Theorem hashf1rn 14384

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 6774	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
2	1	anim2i 628	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → (𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶𝐵))
3	2	ancomd 466	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → (𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉))
4		fex 7224	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V)
5	3, 4	syl 18	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹 ∈ V)
6		f1o2ndf1 8113	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → (2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹)
7		df-2nd 7983	. . . . . . . . 9 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
8	7	funmpt2 6575	. . . . . . . 8 ⊢ Fun 2nd
9		resfunexg 7213	. . . . . . . 8 ⊢ ((Fun 2nd ∧ 𝐹 ∈ V) → (2nd ↾ 𝐹) ∈ V)
10	8, 5, 9	sylancr 598	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → (2nd ↾ 𝐹) ∈ V)
11		f1oeq1 6808	. . . . . . . . . 10 ⊢ ((2nd ↾ 𝐹) = 𝑓 → ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 ↔ 𝑓:𝐹–1-1-onto→ran 𝐹))
12	11	biimpd 232	. . . . . . . . 9 ⊢ ((2nd ↾ 𝐹) = 𝑓 → ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 → 𝑓:𝐹–1-1-onto→ran 𝐹))
13	12	eqcoms 2771	. . . . . . . 8 ⊢ (𝑓 = (2nd ↾ 𝐹) → ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 → 𝑓:𝐹–1-1-onto→ran 𝐹))
14	13	adantl 486	. . . . . . 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 3554	. . . . . 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 417	. . . . 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 89	. . . 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 39	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝐴 ∈ 𝑉 → ∃𝑓 𝑓:𝐹–1-1-onto→ran 𝐹))
19	18	impcom 412	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → ∃𝑓 𝑓:𝐹–1-1-onto→ran 𝐹)
20		hasheqf1oi 14383	. 2 ⊢ (𝐹 ∈ V → (∃𝑓 𝑓:𝐹–1-1-onto→ran 𝐹 → (♯‘𝐹) = (♯‘ran 𝐹)))
21	5, 19, 20	sylc 66	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → (♯‘𝐹) = (♯‘ran 𝐹))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143  Vcvv 3455  {csn 4589  ∪ cuni 4872  ran crn 5662   ↾ cres 5663  Fun wfun 6530  ⟶wf 6532  –1-1→wf1 6533  –1-1-onto→wf1o 6535  ‘cfv 6536  2^nd c2nd 7981  ♯chash 14362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9921  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-nn 12229  df-n0 12500  df-z 12587  df-uz 12858  df-hash 14363

This theorem is referenced by:  hashimarn  14473  hashf1dmrn  14476  usgrsizedg  29565  cyclnumvtx  30149  cycpmco2lem5  33450  cycpmconjslem2  33475  cyc3conja  33477  frlmdim  34001  ply1degltdim  34013  extdgfialglem1  34082  sticksstones2  42914