Theorem fihashf1rn 11040

Description: The size of a finite set which is a one-to-one function is equal to the size of the function's range. (Contributed by Jim Kingdon, 21-Feb-2022.)

Assertion

Ref	Expression
fihashf1rn	⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → (♯‘𝐹) = (♯‘ran 𝐹))

Proof of Theorem fihashf1rn

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1fn 5541	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴)
2		simpl 109	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ∈ Fin)
3		fnfi 7126	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → 𝐹 ∈ Fin)
4	1, 2, 3	syl2an2 596	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹 ∈ Fin)
5		f1o2ndf1 6388	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → (2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹)
6		df-2nd 6299	. . . . . . . . 9 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
7	6	funmpt2 5363	. . . . . . . 8 ⊢ Fun 2nd
8		f1f 5539	. . . . . . . . . . 11 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
9	8	anim2i 342	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → (𝐴 ∈ Fin ∧ 𝐹:𝐴⟶𝐵))
10	9	ancomd 267	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → (𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ Fin))
11		fex 5878	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ Fin) → 𝐹 ∈ V)
12	10, 11	syl 14	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹 ∈ V)
13		resfunexg 5870	. . . . . . . 8 ⊢ ((Fun 2nd ∧ 𝐹 ∈ V) → (2nd ↾ 𝐹) ∈ V)
14	7, 12, 13	sylancr 414	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → (2nd ↾ 𝐹) ∈ V)
15		f1oeq1 5568	. . . . . . . . . 10 ⊢ ((2nd ↾ 𝐹) = 𝑓 → ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 ↔ 𝑓:𝐹–1-1-onto→ran 𝐹))
16	15	biimpd 144	. . . . . . . . 9 ⊢ ((2nd ↾ 𝐹) = 𝑓 → ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 → 𝑓:𝐹–1-1-onto→ran 𝐹))
17	16	eqcoms 2232	. . . . . . . 8 ⊢ (𝑓 = (2nd ↾ 𝐹) → ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 → 𝑓:𝐹–1-1-onto→ran 𝐹))
18	17	adantl 277	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑓 = (2nd ↾ 𝐹)) → ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 → 𝑓:𝐹–1-1-onto→ran 𝐹))
19	14, 18	spcimedv 2890	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 → ∃𝑓 𝑓:𝐹–1-1-onto→ran 𝐹))
20	19	ex 115	. . . . 5 ⊢ (𝐴 ∈ Fin → (𝐹:𝐴–1-1→𝐵 → ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 → ∃𝑓 𝑓:𝐹–1-1-onto→ran 𝐹)))
21	20	com13 80	. . . 4 ⊢ ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 → (𝐹:𝐴–1-1→𝐵 → (𝐴 ∈ Fin → ∃𝑓 𝑓:𝐹–1-1-onto→ran 𝐹)))
22	5, 21	mpcom 36	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝐴 ∈ Fin → ∃𝑓 𝑓:𝐹–1-1-onto→ran 𝐹))
23	22	impcom 125	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → ∃𝑓 𝑓:𝐹–1-1-onto→ran 𝐹)
24		fihasheqf1oi 11039	. . . 4 ⊢ ((𝐹 ∈ Fin ∧ 𝑓:𝐹–1-1-onto→ran 𝐹) → (♯‘𝐹) = (♯‘ran 𝐹))
25	24	ex 115	. . 3 ⊢ (𝐹 ∈ Fin → (𝑓:𝐹–1-1-onto→ran 𝐹 → (♯‘𝐹) = (♯‘ran 𝐹)))
26	25	exlimdv 1865	. 2 ⊢ (𝐹 ∈ Fin → (∃𝑓 𝑓:𝐹–1-1-onto→ran 𝐹 → (♯‘𝐹) = (♯‘ran 𝐹)))
27	4, 23, 26	sylc 62	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → (♯‘𝐹) = (♯‘ran 𝐹))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395  ∃wex 1538   ∈ wcel 2200  Vcvv 2800  {csn 3667  ∪ cuni 3891  ran crn 4724   ↾ cres 4725  Fun wfun 5318   Fn wfn 5319  ⟶wf 5320  –1-1→wf1 5321  –1-1-onto→wf1o 5323  ‘cfv 5324  2^nd c2nd 6297  Fincfn 6904  ♯chash 11027

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-2nd 6299  df-recs 6466  df-frec 6552  df-1o 6577  df-er 6697  df-en 6905  df-dom 6906  df-fin 6907  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-inn 9134  df-n0 9393  df-z 9470  df-uz 9746  df-ihash 11028

This theorem is referenced by: (None)