Theorem hashf1rn 14273

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 Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 4-May-2021.)

Assertion

Ref	Expression
hashf1rn	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → (♯‘𝐹) = (♯‘ran 𝐹))

Proof of Theorem hashf1rn

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1f 6728	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
2	1	anim2i 617	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → (𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶𝐵))
3	2	ancomd 461	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → (𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉))
4		fex 7170	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V)
5	3, 4	syl 17	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹 ∈ V)
6		f1o2ndf1 8062	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → (2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹)
7		df-2nd 7932	. . . . . . . . 9 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
8	7	funmpt2 6529	. . . . . . . 8 ⊢ Fun 2nd
9		resfunexg 7159	. . . . . . . 8 ⊢ ((Fun 2nd ∧ 𝐹 ∈ V) → (2nd ↾ 𝐹) ∈ V)
10	8, 5, 9	sylancr 587	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → (2nd ↾ 𝐹) ∈ V)
11		f1oeq1 6760	. . . . . . . . . 10 ⊢ ((2nd ↾ 𝐹) = 𝑓 → ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 ↔ 𝑓:𝐹–1-1-onto→ran 𝐹))
12	11	biimpd 229	. . . . . . . . 9 ⊢ ((2nd ↾ 𝐹) = 𝑓 → ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 → 𝑓:𝐹–1-1-onto→ran 𝐹))
13	12	eqcoms 2742	. . . . . . . 8 ⊢ (𝑓 = (2nd ↾ 𝐹) → ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 → 𝑓:𝐹–1-1-onto→ran 𝐹))
14	13	adantl 481	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑓 = (2nd ↾ 𝐹)) → ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 → 𝑓:𝐹–1-1-onto→ran 𝐹))
15	10, 14	spcimedv 3547	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 → ∃𝑓 𝑓:𝐹–1-1-onto→ran 𝐹))
16	15	ex 412	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐹:𝐴–1-1→𝐵 → ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 → ∃𝑓 𝑓:𝐹–1-1-onto→ran 𝐹)))
17	16	com13 88	. . . 4 ⊢ ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 → (𝐹:𝐴–1-1→𝐵 → (𝐴 ∈ 𝑉 → ∃𝑓 𝑓:𝐹–1-1-onto→ran 𝐹)))
18	6, 17	mpcom 38	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝐴 ∈ 𝑉 → ∃𝑓 𝑓:𝐹–1-1-onto→ran 𝐹))
19	18	impcom 407	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → ∃𝑓 𝑓:𝐹–1-1-onto→ran 𝐹)
20		hasheqf1oi 14272	. 2 ⊢ (𝐹 ∈ V → (∃𝑓 𝑓:𝐹–1-1-onto→ran 𝐹 → (♯‘𝐹) = (♯‘ran 𝐹)))
21	5, 19, 20	sylc 65	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → (♯‘𝐹) = (♯‘ran 𝐹))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  Vcvv 3438  {csn 4578  ∪ cuni 4861  ran crn 5623   ↾ cres 5624  Fun wfun 6484  ⟶wf 6486  –1-1→wf1 6487  –1-1-onto→wf1o 6489  ‘cfv 6490  2^nd c2nd 7930  ♯chash 14251

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8633  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-card 9849  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-nn 12144  df-n0 12400  df-z 12487  df-uz 12750  df-hash 14252

This theorem is referenced by:  hashimarn  14361  hashf1dmrn  14364  usgrsizedg  29237  cyclnumvtx  29822  cycpmco2lem5  33161  cycpmconjslem2  33186  cyc3conja  33188  frlmdim  33717  ply1degltdim  33729  extdgfialglem1  33798  sticksstones2  42340