Theorem reprinrn 35005

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 6758	. . . . 5 ⊢ (𝑐:(0..^𝑆)⟶(𝐴 ∩ 𝐵) ↔ (𝑐:(0..^𝑆)⟶𝐴 ∧ 𝑐:(0..^𝑆)⟶𝐵))
2		df-f 6540	. . . . . . 7 ⊢ (𝑐:(0..^𝑆)⟶𝐵 ↔ (𝑐 Fn (0..^𝑆) ∧ ran 𝑐 ⊆ 𝐵))
3		ffn 6705	. . . . . . . . . 10 ⊢ (𝑐:(0..^𝑆)⟶𝐴 → 𝑐 Fn (0..^𝑆))
4	3	adantl 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐:(0..^𝑆)⟶𝐴) → 𝑐 Fn (0..^𝑆))
5	4	biantrurd 541	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐:(0..^𝑆)⟶𝐴) → (ran 𝑐 ⊆ 𝐵 ↔ (𝑐 Fn (0..^𝑆) ∧ ran 𝑐 ⊆ 𝐵)))
6	5	bicomd 226	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐:(0..^𝑆)⟶𝐴) → ((𝑐 Fn (0..^𝑆) ∧ ran 𝑐 ⊆ 𝐵) ↔ ran 𝑐 ⊆ 𝐵))
7	2, 6	bitrid 286	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐:(0..^𝑆)⟶𝐴) → (𝑐:(0..^𝑆)⟶𝐵 ↔ ran 𝑐 ⊆ 𝐵))
8	7	pm5.32da 589	. . . . 5 ⊢ (𝜑 → ((𝑐:(0..^𝑆)⟶𝐴 ∧ 𝑐:(0..^𝑆)⟶𝐵) ↔ (𝑐:(0..^𝑆)⟶𝐴 ∧ ran 𝑐 ⊆ 𝐵)))
9	1, 8	bitrid 286	. . . 4 ⊢ (𝜑 → (𝑐:(0..^𝑆)⟶(𝐴 ∩ 𝐵) ↔ (𝑐:(0..^𝑆)⟶𝐴 ∧ ran 𝑐 ⊆ 𝐵)))
10		nnex 12234	. . . . . . . 8 ⊢ ℕ ∈ V
11	10	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℕ ∈ V)
12		reprval.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℕ)
13	11, 12	ssexd 5295	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V)
14		inex1g 5288	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ∩ 𝐵) ∈ V)
15	13, 14	syl 18	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ V)
16		ovex 7443	. . . . 5 ⊢ (0..^𝑆) ∈ V
17		elmapg 8832	. . . . 5 ⊢ (((𝐴 ∩ 𝐵) ∈ V ∧ (0..^𝑆) ∈ V) → (𝑐 ∈ ((𝐴 ∩ 𝐵) ↑m (0..^𝑆)) ↔ 𝑐:(0..^𝑆)⟶(𝐴 ∩ 𝐵)))
18	15, 16, 17	sylancl 597	. . . 4 ⊢ (𝜑 → (𝑐 ∈ ((𝐴 ∩ 𝐵) ↑m (0..^𝑆)) ↔ 𝑐:(0..^𝑆)⟶(𝐴 ∩ 𝐵)))
19		elmapg 8832	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ (0..^𝑆) ∈ V) → (𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ↔ 𝑐:(0..^𝑆)⟶𝐴))
20	13, 16, 19	sylancl 597	. . . . 5 ⊢ (𝜑 → (𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ↔ 𝑐:(0..^𝑆)⟶𝐴))
21	20	anbi1d 642	. . . 4 ⊢ (𝜑 → ((𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∧ ran 𝑐 ⊆ 𝐵) ↔ (𝑐:(0..^𝑆)⟶𝐴 ∧ ran 𝑐 ⊆ 𝐵)))
22	9, 18, 21	3bitr4d 314	. . 3 ⊢ (𝜑 → (𝑐 ∈ ((𝐴 ∩ 𝐵) ↑m (0..^𝑆)) ↔ (𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∧ ran 𝑐 ⊆ 𝐵)))
23	22	anbi1d 642	. 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 4189	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
25	24, 12	sstrid 3948	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ ℕ)
26		reprval.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
27		reprval.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ ℕ0)
28	25, 26, 27	reprval 34997	. . . 4 ⊢ (𝜑 → ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀) = {𝑐 ∈ ((𝐴 ∩ 𝐵) ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀})
29	28	eleq2d 2849	. . 3 ⊢ (𝜑 → (𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀) ↔ 𝑐 ∈ {𝑐 ∈ ((𝐴 ∩ 𝐵) ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀}))
30		rabid 3437	. . 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 290	. 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 34997	. . . . . 6 ⊢ (𝜑 → (𝐴(repr‘𝑆)𝑀) = {𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀})
33	32	eleq2d 2849	. . . . 5 ⊢ (𝜑 → (𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ↔ 𝑐 ∈ {𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀}))
34		rabid 3437	. . . . 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 290	. . . 4 ⊢ (𝜑 → (𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ↔ (𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀)))
36	35	anbi1d 642	. . 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 658	. . 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 290	. 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 314	1 ⊢ (𝜑 → (𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀) ↔ (𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ran 𝑐 ⊆ 𝐵)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  {crab 3416  Vcvv 3455   ∩ cin 3904   ⊆ wss 3905  ran crn 5662   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ↑_m cmap 8820  0cc0 11095  ℕcn 12228  ℕ₀cn0 12499  ℤcz 12586  ..^cfzo 13678  Σcsu 15733  reprcrepr 34995

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-addcl 11155

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-map 8822  df-neg 11439  df-nn 12229  df-z 12587  df-seq 14034  df-sum 15734  df-repr 34996

This theorem is referenced by:  hashreprin  35007