Theorem f1dmvrnfibi 6885

Description: A one-to-one function whose domain is a set is finite if and only if its range is finite. See also f1vrnfibi 6886. (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 5378	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → Rel 𝐹)
2	1	ad2antlr 481	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝐹 ∈ Fin) → Rel 𝐹)
3		f1cnv 5437	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → ◡𝐹:ran 𝐹–1-1-onto→𝐴)
4		f1ofun 5415	. . . . 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 481	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝐹 ∈ Fin) → Fun ◡𝐹)
7		simpr 109	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝐹 ∈ Fin) → 𝐹 ∈ Fin)
8		funrnfi 6883	. . 3 ⊢ ((Rel 𝐹 ∧ Fun ◡𝐹 ∧ 𝐹 ∈ Fin) → ran 𝐹 ∈ Fin)
9	2, 6, 7, 8	syl3anc 1220	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝐹 ∈ Fin) → ran 𝐹 ∈ Fin)
10		simpr 109	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) ∧ ran 𝐹 ∈ Fin) → ran 𝐹 ∈ Fin)
11		f1dm 5379	. . . . . . . 8 ⊢ (𝐹:𝐴–1-1→𝐵 → dom 𝐹 = 𝐴)
12		f1f1orn 5424	. . . . . . . 8 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→ran 𝐹)
13		eleq1 2220	. . . . . . . . . . . 12 ⊢ (𝐴 = dom 𝐹 → (𝐴 ∈ 𝑉 ↔ dom 𝐹 ∈ 𝑉))
14		f1oeq2 5403	. . . . . . . . . . . 12 ⊢ (𝐴 = dom 𝐹 → (𝐹:𝐴–1-1-onto→ran 𝐹 ↔ 𝐹:dom 𝐹–1-1-onto→ran 𝐹))
15	13, 14	anbi12d 465	. . . . . . . . . . 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 2160	. . . . . . . . . 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 143	. . . . . . . . 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 1421	. . . . . . . 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 124	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-onto→ran 𝐹))
21	20	adantr 274	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) ∧ ran 𝐹 ∈ Fin) → (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-onto→ran 𝐹))
22		f1oeng 6699	. . . . 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 6816	. . . 4 ⊢ ((ran 𝐹 ∈ Fin ∧ dom 𝐹 ≈ ran 𝐹) → dom 𝐹 ∈ Fin)
25	10, 23, 24	syl2anc 409	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) ∧ ran 𝐹 ∈ Fin) → dom 𝐹 ∈ Fin)
26		f1fun 5377	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → Fun 𝐹)
27	26	ad2antlr 481	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) ∧ ran 𝐹 ∈ Fin) → Fun 𝐹)
28		fundmfibi 6880	. . . 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 166	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) ∧ ran 𝐹 ∈ Fin) → 𝐹 ∈ Fin)
31	9, 30	impbida 586	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → (𝐹 ∈ Fin ↔ ran 𝐹 ∈ Fin))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1335   ∈ wcel 2128   class class class wbr 3965  ^◡ccnv 4584  dom cdm 4585  ran crn 4586  Rel wrel 4590  Fun wfun 5163  –1-1→wf1 5166  –1-1-onto→wf1o 5168   ≈ cen 6680  Fincfn 6682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495  ax-iinf 4546

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4549  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-fv 5177  df-1st 6085  df-2nd 6086  df-1o 6360  df-er 6477  df-en 6683  df-fin 6685

This theorem is referenced by:  f1vrnfibi  6886