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Theorem f1dmvrnfibi 9338
Description: A one-to-one function whose domain is a set is finite if and only if its range is finite. See also f1vrnfibi 9339. (Contributed by AV, 10-Jan-2020.)
Assertion
Ref Expression
f1dmvrnfibi ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐡) β†’ (𝐹 ∈ Fin ↔ ran 𝐹 ∈ Fin))

Proof of Theorem f1dmvrnfibi
StepHypRef Expression
1 rnfi 9337 . 2 (𝐹 ∈ Fin β†’ ran 𝐹 ∈ Fin)
2 simpr 485 . . . . 5 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐡) ∧ ran 𝐹 ∈ Fin) β†’ ran 𝐹 ∈ Fin)
3 f1dm 6791 . . . . . . . . 9 (𝐹:𝐴–1-1→𝐡 β†’ dom 𝐹 = 𝐴)
4 f1f1orn 6844 . . . . . . . . 9 (𝐹:𝐴–1-1→𝐡 β†’ 𝐹:𝐴–1-1-ontoβ†’ran 𝐹)
5 eleq1 2821 . . . . . . . . . . . . 13 (𝐴 = dom 𝐹 β†’ (𝐴 ∈ 𝑉 ↔ dom 𝐹 ∈ 𝑉))
6 f1oeq2 6822 . . . . . . . . . . . . 13 (𝐴 = dom 𝐹 β†’ (𝐹:𝐴–1-1-ontoβ†’ran 𝐹 ↔ 𝐹:dom 𝐹–1-1-ontoβ†’ran 𝐹))
75, 6anbi12d 631 . . . . . . . . . . . 12 (𝐴 = dom 𝐹 β†’ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1-ontoβ†’ran 𝐹) ↔ (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-ontoβ†’ran 𝐹)))
87eqcoms 2740 . . . . . . . . . . 11 (dom 𝐹 = 𝐴 β†’ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1-ontoβ†’ran 𝐹) ↔ (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-ontoβ†’ran 𝐹)))
98biimpd 228 . . . . . . . . . 10 (dom 𝐹 = 𝐴 β†’ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1-ontoβ†’ran 𝐹) β†’ (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-ontoβ†’ran 𝐹)))
109expcomd 417 . . . . . . . . 9 (dom 𝐹 = 𝐴 β†’ (𝐹:𝐴–1-1-ontoβ†’ran 𝐹 β†’ (𝐴 ∈ 𝑉 β†’ (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-ontoβ†’ran 𝐹))))
113, 4, 10sylc 65 . . . . . . . 8 (𝐹:𝐴–1-1→𝐡 β†’ (𝐴 ∈ 𝑉 β†’ (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-ontoβ†’ran 𝐹)))
1211impcom 408 . . . . . . 7 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐡) β†’ (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-ontoβ†’ran 𝐹))
1312adantr 481 . . . . . 6 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐡) ∧ ran 𝐹 ∈ Fin) β†’ (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-ontoβ†’ran 𝐹))
14 f1oeng 8969 . . . . . 6 ((dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-ontoβ†’ran 𝐹) β†’ dom 𝐹 β‰ˆ ran 𝐹)
1513, 14syl 17 . . . . 5 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐡) ∧ ran 𝐹 ∈ Fin) β†’ dom 𝐹 β‰ˆ ran 𝐹)
16 enfii 9191 . . . . 5 ((ran 𝐹 ∈ Fin ∧ dom 𝐹 β‰ˆ ran 𝐹) β†’ dom 𝐹 ∈ Fin)
172, 15, 16syl2anc 584 . . . 4 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐡) ∧ ran 𝐹 ∈ Fin) β†’ dom 𝐹 ∈ Fin)
18 f1fun 6789 . . . . . 6 (𝐹:𝐴–1-1→𝐡 β†’ Fun 𝐹)
1918ad2antlr 725 . . . . 5 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐡) ∧ ran 𝐹 ∈ Fin) β†’ Fun 𝐹)
20 fundmfibi 9333 . . . . 5 (Fun 𝐹 β†’ (𝐹 ∈ Fin ↔ dom 𝐹 ∈ Fin))
2119, 20syl 17 . . . 4 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐡) ∧ ran 𝐹 ∈ Fin) β†’ (𝐹 ∈ Fin ↔ dom 𝐹 ∈ Fin))
2217, 21mpbird 256 . . 3 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐡) ∧ ran 𝐹 ∈ Fin) β†’ 𝐹 ∈ Fin)
2322ex 413 . 2 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐡) β†’ (ran 𝐹 ∈ Fin β†’ 𝐹 ∈ Fin))
241, 23impbid2 225 1 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐡) β†’ (𝐹 ∈ Fin ↔ ran 𝐹 ∈ Fin))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   class class class wbr 5148  dom cdm 5676  ran crn 5677  Fun wfun 6537  β€“1-1β†’wf1 6540  β€“1-1-ontoβ†’wf1o 6542   β‰ˆ cen 8938  Fincfn 8941
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-om 7858  df-1st 7977  df-2nd 7978  df-1o 8468  df-er 8705  df-en 8942  df-dom 8943  df-fin 8945
This theorem is referenced by:  f1vrnfibi  9339  fmtnoinf  46289
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