Theorem fihashf1rn 10859

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 5461	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴)
2		simpl 109	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ∈ Fin)
3		fnfi 6995	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → 𝐹 ∈ Fin)
4	1, 2, 3	syl2an2 594	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹 ∈ Fin)
5		f1o2ndf1 6281	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → (2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹)
6		df-2nd 6194	. . . . . . . . 9 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
7	6	funmpt2 5293	. . . . . . . 8 ⊢ Fun 2nd
8		f1f 5459	. . . . . . . . . . 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 5787	. . . . . . . . 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 5779	. . . . . . . 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 5488	. . . . . . . . . 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 2196	. . . . . . . 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 2846	. . . . . 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 10858	. . . 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 1830	. 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 1364  ∃wex 1503   ∈ wcel 2164  Vcvv 2760  {csn 3618  ∪ cuni 3835  ran crn 4660   ↾ cres 4661  Fun wfun 5248   Fn wfn 5249  ⟶wf 5250  –1-1→wf1 5251  –1-1-onto→wf1o 5253  ‘cfv 5254  2^nd c2nd 6192  Fincfn 6794  ♯chash 10846

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-2nd 6194  df-recs 6358  df-frec 6444  df-1o 6469  df-er 6587  df-en 6795  df-dom 6796  df-fin 6797  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-inn 8983  df-n0 9241  df-z 9318  df-uz 9593  df-ihash 10847

This theorem is referenced by: (None)