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Theorem f1dmvrnfibi 6943
Description: A one-to-one function whose domain is a set is finite if and only if its range is finite. See also f1vrnfibi 6944. (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 f1rel 5426 . . . 4 (𝐹:𝐴–1-1→𝐡 β†’ Rel 𝐹)
21ad2antlr 489 . . 3 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐡) ∧ 𝐹 ∈ Fin) β†’ Rel 𝐹)
3 f1cnv 5486 . . . . 5 (𝐹:𝐴–1-1→𝐡 β†’ ◑𝐹:ran 𝐹–1-1-onto→𝐴)
4 f1ofun 5464 . . . . 5 (◑𝐹:ran 𝐹–1-1-onto→𝐴 β†’ Fun ◑𝐹)
53, 4syl 14 . . . 4 (𝐹:𝐴–1-1→𝐡 β†’ Fun ◑𝐹)
65ad2antlr 489 . . 3 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐡) ∧ 𝐹 ∈ Fin) β†’ Fun ◑𝐹)
7 simpr 110 . . 3 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐡) ∧ 𝐹 ∈ Fin) β†’ 𝐹 ∈ Fin)
8 funrnfi 6941 . . 3 ((Rel 𝐹 ∧ Fun ◑𝐹 ∧ 𝐹 ∈ Fin) β†’ ran 𝐹 ∈ Fin)
92, 6, 7, 8syl3anc 1238 . 2 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐡) ∧ 𝐹 ∈ Fin) β†’ ran 𝐹 ∈ Fin)
10 simpr 110 . . . 4 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐡) ∧ ran 𝐹 ∈ Fin) β†’ ran 𝐹 ∈ Fin)
11 f1dm 5427 . . . . . . . 8 (𝐹:𝐴–1-1→𝐡 β†’ dom 𝐹 = 𝐴)
12 f1f1orn 5473 . . . . . . . 8 (𝐹:𝐴–1-1→𝐡 β†’ 𝐹:𝐴–1-1-ontoβ†’ran 𝐹)
13 eleq1 2240 . . . . . . . . . . . 12 (𝐴 = dom 𝐹 β†’ (𝐴 ∈ 𝑉 ↔ dom 𝐹 ∈ 𝑉))
14 f1oeq2 5451 . . . . . . . . . . . 12 (𝐴 = dom 𝐹 β†’ (𝐹:𝐴–1-1-ontoβ†’ran 𝐹 ↔ 𝐹:dom 𝐹–1-1-ontoβ†’ran 𝐹))
1513, 14anbi12d 473 . . . . . . . . . . 11 (𝐴 = dom 𝐹 β†’ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1-ontoβ†’ran 𝐹) ↔ (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-ontoβ†’ran 𝐹)))
1615eqcoms 2180 . . . . . . . . . 10 (dom 𝐹 = 𝐴 β†’ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1-ontoβ†’ran 𝐹) ↔ (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-ontoβ†’ran 𝐹)))
1716biimpd 144 . . . . . . . . 9 (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)))
1817expcomd 1441 . . . . . . . 8 (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))))
1911, 12, 18sylc 62 . . . . . . 7 (𝐹:𝐴–1-1→𝐡 β†’ (𝐴 ∈ 𝑉 β†’ (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-ontoβ†’ran 𝐹)))
2019impcom 125 . . . . . 6 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐡) β†’ (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-ontoβ†’ran 𝐹))
2120adantr 276 . . . . 5 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐡) ∧ ran 𝐹 ∈ Fin) β†’ (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-ontoβ†’ran 𝐹))
22 f1oeng 6757 . . . . 5 ((dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-ontoβ†’ran 𝐹) β†’ dom 𝐹 β‰ˆ ran 𝐹)
2321, 22syl 14 . . . 4 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐡) ∧ ran 𝐹 ∈ Fin) β†’ dom 𝐹 β‰ˆ ran 𝐹)
24 enfii 6874 . . . 4 ((ran 𝐹 ∈ Fin ∧ dom 𝐹 β‰ˆ ran 𝐹) β†’ dom 𝐹 ∈ Fin)
2510, 23, 24syl2anc 411 . . 3 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐡) ∧ ran 𝐹 ∈ Fin) β†’ dom 𝐹 ∈ Fin)
26 f1fun 5425 . . . . 5 (𝐹:𝐴–1-1→𝐡 β†’ Fun 𝐹)
2726ad2antlr 489 . . . 4 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐡) ∧ ran 𝐹 ∈ Fin) β†’ Fun 𝐹)
28 fundmfibi 6938 . . . 4 (Fun 𝐹 β†’ (𝐹 ∈ Fin ↔ dom 𝐹 ∈ Fin))
2927, 28syl 14 . . 3 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐡) ∧ ran 𝐹 ∈ Fin) β†’ (𝐹 ∈ Fin ↔ dom 𝐹 ∈ Fin))
3025, 29mpbird 167 . 2 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐡) ∧ ran 𝐹 ∈ Fin) β†’ 𝐹 ∈ Fin)
319, 30impbida 596 1 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐡) β†’ (𝐹 ∈ Fin ↔ ran 𝐹 ∈ Fin))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148   class class class wbr 4004  β—‘ccnv 4626  dom cdm 4627  ran crn 4628  Rel wrel 4632  Fun wfun 5211  β€“1-1β†’wf1 5214  β€“1-1-ontoβ†’wf1o 5216   β‰ˆ cen 6738  Fincfn 6740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-1st 6141  df-2nd 6142  df-1o 6417  df-er 6535  df-en 6741  df-fin 6743
This theorem is referenced by:  f1vrnfibi  6944
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