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Theorem f1dmvrnfibi 9336
Description: A one-to-one function whose domain is a set is finite if and only if its range is finite. See also f1vrnfibi 9337. (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 9335 . 2 (𝐹 ∈ Fin β†’ ran 𝐹 ∈ Fin)
2 simpr 486 . . . . 5 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐡) ∧ ran 𝐹 ∈ Fin) β†’ ran 𝐹 ∈ Fin)
3 f1dm 6792 . . . . . . . . 9 (𝐹:𝐴–1-1→𝐡 β†’ dom 𝐹 = 𝐴)
4 f1f1orn 6845 . . . . . . . . 9 (𝐹:𝐴–1-1→𝐡 β†’ 𝐹:𝐴–1-1-ontoβ†’ran 𝐹)
5 eleq1 2822 . . . . . . . . . . . . 13 (𝐴 = dom 𝐹 β†’ (𝐴 ∈ 𝑉 ↔ dom 𝐹 ∈ 𝑉))
6 f1oeq2 6823 . . . . . . . . . . . . 13 (𝐴 = dom 𝐹 β†’ (𝐹:𝐴–1-1-ontoβ†’ran 𝐹 ↔ 𝐹:dom 𝐹–1-1-ontoβ†’ran 𝐹))
75, 6anbi12d 632 . . . . . . . . . . . 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 2741 . . . . . . . . . . 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 418 . . . . . . . . 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 409 . . . . . . 7 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐡) β†’ (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-ontoβ†’ran 𝐹))
1312adantr 482 . . . . . 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 8967 . . . . . 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 9189 . . . . 5 ((ran 𝐹 ∈ Fin ∧ dom 𝐹 β‰ˆ ran 𝐹) β†’ dom 𝐹 ∈ Fin)
172, 15, 16syl2anc 585 . . . 4 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐡) ∧ ran 𝐹 ∈ Fin) β†’ dom 𝐹 ∈ Fin)
18 f1fun 6790 . . . . . 6 (𝐹:𝐴–1-1→𝐡 β†’ Fun 𝐹)
1918ad2antlr 726 . . . . 5 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐡) ∧ ran 𝐹 ∈ Fin) β†’ Fun 𝐹)
20 fundmfibi 9331 . . . . 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 257 . . 3 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐡) ∧ ran 𝐹 ∈ Fin) β†’ 𝐹 ∈ Fin)
2322ex 414 . 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 397   = wceq 1542   ∈ wcel 2107   class class class wbr 5149  dom cdm 5677  ran crn 5678  Fun wfun 6538  β€“1-1β†’wf1 6541  β€“1-1-ontoβ†’wf1o 6543   β‰ˆ cen 8936  Fincfn 8939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-om 7856  df-1st 7975  df-2nd 7976  df-1o 8466  df-er 8703  df-en 8940  df-dom 8941  df-fin 8943
This theorem is referenced by:  f1vrnfibi  9337  fmtnoinf  46204
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