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⟶1-1𝐵) → (𝐹 ∈ Fin ↔ ran 𝐹 ∈ Fin))

Proof of Theorem f1dmvrnfibi

Step	Hyp	Ref	Expression
1		rnfi 9357	. 2 ⊢ (𝐹 ∈ Fin → ran 𝐹 ∈ Fin)
2		simpr 490	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶1-1𝐵) ∧ ran 𝐹 ∈ Fin) → ran 𝐹 ∈ Fin)
3		f1dm 6806	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶1-1𝐵 → dom 𝐹 = 𝐴)
4		f1f1orn 6858	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶1-1𝐵 → 𝐹:𝐴⟶1-1ontoran 𝐹)
5		eleq1 2852	. . . . . . . . . . . . 13 ⊢ (𝐴 = dom 𝐹 → (𝐴 ∈ 𝑉 ↔ dom 𝐹 ∈ 𝑉))
6		f1oeq2 6835	. . . . . . . . . . . . 13 ⊢ (𝐴 = dom 𝐹 → (𝐹:𝐴⟶1-1ontoran 𝐹 ↔ 𝐹:dom 𝐹⟶1-1ontoran 𝐹))
7	5, 6	anbi12d 644	. . . . . . . . . . . 12 ⊢ (𝐴 = dom 𝐹 → ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶1-1ontoran 𝐹) ↔ (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹⟶1-1ontoran 𝐹)))
8	7	eqcoms 2771	. . . . . . . . . . 11 ⊢ (dom 𝐹 = 𝐴 → ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶1-1ontoran 𝐹) ↔ (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹⟶1-1ontoran 𝐹)))
9	8	biimpd 232	. . . . . . . . . 10 ⊢ (dom 𝐹 = 𝐴 → ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶1-1ontoran 𝐹) → (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹⟶1-1ontoran 𝐹)))
10	9	expcomd 422	. . . . . . . . 9 ⊢ (dom 𝐹 = 𝐴 → (𝐹:𝐴⟶1-1ontoran 𝐹 → (𝐴 ∈ 𝑉 → (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹⟶1-1ontoran 𝐹))))
11	3, 4, 10	sylc 66	. . . . . . . 8 ⊢ (𝐹:𝐴⟶1-1𝐵 → (𝐴 ∈ 𝑉 → (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹⟶1-1ontoran 𝐹)))
12	11	impcom 413	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶1-1𝐵) → (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹⟶1-1ontoran 𝐹))
13	12	adantr 486	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶1-1𝐵) ∧ ran 𝐹 ∈ Fin) → (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹⟶1-1ontoran 𝐹))
14		f1oeng 9025	. . . . . 6 ⊢ ((dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹⟶1-1ontoran 𝐹) → dom 𝐹 ≈ ran 𝐹)
15	13, 14	syl 18	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶1-1𝐵) ∧ ran 𝐹 ∈ Fin) → dom 𝐹 ≈ ran 𝐹)
16		enfii 9229	. . . . 5 ⊢ ((ran 𝐹 ∈ Fin ∧ dom 𝐹 ≈ ran 𝐹) → dom 𝐹 ∈ Fin)
17	2, 15, 16	syl2anc 596	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶1-1𝐵) ∧ ran 𝐹 ∈ Fin) → dom 𝐹 ∈ Fin)
18		f1fun 6802	. . . . . 6 ⊢ (𝐹:𝐴⟶1-1𝐵 → Fun 𝐹)
19	18	ad2antlr 740	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶1-1𝐵) ∧ ran 𝐹 ∈ Fin) → Fun 𝐹)
20		fundmfibi 9353	. . . . 5 ⊢ (Fun 𝐹 → (𝐹 ∈ Fin ↔ dom 𝐹 ∈ Fin))
21	19, 20	syl 18	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶1-1𝐵) ∧ ran 𝐹 ∈ Fin) → (𝐹 ∈ Fin ↔ dom 𝐹 ∈ Fin))
22	17, 21	mpbird 260	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶1-1𝐵) ∧ ran 𝐹 ∈ Fin) → 𝐹 ∈ Fin)
23	22	ex 418	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶1-1𝐵) → (ran 𝐹 ∈ Fin → 𝐹 ∈ Fin))
24	1, 23	impbid2 229	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶1-1𝐵) → (𝐹 ∈ Fin ↔ ran 𝐹 ∈ Fin))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147   class class class wbr 5106  dom cdm 5655  ran crn 5656  Fun wfun 6552  ⟶1-1wf1 6555  ⟶1-1ontowf1o 6557   ≈ cen 8998  Fincfn 9001

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-om 7902  df-1st 8025  df-2nd 8026  df-1o 8504  df-en 9002  df-dom 9003  df-fin 9005

This theorem is used by:  f1vrnfibi  9359  fmtnoinf  48826