Theorem sseqn 11196

Description: Two ways to express the subsets of a class of a given size. It might seem that {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁} would suffice, but that would require the converse of hashcl 11139 or something similar. Although each side of the equality would be well defined if we changed 𝑁 ∈ ℕ₀ to 𝑁 ∈ ℤ, they would give different results for the (degenerate) case of a negative size, as shown at ssenneg 11197 and sshashneg 11198. (Contributed by Jim Kingdon, 22-May-2026.)

Assertion

Ref	Expression
sseqn	⊢ (𝑁 ∈ ℕ0 → {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≈ (1...𝑁)} = {𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝑁})

Distinct variable group:   𝑥,𝑁

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem sseqn

Step	Hyp	Ref	Expression
1		1zzd 9600	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝒫 𝐴) ∧ 𝑥 ≈ (1...𝑁)) → 1 ∈ ℤ)
2		simpll 527	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝒫 𝐴) ∧ 𝑥 ≈ (1...𝑁)) → 𝑁 ∈ ℕ0)
3	2	nn0zd 9694	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝒫 𝐴) ∧ 𝑥 ≈ (1...𝑁)) → 𝑁 ∈ ℤ)
4	1, 3	fzfigd 10789	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝒫 𝐴) ∧ 𝑥 ≈ (1...𝑁)) → (1...𝑁) ∈ Fin)
5		enfii 7128	. . . . . . 7 ⊢ (((1...𝑁) ∈ Fin ∧ 𝑥 ≈ (1...𝑁)) → 𝑥 ∈ Fin)
6	4, 5	sylancom 420	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝒫 𝐴) ∧ 𝑥 ≈ (1...𝑁)) → 𝑥 ∈ Fin)
7		simpr 110	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝒫 𝐴) ∧ 𝑥 ≈ (1...𝑁)) → 𝑥 ≈ (1...𝑁))
8		hashen 11142	. . . . . . . . 9 ⊢ ((𝑥 ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((♯‘𝑥) = (♯‘(1...𝑁)) ↔ 𝑥 ≈ (1...𝑁)))
9	6, 4, 8	syl2anc 411	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝒫 𝐴) ∧ 𝑥 ≈ (1...𝑁)) → ((♯‘𝑥) = (♯‘(1...𝑁)) ↔ 𝑥 ≈ (1...𝑁)))
10	7, 9	mpbird 167	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝒫 𝐴) ∧ 𝑥 ≈ (1...𝑁)) → (♯‘𝑥) = (♯‘(1...𝑁)))
11		hashfz1 11141	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = 𝑁)
12	2, 11	syl 14	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝒫 𝐴) ∧ 𝑥 ≈ (1...𝑁)) → (♯‘(1...𝑁)) = 𝑁)
13	10, 12	eqtrd 2265	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝒫 𝐴) ∧ 𝑥 ≈ (1...𝑁)) → (♯‘𝑥) = 𝑁)
14	6, 13	jca 306	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝒫 𝐴) ∧ 𝑥 ≈ (1...𝑁)) → (𝑥 ∈ Fin ∧ (♯‘𝑥) = 𝑁))
15		simprr 533	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝒫 𝐴) ∧ (𝑥 ∈ Fin ∧ (♯‘𝑥) = 𝑁)) → (♯‘𝑥) = 𝑁)
16	15	oveq2d 6065	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝒫 𝐴) ∧ (𝑥 ∈ Fin ∧ (♯‘𝑥) = 𝑁)) → (1...(♯‘𝑥)) = (1...𝑁))
17		isfinite4im 11150	. . . . . . . 8 ⊢ (𝑥 ∈ Fin → (1...(♯‘𝑥)) ≈ 𝑥)
18	17	ad2antrl 490	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝒫 𝐴) ∧ (𝑥 ∈ Fin ∧ (♯‘𝑥) = 𝑁)) → (1...(♯‘𝑥)) ≈ 𝑥)
19	16, 18	eqbrtrrd 4132	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝒫 𝐴) ∧ (𝑥 ∈ Fin ∧ (♯‘𝑥) = 𝑁)) → (1...𝑁) ≈ 𝑥)
20	19	ensymd 7022	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝒫 𝐴) ∧ (𝑥 ∈ Fin ∧ (♯‘𝑥) = 𝑁)) → 𝑥 ≈ (1...𝑁))
21	14, 20	impbida 600	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝒫 𝐴) → (𝑥 ≈ (1...𝑁) ↔ (𝑥 ∈ Fin ∧ (♯‘𝑥) = 𝑁)))
22	21	pm5.32da 452	. . 3 ⊢ (𝑁 ∈ ℕ0 → ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ≈ (1...𝑁)) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (𝑥 ∈ Fin ∧ (♯‘𝑥) = 𝑁))))
23		elin 3401	. . . . 5 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ Fin))
24	23	anbi1i 458	. . . 4 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑥) = 𝑁) ↔ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ Fin) ∧ (♯‘𝑥) = 𝑁))
25		anass 401	. . . 4 ⊢ (((𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ Fin) ∧ (♯‘𝑥) = 𝑁) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (𝑥 ∈ Fin ∧ (♯‘𝑥) = 𝑁)))
26	24, 25	bitri 184	. . 3 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑥) = 𝑁) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (𝑥 ∈ Fin ∧ (♯‘𝑥) = 𝑁)))
27	22, 26	bitr4di 198	. 2 ⊢ (𝑁 ∈ ℕ0 → ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ≈ (1...𝑁)) ↔ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑥) = 𝑁)))
28	27	rabbidva2 2796	1 ⊢ (𝑁 ∈ ℕ0 → {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≈ (1...𝑁)} = {𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝑁})

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2203  {crab 2524   ∩ cin 3209  𝒫 cpw 3668   class class class wbr 4108  ‘cfv 5351  (class class class)co 6049   ≈ cen 6972  Fincfn 6974  1c1 8124  ℕ₀cn0 9492  ...cfz 10338  ♯chash 11133

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-addcom 8223  ax-addass 8225  ax-distr 8227  ax-i2m1 8228  ax-0lt1 8229  ax-0id 8231  ax-rnegex 8232  ax-cnre 8234  ax-pre-ltirr 8235  ax-pre-ltwlin 8236  ax-pre-lttrn 8237  ax-pre-apti 8238  ax-pre-ltadd 8239

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-frec 6621  df-1o 6646  df-er 6766  df-en 6975  df-dom 6976  df-fin 6977  df-pnf 8306  df-mnf 8307  df-xr 8308  df-ltxr 8309  df-le 8310  df-sub 8442  df-neg 8443  df-inn 9234  df-n0 9493  df-z 9574  df-uz 9850  df-fz 10339  df-ihash 11134

This theorem is referenced by: (None)