Theorem hashf1rn 14537

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⟶1-1𝐵) → (♯‘𝐹) = (♯‘ran 𝐹))

Proof of Theorem hashf1rn

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1f 6808	. . . . 5 ⊢ (𝐹:𝐴⟶1-1𝐵 → 𝐹:𝐴⟶𝐵)
2	1	anim2i 629	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶1-1𝐵) → (𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶𝐵))
3	2	ancomd 467	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶1-1𝐵) → (𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉))
4		fex 7263	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V)
5	3, 4	syl 18	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶1-1𝐵) → 𝐹 ∈ V)
6		f1o2ndf1 8170	. . . 4 ⊢ (𝐹:𝐴⟶1-1𝐵 → (2nd ↾ 𝐹):𝐹⟶1-1ontoran 𝐹)
7		df-2nd 8036	. . . . . . . . 9 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
8	7	funmpt2 6607	. . . . . . . 8 ⊢ Fun 2nd
9		resfunexg 7252	. . . . . . . 8 ⊢ ((Fun 2nd ∧ 𝐹 ∈ V) → (2nd ↾ 𝐹) ∈ V)
10	8, 5, 9	sylancr 599	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶1-1𝐵) → (2nd ↾ 𝐹) ∈ V)
11		f1oeq1 6842	. . . . . . . . . 10 ⊢ ((2nd ↾ 𝐹) = 𝑓 → ((2nd ↾ 𝐹):𝐹⟶1-1ontoran 𝐹 ↔ 𝑓:𝐹⟶1-1ontoran 𝐹))
12	11	biimpd 232	. . . . . . . . 9 ⊢ ((2nd ↾ 𝐹) = 𝑓 → ((2nd ↾ 𝐹):𝐹⟶1-1ontoran 𝐹 → 𝑓:𝐹⟶1-1ontoran 𝐹))
13	12	eqcoms 2771	. . . . . . . 8 ⊢ (𝑓 = (2nd ↾ 𝐹) → ((2nd ↾ 𝐹):𝐹⟶1-1ontoran 𝐹 → 𝑓:𝐹⟶1-1ontoran 𝐹))
14	13	adantl 487	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶1-1𝐵) ∧ 𝑓 = (2nd ↾ 𝐹)) → ((2nd ↾ 𝐹):𝐹⟶1-1ontoran 𝐹 → 𝑓:𝐹⟶1-1ontoran 𝐹))
15	10, 14	spcimedv 3553	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶1-1𝐵) → ((2nd ↾ 𝐹):𝐹⟶1-1ontoran 𝐹 → ∃𝑓 𝑓:𝐹⟶1-1ontoran 𝐹))
16	15	ex 418	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐹:𝐴⟶1-1𝐵 → ((2nd ↾ 𝐹):𝐹⟶1-1ontoran 𝐹 → ∃𝑓 𝑓:𝐹⟶1-1ontoran 𝐹)))
17	16	com13 89	. . . 4 ⊢ ((2nd ↾ 𝐹):𝐹⟶1-1ontoran 𝐹 → (𝐹:𝐴⟶1-1𝐵 → (𝐴 ∈ 𝑉 → ∃𝑓 𝑓:𝐹⟶1-1ontoran 𝐹)))
18	6, 17	mpcom 39	. . 3 ⊢ (𝐹:𝐴⟶1-1𝐵 → (𝐴 ∈ 𝑉 → ∃𝑓 𝑓:𝐹⟶1-1ontoran 𝐹))
19	18	impcom 413	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶1-1𝐵) → ∃𝑓 𝑓:𝐹⟶1-1ontoran 𝐹)
20		hasheqf1oi 14536	. 2 ⊢ (𝐹 ∈ V → (∃𝑓 𝑓:𝐹⟶1-1ontoran 𝐹 → (♯‘𝐹) = (♯‘ran 𝐹)))
21	5, 19, 20	sylc 66	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶1-1𝐵) → (♯‘𝐹) = (♯‘ran 𝐹))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  Vcvv 3454  {csn 4588  ∪ cuni 4872  ran crn 5659   ↾ cres 5660  Fun wfun 6560  ⟶wf 6562  ⟶1-1wf1 6563  ⟶1-1ontowf1o 6565  ‘cfv 6566  2^nd c2nd 8034  ♯chash 14515

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 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316

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-nel 3066  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 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-er 8755  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-card 10057  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-n0 12647  df-z 12734  df-uz 13006  df-hash 14516

This theorem is used by:  hashimarn  14626  hashf1dmrn  14629  usgrsizedg  29853  cyclnumvtx  30445  cycpmco2lem5  33790  cycpmconjslem2  33815  cyc3conja  33817  frlmdim  34354  ply1degltdim  34366  extdgfialglem1  34435  sticksstones2  43463