Theorem f1dmvrnfibi 7186

Description: A one-to-one function whose domain is a set is finite if and only if its range is finite. See also f1vrnfibi 7187. (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 5555	. . . 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 5616	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → ◡𝐹:ran 𝐹–1-1-onto→𝐴)
4		f1ofun 5594	. . . . 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 7184	. . 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 5556	. . . . . . . 8 ⊢ (𝐹:𝐴–1-1→𝐵 → dom 𝐹 = 𝐴)
12		f1f1orn 5603	. . . . . . . 8 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→ran 𝐹)
13		eleq1 2294	. . . . . . . . . . . 12 ⊢ (𝐴 = dom 𝐹 → (𝐴 ∈ 𝑉 ↔ dom 𝐹 ∈ 𝑉))
14		f1oeq2 5581	. . . . . . . . . . . 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 2234	. . . . . . . . . 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 6973	. . . . 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 7104	. . . 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 5554	. . . . 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 7180	. . . 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 2202   class class class wbr 4093  ^◡ccnv 4730  dom cdm 4731  ran crn 4732  Rel wrel 4736  Fun wfun 5327  –1-1→wf1 5330  –1-1-onto→wf1o 5332   ≈ cen 6950  Fincfn 6952

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-1st 6312  df-2nd 6313  df-1o 6625  df-er 6745  df-en 6953  df-fin 6955

This theorem is referenced by:  f1vrnfibi  7187