Theorem f1dmvrnfibi 7213

Description: A one-to-one function whose domain is a set is finite if and only if its range is finite. See also f1vrnfibi 7214. (Contributed by AV, 10-Jan-2020.)

Assertion

Ref	Expression
f1dmvrnfibi	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → (𝐹 ∈ Fin ↔ ran 𝐹 ∈ Fin))

Proof of Theorem f1dmvrnfibi

Step	Hyp	Ref	Expression
1		f1rel 5579	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → Rel 𝐹)
2	1	ad2antlr 489	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝐹 ∈ Fin) → Rel 𝐹)
3		f1cnv 5640	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → ◡𝐹:ran 𝐹–1-1-onto→𝐴)
4		f1ofun 5618	. . . . 5 ⊢ (◡𝐹:ran 𝐹–1-1-onto→𝐴 → Fun ◡𝐹)
5	3, 4	syl 14	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → Fun ◡𝐹)
6	5	ad2antlr 489	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝐹 ∈ Fin) → Fun ◡𝐹)
7		simpr 110	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝐹 ∈ Fin) → 𝐹 ∈ Fin)
8		funrnfi 7211	. . 3 ⊢ ((Rel 𝐹 ∧ Fun ◡𝐹 ∧ 𝐹 ∈ Fin) → ran 𝐹 ∈ Fin)
9	2, 6, 7, 8	syl3anc 1274	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝐹 ∈ Fin) → ran 𝐹 ∈ Fin)
10		simpr 110	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) ∧ ran 𝐹 ∈ Fin) → ran 𝐹 ∈ Fin)
11		f1dm 5580	. . . . . . . 8 ⊢ (𝐹:𝐴–1-1→𝐵 → dom 𝐹 = 𝐴)
12		f1f1orn 5627	. . . . . . . 8 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→ran 𝐹)
13		eleq1 2297	. . . . . . . . . . . 12 ⊢ (𝐴 = dom 𝐹 → (𝐴 ∈ 𝑉 ↔ dom 𝐹 ∈ 𝑉))
14		f1oeq2 5605	. . . . . . . . . . . 12 ⊢ (𝐴 = dom 𝐹 → (𝐹:𝐴–1-1-onto→ran 𝐹 ↔ 𝐹:dom 𝐹–1-1-onto→ran 𝐹))
15	13, 14	anbi12d 473	. . . . . . . . . . 11 ⊢ (𝐴 = dom 𝐹 → ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1-onto→ran 𝐹) ↔ (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-onto→ran 𝐹)))
16	15	eqcoms 2237	. . . . . . . . . 10 ⊢ (dom 𝐹 = 𝐴 → ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1-onto→ran 𝐹) ↔ (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-onto→ran 𝐹)))
17	16	biimpd 144	. . . . . . . . 9 ⊢ (dom 𝐹 = 𝐴 → ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1-onto→ran 𝐹) → (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-onto→ran 𝐹)))
18	17	expcomd 1487	. . . . . . . 8 ⊢ (dom 𝐹 = 𝐴 → (𝐹:𝐴–1-1-onto→ran 𝐹 → (𝐴 ∈ 𝑉 → (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-onto→ran 𝐹))))
19	11, 12, 18	sylc 62	. . . . . . 7 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝐴 ∈ 𝑉 → (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-onto→ran 𝐹)))
20	19	impcom 125	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-onto→ran 𝐹))
21	20	adantr 276	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) ∧ ran 𝐹 ∈ Fin) → (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-onto→ran 𝐹))
22		f1oeng 6998	. . . . 5 ⊢ ((dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-onto→ran 𝐹) → dom 𝐹 ≈ ran 𝐹)
23	21, 22	syl 14	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) ∧ ran 𝐹 ∈ Fin) → dom 𝐹 ≈ ran 𝐹)
24		enfii 7131	. . . 4 ⊢ ((ran 𝐹 ∈ Fin ∧ dom 𝐹 ≈ ran 𝐹) → dom 𝐹 ∈ Fin)
25	10, 23, 24	syl2anc 411	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) ∧ ran 𝐹 ∈ Fin) → dom 𝐹 ∈ Fin)
26		f1fun 5578	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → Fun 𝐹)
27	26	ad2antlr 489	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) ∧ ran 𝐹 ∈ Fin) → Fun 𝐹)
28		fundmfibi 7207	. . . 4 ⊢ (Fun 𝐹 → (𝐹 ∈ Fin ↔ dom 𝐹 ∈ Fin))
29	27, 28	syl 14	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) ∧ ran 𝐹 ∈ Fin) → (𝐹 ∈ Fin ↔ dom 𝐹 ∈ Fin))
30	25, 29	mpbird 167	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) ∧ ran 𝐹 ∈ Fin) → 𝐹 ∈ Fin)
31	9, 30	impbida 600	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → (𝐹 ∈ Fin ↔ ran 𝐹 ∈ Fin))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2205   class class class wbr 4111  ^◡ccnv 4750  dom cdm 4751  ran crn 4752  Rel wrel 4756  Fun wfun 5348  –1-1→wf1 5351  –1-1-onto→wf1o 5353   ≈ cen 6975  Fincfn 6977

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-1st 6336  df-2nd 6337  df-1o 6649  df-er 6769  df-en 6978  df-fin 6980

This theorem is referenced by:  f1vrnfibi  7214