Theorem reprinrn 34633

Description: Representations with term in an intersection. (Contributed by Thierry Arnoux, 11-Dec-2021.)

Hypotheses

Ref	Expression
reprval.a	⊢ (𝜑 → 𝐴 ⊆ ℕ)
reprval.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
reprval.s	⊢ (𝜑 → 𝑆 ∈ ℕ0)

Assertion

Ref	Expression
reprinrn	⊢ (𝜑 → (𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀) ↔ (𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ran 𝑐 ⊆ 𝐵)))

Distinct variable groups:   𝐴,𝑐   𝑀,𝑐   𝑆,𝑐   𝜑,𝑐   𝐵,𝑐

Proof of Theorem reprinrn

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fin 6788	. . . . 5 ⊢ (𝑐:(0..^𝑆)⟶(𝐴 ∩ 𝐵) ↔ (𝑐:(0..^𝑆)⟶𝐴 ∧ 𝑐:(0..^𝑆)⟶𝐵))
2		df-f 6565	. . . . . . 7 ⊢ (𝑐:(0..^𝑆)⟶𝐵 ↔ (𝑐 Fn (0..^𝑆) ∧ ran 𝑐 ⊆ 𝐵))
3		ffn 6736	. . . . . . . . . 10 ⊢ (𝑐:(0..^𝑆)⟶𝐴 → 𝑐 Fn (0..^𝑆))
4	3	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐:(0..^𝑆)⟶𝐴) → 𝑐 Fn (0..^𝑆))
5	4	biantrurd 532	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐:(0..^𝑆)⟶𝐴) → (ran 𝑐 ⊆ 𝐵 ↔ (𝑐 Fn (0..^𝑆) ∧ ran 𝑐 ⊆ 𝐵)))
6	5	bicomd 223	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐:(0..^𝑆)⟶𝐴) → ((𝑐 Fn (0..^𝑆) ∧ ran 𝑐 ⊆ 𝐵) ↔ ran 𝑐 ⊆ 𝐵))
7	2, 6	bitrid 283	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐:(0..^𝑆)⟶𝐴) → (𝑐:(0..^𝑆)⟶𝐵 ↔ ran 𝑐 ⊆ 𝐵))
8	7	pm5.32da 579	. . . . 5 ⊢ (𝜑 → ((𝑐:(0..^𝑆)⟶𝐴 ∧ 𝑐:(0..^𝑆)⟶𝐵) ↔ (𝑐:(0..^𝑆)⟶𝐴 ∧ ran 𝑐 ⊆ 𝐵)))
9	1, 8	bitrid 283	. . . 4 ⊢ (𝜑 → (𝑐:(0..^𝑆)⟶(𝐴 ∩ 𝐵) ↔ (𝑐:(0..^𝑆)⟶𝐴 ∧ ran 𝑐 ⊆ 𝐵)))
10		nnex 12272	. . . . . . . 8 ⊢ ℕ ∈ V
11	10	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℕ ∈ V)
12		reprval.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℕ)
13	11, 12	ssexd 5324	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V)
14		inex1g 5319	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ∩ 𝐵) ∈ V)
15	13, 14	syl 17	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ V)
16		ovex 7464	. . . . 5 ⊢ (0..^𝑆) ∈ V
17		elmapg 8879	. . . . 5 ⊢ (((𝐴 ∩ 𝐵) ∈ V ∧ (0..^𝑆) ∈ V) → (𝑐 ∈ ((𝐴 ∩ 𝐵) ↑m (0..^𝑆)) ↔ 𝑐:(0..^𝑆)⟶(𝐴 ∩ 𝐵)))
18	15, 16, 17	sylancl 586	. . . 4 ⊢ (𝜑 → (𝑐 ∈ ((𝐴 ∩ 𝐵) ↑m (0..^𝑆)) ↔ 𝑐:(0..^𝑆)⟶(𝐴 ∩ 𝐵)))
19		elmapg 8879	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ (0..^𝑆) ∈ V) → (𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ↔ 𝑐:(0..^𝑆)⟶𝐴))
20	13, 16, 19	sylancl 586	. . . . 5 ⊢ (𝜑 → (𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ↔ 𝑐:(0..^𝑆)⟶𝐴))
21	20	anbi1d 631	. . . 4 ⊢ (𝜑 → ((𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∧ ran 𝑐 ⊆ 𝐵) ↔ (𝑐:(0..^𝑆)⟶𝐴 ∧ ran 𝑐 ⊆ 𝐵)))
22	9, 18, 21	3bitr4d 311	. . 3 ⊢ (𝜑 → (𝑐 ∈ ((𝐴 ∩ 𝐵) ↑m (0..^𝑆)) ↔ (𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∧ ran 𝑐 ⊆ 𝐵)))
23	22	anbi1d 631	. 2 ⊢ (𝜑 → ((𝑐 ∈ ((𝐴 ∩ 𝐵) ↑m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀) ↔ ((𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∧ ran 𝑐 ⊆ 𝐵) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀)))
24		inss1 4237	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
25	24, 12	sstrid 3995	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ ℕ)
26		reprval.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
27		reprval.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ ℕ0)
28	25, 26, 27	reprval 34625	. . . 4 ⊢ (𝜑 → ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀) = {𝑐 ∈ ((𝐴 ∩ 𝐵) ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀})
29	28	eleq2d 2827	. . 3 ⊢ (𝜑 → (𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀) ↔ 𝑐 ∈ {𝑐 ∈ ((𝐴 ∩ 𝐵) ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀}))
30		rabid 3458	. . 3 ⊢ (𝑐 ∈ {𝑐 ∈ ((𝐴 ∩ 𝐵) ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀} ↔ (𝑐 ∈ ((𝐴 ∩ 𝐵) ↑m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀))
31	29, 30	bitrdi 287	. 2 ⊢ (𝜑 → (𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀) ↔ (𝑐 ∈ ((𝐴 ∩ 𝐵) ↑m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀)))
32	12, 26, 27	reprval 34625	. . . . . 6 ⊢ (𝜑 → (𝐴(repr‘𝑆)𝑀) = {𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀})
33	32	eleq2d 2827	. . . . 5 ⊢ (𝜑 → (𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ↔ 𝑐 ∈ {𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀}))
34		rabid 3458	. . . . 5 ⊢ (𝑐 ∈ {𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀} ↔ (𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀))
35	33, 34	bitrdi 287	. . . 4 ⊢ (𝜑 → (𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ↔ (𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀)))
36	35	anbi1d 631	. . 3 ⊢ (𝜑 → ((𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ran 𝑐 ⊆ 𝐵) ↔ ((𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀) ∧ ran 𝑐 ⊆ 𝐵)))
37		an32 646	. . 3 ⊢ (((𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀) ∧ ran 𝑐 ⊆ 𝐵) ↔ ((𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∧ ran 𝑐 ⊆ 𝐵) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀))
38	36, 37	bitrdi 287	. 2 ⊢ (𝜑 → ((𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ran 𝑐 ⊆ 𝐵) ↔ ((𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∧ ran 𝑐 ⊆ 𝐵) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀)))
39	23, 31, 38	3bitr4d 311	1 ⊢ (𝜑 → (𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀) ↔ (𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ran 𝑐 ⊆ 𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {crab 3436  Vcvv 3480   ∩ cin 3950   ⊆ wss 3951  ran crn 5686   Fn wfn 6556  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431   ↑_m cmap 8866  0cc0 11155  ℕcn 12266  ℕ₀cn0 12526  ℤcz 12613  ..^cfzo 13694  Σcsu 15722  reprcrepr 34623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-addcl 11215

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-map 8868  df-neg 11495  df-nn 12267  df-z 12614  df-seq 14043  df-sum 15723  df-repr 34624

This theorem is referenced by:  hashreprin  34635