Theorem sseqn 11262

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 11203 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 11263 and sshashneg 11264. (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 9654	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝒫 𝐴) ∧ 𝑥 ≈ (1...𝑁)) → 1 ∈ ℤ)
2		simpll 531	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝒫 𝐴) ∧ 𝑥 ≈ (1...𝑁)) → 𝑁 ∈ ℕ0)
3	2	nn0zd 9749	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝒫 𝐴) ∧ 𝑥 ≈ (1...𝑁)) → 𝑁 ∈ ℤ)
4	1, 3	fzfigd 10851	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝒫 𝐴) ∧ 𝑥 ≈ (1...𝑁)) → (1...𝑁) ∈ Fin)
5		enfii 7170	. . . . . . 7 ⊢ (((1...𝑁) ∈ Fin ∧ 𝑥 ≈ (1...𝑁)) → 𝑥 ∈ Fin)
6	4, 5	sylancom 424	. . . . . 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 11206	. . . . . . . . 9 ⊢ ((𝑥 ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((♯‘𝑥) = (♯‘(1...𝑁)) ↔ 𝑥 ≈ (1...𝑁)))
9	6, 4, 8	syl2anc 415	. . . . . . . 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 11205	. . . . . . . 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 2271	. . . . . 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 537	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝒫 𝐴) ∧ (𝑥 ∈ Fin ∧ (♯‘𝑥) = 𝑁)) → (♯‘𝑥) = 𝑁)
16	15	oveq2d 6095	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝒫 𝐴) ∧ (𝑥 ∈ Fin ∧ (♯‘𝑥) = 𝑁)) → (1...(♯‘𝑥)) = (1...𝑁))
17		isfinite4im 11214	. . . . . . . 8 ⊢ (𝑥 ∈ Fin → (1...(♯‘𝑥)) ≈ 𝑥)
18	17	ad2antrl 494	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝒫 𝐴) ∧ (𝑥 ∈ Fin ∧ (♯‘𝑥) = 𝑁)) → (1...(♯‘𝑥)) ≈ 𝑥)
19	16, 18	eqbrtrrd 4152	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝒫 𝐴) ∧ (𝑥 ∈ Fin ∧ (♯‘𝑥) = 𝑁)) → (1...𝑁) ≈ 𝑥)
20	19	ensymd 7064	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝒫 𝐴) ∧ (𝑥 ∈ Fin ∧ (♯‘𝑥) = 𝑁)) → 𝑥 ≈ (1...𝑁))
21	14, 20	impbida 604	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝒫 𝐴) → (𝑥 ≈ (1...𝑁) ↔ (𝑥 ∈ Fin ∧ (♯‘𝑥) = 𝑁)))
22	21	pm5.32da 456	. . 3 ⊢ (𝑁 ∈ ℕ0 → ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ≈ (1...𝑁)) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (𝑥 ∈ Fin ∧ (♯‘𝑥) = 𝑁))))
23		elin 3412	. . . . 5 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ Fin))
24	23	anbi1i 462	. . . 4 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ (♯‘𝑥) = 𝑁) ↔ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ Fin) ∧ (♯‘𝑥) = 𝑁))
25		anass 405	. . . 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 2805	1 ⊢ (𝑁 ∈ ℕ0 → {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≈ (1...𝑁)} = {𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝑁})

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  {crab 2532   ∩ cin 3219  𝒫 cpw 3688   class class class wbr 4128  ‘cfv 5375  (class class class)co 6079   ≈ cen 7014  Fincfn 7016  1c1 8174  ℕ₀cn0 9546  ...cfz 10394  ♯chash 11197

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-addass 8275  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-0id 8281  ax-rnegex 8282  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-frec 6656  df-1o 6681  df-er 6801  df-en 7017  df-dom 7018  df-fin 7019  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-inn 9288  df-n0 9547  df-z 9628  df-uz 9905  df-fz 10395  df-ihash 11198

This theorem is referenced by: (None)