Theorem sseqn 11207

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 11148 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 11208 and sshashneg 11209. (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 9606	. . . . . . . 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 9701	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝒫 𝐴) ∧ 𝑥 ≈ (1...𝑁)) → 𝑁 ∈ ℤ)
4	1, 3	fzfigd 10797	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝒫 𝐴) ∧ 𝑥 ≈ (1...𝑁)) → (1...𝑁) ∈ Fin)
5		enfii 7131	. . . . . . 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 11151	. . . . . . . . 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 11150	. . . . . . . 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 2267	. . . . . 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 6068	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝒫 𝐴) ∧ (𝑥 ∈ Fin ∧ (♯‘𝑥) = 𝑁)) → (1...(♯‘𝑥)) = (1...𝑁))
17		isfinite4im 11159	. . . . . . . 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 4135	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝒫 𝐴) ∧ (𝑥 ∈ Fin ∧ (♯‘𝑥) = 𝑁)) → (1...𝑁) ≈ 𝑥)
20	19	ensymd 7025	. . . . 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 3404	. . . . 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 2799	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 2205  {crab 2526   ∩ cin 3212  𝒫 cpw 3671   class class class wbr 4111  ‘cfv 5354  (class class class)co 6052   ≈ cen 6975  Fincfn 6977  1c1 8130  ℕ₀cn0 9498  ...cfz 10345  ♯chash 11142

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-0id 8237  ax-rnegex 8238  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-frec 6624  df-1o 6649  df-er 6769  df-en 6978  df-dom 6979  df-fin 6980  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-inn 9240  df-n0 9499  df-z 9580  df-uz 9857  df-fz 10346  df-ihash 11143

This theorem is referenced by: (None)