . . . 12 (𝑥𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
2524adantl 485 . . . . . . . . . . 11 ((𝑓:𝐴𝐵𝑥𝐴) → (( I ↾ 𝐴)‘𝑥) = 𝑥)
2623, 25eqeq12d 2817 . . . . . . . . . 10 ((𝑓:𝐴𝐵𝑥𝐴) → (((𝐹𝑓)‘𝑥) = (( I ↾ 𝐴)‘𝑥) ↔ (𝐹‘(𝑓𝑥)) = 𝑥))
2726ralbidva 3164 . . . . . . . . 9 (𝑓:𝐴𝐵 → (∀𝑥𝐴 ((𝐹𝑓)‘𝑥) = (( I ↾ 𝐴)‘𝑥) ↔ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥))
2827biimprd 251 . . . . . . . 8 (𝑓:𝐴𝐵 → (∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥 → ∀𝑥𝐴 ((𝐹𝑓)‘𝑥) = (( I ↾ 𝐴)‘𝑥)))
2928adantl 485 . . . . . . 7 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ 𝑓:𝐴𝐵) → (∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥 → ∀𝑥𝐴 ((𝐹𝑓)‘𝑥) = (( I ↾ 𝐴)‘𝑥)))
3029impr 458 . . . . . 6 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → ∀𝑥𝐴 ((𝐹𝑓)‘𝑥) = (( I ↾ 𝐴)‘𝑥))
31 simpl1 1188 . . . . . . . 8 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → 𝐹 Fn 𝐵)
32 ffn 6491 . . . . . . . . 9 (𝑓:𝐴𝐵𝑓 Fn 𝐴)
3332ad2antrl 727 . . . . . . . 8 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → 𝑓 Fn 𝐴)
34 frn 6497 . . . . . . . . 9 (𝑓:𝐴𝐵 → ran 𝑓𝐵)
3534ad2antrl 727 . . . . . . . 8 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → ran 𝑓𝐵)
36 fnco 6441 . . . . . . . 8 ((𝐹 Fn 𝐵𝑓 Fn 𝐴 ∧ ran 𝑓𝐵) → (𝐹𝑓) Fn 𝐴)
3731, 33, 35, 36syl3anc 1368 . . . . . . 7 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → (𝐹𝑓) Fn 𝐴)
38 fnresi 6452 . . . . . . 7 ( I ↾ 𝐴) Fn 𝐴
39 eqfnfv 6783 . . . . . . 7 (((𝐹𝑓) Fn 𝐴 ∧ ( I ↾ 𝐴) Fn 𝐴) → ((𝐹𝑓) = ( I ↾ 𝐴) ↔ ∀𝑥𝐴 ((𝐹𝑓)‘𝑥) = (( I ↾ 𝐴)‘𝑥)))
4037, 38, 39sylancl 589 . . . . . 6 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → ((𝐹𝑓) = ( I ↾ 𝐴) ↔ ∀𝑥𝐴 ((𝐹𝑓)‘𝑥) = (( I ↾ 𝐴)‘𝑥)))
4130, 40mpbird 260 . . . . 5 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → (𝐹𝑓) = ( I ↾ 𝐴))
4241imaeq1d 5899 . . . 4 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → ((𝐹𝑓) “ 𝐴) = (( I ↾ 𝐴) “ 𝐴))
43 imaco 6075 . . . 4 ((𝐹𝑓) “ 𝐴) = (𝐹 “ (𝑓𝐴))
44 ssid 3940 . . . . 5 𝐴𝐴
45 resiima 5915 . . . . 5 (𝐴𝐴 → (( I ↾ 𝐴) “ 𝐴) = 𝐴)
4644, 45ax-mp 5 . . . 4 (( I ↾ 𝐴) “ 𝐴) = 𝐴
4742, 43, 463eqtr3g 2859 . . 3 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → (𝐹 “ (𝑓𝐴)) = 𝐴)
48 imaeq2 5896 . . . . 5 (𝑐 = (𝑓𝐴) → (𝐹𝑐) = (𝐹 “ (𝑓𝐴)))
4948eqeq1d 2803 . . . 4 (𝑐 = (𝑓𝐴) → ((𝐹𝑐) = 𝐴 ↔ (𝐹 “ (𝑓𝐴)) = 𝐴))
5049rspcev 3574 . . 3 (((𝑓𝐴) ∈ (𝒫 𝐵 ∩ Fin) ∧ (𝐹 “ (𝑓𝐴)) = 𝐴) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)(𝐹𝑐) = 𝐴)
5122, 47, 50syl2anc 587 . 2 (((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹‘(𝑓𝑥)) = 𝑥)) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)(𝐹𝑐) = 𝐴)
5210, 51exlimddv 1936 1 ((𝐹 Fn 𝐵𝐴 ⊆ ran 𝐹𝐴 ∈ Fin) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)(𝐹𝑐) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wex 1781  wcel 2112  wral 3109  wrex 3110  cin 3883  wss 3884  𝒫 cpw 4500   I cid 5427  ran crn 5524  cres 5525  cima 5526  ccom 5527  Fun wfun 6322   Fn wfn 6323  wf 6324  cfv 6328  Fincfn 8496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-om 7565  df-1o 8089  df-er 8276  df-en 8497  df-dom 8498  df-fin 8500
This theorem is referenced by:  fodomfi2  9475  cmpfi  22016  elrfirn  39623  lmhmfgsplit  40017  hbtlem6  40060
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