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Theorem relfi 29753
 Description: A relation (set) is finite if and only if both its domain and range are finite. (Contributed by Thierry Arnoux, 27-Aug-2017.)
Assertion
Ref Expression
relfi (Rel 𝐴 → (𝐴 ∈ Fin ↔ (dom 𝐴 ∈ Fin ∧ ran 𝐴 ∈ Fin)))

Proof of Theorem relfi
StepHypRef Expression
1 dmfi 8404 . . 3 (𝐴 ∈ Fin → dom 𝐴 ∈ Fin)
2 rnfi 8409 . . 3 (𝐴 ∈ Fin → ran 𝐴 ∈ Fin)
31, 2jca 501 . 2 (𝐴 ∈ Fin → (dom 𝐴 ∈ Fin ∧ ran 𝐴 ∈ Fin))
4 xpfi 8391 . . . 4 ((dom 𝐴 ∈ Fin ∧ ran 𝐴 ∈ Fin) → (dom 𝐴 × ran 𝐴) ∈ Fin)
5 relssdmrn 5799 . . . 4 (Rel 𝐴𝐴 ⊆ (dom 𝐴 × ran 𝐴))
6 ssfi 8340 . . . 4 (((dom 𝐴 × ran 𝐴) ∈ Fin ∧ 𝐴 ⊆ (dom 𝐴 × ran 𝐴)) → 𝐴 ∈ Fin)
74, 5, 6syl2anr 584 . . 3 ((Rel 𝐴 ∧ (dom 𝐴 ∈ Fin ∧ ran 𝐴 ∈ Fin)) → 𝐴 ∈ Fin)
87ex 397 . 2 (Rel 𝐴 → ((dom 𝐴 ∈ Fin ∧ ran 𝐴 ∈ Fin) → 𝐴 ∈ Fin))
93, 8impbid2 216 1 (Rel 𝐴 → (𝐴 ∈ Fin ↔ (dom 𝐴 ∈ Fin ∧ ran 𝐴 ∈ Fin)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   ∈ wcel 2145   ⊆ wss 3723   × cxp 5248  dom cdm 5250  ran crn 5251  Rel wrel 5255  Fincfn 8113 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-1st 7319  df-2nd 7320  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-1o 7717  df-oadd 7721  df-er 7900  df-en 8114  df-dom 8115  df-fin 8117 This theorem is referenced by:  fpwrelmapffslem  29847
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