Theorem reprinrn 32598

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 6654	. . . . 5 ⊢ (𝑐:(0..^𝑆)⟶(𝐴 ∩ 𝐵) ↔ (𝑐:(0..^𝑆)⟶𝐴 ∧ 𝑐:(0..^𝑆)⟶𝐵))
2		df-f 6437	. . . . . . 7 ⊢ (𝑐:(0..^𝑆)⟶𝐵 ↔ (𝑐 Fn (0..^𝑆) ∧ ran 𝑐 ⊆ 𝐵))
3		ffn 6600	. . . . . . . . . 10 ⊢ (𝑐:(0..^𝑆)⟶𝐴 → 𝑐 Fn (0..^𝑆))
4	3	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐:(0..^𝑆)⟶𝐴) → 𝑐 Fn (0..^𝑆))
5	4	biantrurd 533	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐:(0..^𝑆)⟶𝐴) → (ran 𝑐 ⊆ 𝐵 ↔ (𝑐 Fn (0..^𝑆) ∧ ran 𝑐 ⊆ 𝐵)))
6	5	bicomd 222	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐:(0..^𝑆)⟶𝐴) → ((𝑐 Fn (0..^𝑆) ∧ ran 𝑐 ⊆ 𝐵) ↔ ran 𝑐 ⊆ 𝐵))
7	2, 6	syl5bb 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	syl5bb 283	. . . 4 ⊢ (𝜑 → (𝑐:(0..^𝑆)⟶(𝐴 ∩ 𝐵) ↔ (𝑐:(0..^𝑆)⟶𝐴 ∧ ran 𝑐 ⊆ 𝐵)))
10		nnex 11979	. . . . . . . 8 ⊢ ℕ ∈ V
11	10	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℕ ∈ V)
12		reprval.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℕ)
13	11, 12	ssexd 5248	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V)
14		inex1g 5243	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ∩ 𝐵) ∈ V)
15	13, 14	syl 17	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ V)
16		ovex 7308	. . . . 5 ⊢ (0..^𝑆) ∈ V
17		elmapg 8628	. . . . 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 8628	. . . . . 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 630	. . . 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 630	. 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 4162	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
25	24, 12	sstrid 3932	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ ℕ)
26		reprval.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
27		reprval.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ ℕ0)
28	25, 26, 27	reprval 32590	. . . 4 ⊢ (𝜑 → ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀) = {𝑐 ∈ ((𝐴 ∩ 𝐵) ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀})
29	28	eleq2d 2824	. . 3 ⊢ (𝜑 → (𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀) ↔ 𝑐 ∈ {𝑐 ∈ ((𝐴 ∩ 𝐵) ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀}))
30		rabid 3310	. . 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 32590	. . . . . 6 ⊢ (𝜑 → (𝐴(repr‘𝑆)𝑀) = {𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀})
33	32	eleq2d 2824	. . . . 5 ⊢ (𝜑 → (𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ↔ 𝑐 ∈ {𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀}))
34		rabid 3310	. . . . 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 630	. . 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 643	. . 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 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  {crab 3068  Vcvv 3432   ∩ cin 3886   ⊆ wss 3887  ran crn 5590   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ↑_m cmap 8615  0cc0 10871  ℕcn 11973  ℕ₀cn0 12233  ℤcz 12319  ..^cfzo 13382  Σcsu 15397  reprcrepr 32588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-addcl 10931

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-map 8617  df-neg 11208  df-nn 11974  df-z 12320  df-seq 13722  df-sum 15398  df-repr 32589

This theorem is referenced by:  hashreprin  32600