Theorem relfi 32586

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

Step	Hyp	Ref	Expression
1		dmfi 9228	. . 3 ⊢ (𝐴 ∈ Fin → dom 𝐴 ∈ Fin)
2		rnfi 9233	. . 3 ⊢ (𝐴 ∈ Fin → ran 𝐴 ∈ Fin)
3	1, 2	jca 511	. 2 ⊢ (𝐴 ∈ Fin → (dom 𝐴 ∈ Fin ∧ ran 𝐴 ∈ Fin))
4		xpfi 9213	. . . 4 ⊢ ((dom 𝐴 ∈ Fin ∧ ran 𝐴 ∈ Fin) → (dom 𝐴 × ran 𝐴) ∈ Fin)
5		relssdmrn 6223	. . . 4 ⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴))
6		ssfi 9091	. . . 4 ⊢ (((dom 𝐴 × ran 𝐴) ∈ Fin ∧ 𝐴 ⊆ (dom 𝐴 × ran 𝐴)) → 𝐴 ∈ Fin)
7	4, 5, 6	syl2anr 597	. . 3 ⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∈ Fin ∧ ran 𝐴 ∈ Fin)) → 𝐴 ∈ Fin)
8	7	ex 412	. 2 ⊢ (Rel 𝐴 → ((dom 𝐴 ∈ Fin ∧ ran 𝐴 ∈ Fin) → 𝐴 ∈ Fin))
9	3, 8	impbid2 226	1 ⊢ (Rel 𝐴 → (𝐴 ∈ Fin ↔ (dom 𝐴 ∈ Fin ∧ ran 𝐴 ∈ Fin)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2113   ⊆ wss 3898   × cxp 5619  dom cdm 5621  ran crn 5622  Rel wrel 5626  Fincfn 8877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7676

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-om 7805  df-1st 7929  df-2nd 7930  df-1o 8393  df-en 8878  df-dom 8879  df-fin 8881

This theorem is referenced by:  fpwrelmapffslem  32721