Theorem f1dmvrnfibi 9358

Description: A one-to-one function whose domain is a set is finite if and only if its range is finite. See also f1vrnfibi 9359. (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		rnfi 9357	. 2 ⊢ (𝐹 ∈ Fin → ran 𝐹 ∈ Fin)
2		simpr 484	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) ∧ ran 𝐹 ∈ Fin) → ran 𝐹 ∈ Fin)
3		f1dm 6783	. . . . . . . . 9 ⊢ (𝐹:𝐴–1-1→𝐵 → dom 𝐹 = 𝐴)
4		f1f1orn 6834	. . . . . . . . 9 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→ran 𝐹)
5		eleq1 2823	. . . . . . . . . . . . 13 ⊢ (𝐴 = dom 𝐹 → (𝐴 ∈ 𝑉 ↔ dom 𝐹 ∈ 𝑉))
6		f1oeq2 6812	. . . . . . . . . . . . 13 ⊢ (𝐴 = dom 𝐹 → (𝐹:𝐴–1-1-onto→ran 𝐹 ↔ 𝐹:dom 𝐹–1-1-onto→ran 𝐹))
7	5, 6	anbi12d 632	. . . . . . . . . . . 12 ⊢ (𝐴 = dom 𝐹 → ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1-onto→ran 𝐹) ↔ (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-onto→ran 𝐹)))
8	7	eqcoms 2744	. . . . . . . . . . 11 ⊢ (dom 𝐹 = 𝐴 → ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1-onto→ran 𝐹) ↔ (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-onto→ran 𝐹)))
9	8	biimpd 229	. . . . . . . . . 10 ⊢ (dom 𝐹 = 𝐴 → ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1-onto→ran 𝐹) → (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-onto→ran 𝐹)))
10	9	expcomd 416	. . . . . . . . 9 ⊢ (dom 𝐹 = 𝐴 → (𝐹:𝐴–1-1-onto→ran 𝐹 → (𝐴 ∈ 𝑉 → (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-onto→ran 𝐹))))
11	3, 4, 10	sylc 65	. . . . . . . 8 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝐴 ∈ 𝑉 → (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-onto→ran 𝐹)))
12	11	impcom 407	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-onto→ran 𝐹))
13	12	adantr 480	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) ∧ ran 𝐹 ∈ Fin) → (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-onto→ran 𝐹))
14		f1oeng 8990	. . . . . 6 ⊢ ((dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-onto→ran 𝐹) → dom 𝐹 ≈ ran 𝐹)
15	13, 14	syl 17	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) ∧ ran 𝐹 ∈ Fin) → dom 𝐹 ≈ ran 𝐹)
16		enfii 9205	. . . . 5 ⊢ ((ran 𝐹 ∈ Fin ∧ dom 𝐹 ≈ ran 𝐹) → dom 𝐹 ∈ Fin)
17	2, 15, 16	syl2anc 584	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) ∧ ran 𝐹 ∈ Fin) → dom 𝐹 ∈ Fin)
18		f1fun 6781	. . . . . 6 ⊢ (𝐹:𝐴–1-1→𝐵 → Fun 𝐹)
19	18	ad2antlr 727	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) ∧ ran 𝐹 ∈ Fin) → Fun 𝐹)
20		fundmfibi 9353	. . . . 5 ⊢ (Fun 𝐹 → (𝐹 ∈ Fin ↔ dom 𝐹 ∈ Fin))
21	19, 20	syl 17	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) ∧ ran 𝐹 ∈ Fin) → (𝐹 ∈ Fin ↔ dom 𝐹 ∈ Fin))
22	17, 21	mpbird 257	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) ∧ ran 𝐹 ∈ Fin) → 𝐹 ∈ Fin)
23	22	ex 412	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → (ran 𝐹 ∈ Fin → 𝐹 ∈ Fin))
24	1, 23	impbid2 226	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → (𝐹 ∈ Fin ↔ ran 𝐹 ∈ Fin))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   class class class wbr 5124  dom cdm 5659  ran crn 5660  Fun wfun 6530  –1-1→wf1 6533  –1-1-onto→wf1o 6535   ≈ cen 8961  Fincfn 8964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-om 7867  df-1st 7993  df-2nd 7994  df-1o 8485  df-en 8965  df-dom 8966  df-fin 8968

This theorem is referenced by:  f1vrnfibi  9359  fmtnoinf  47517