Theorem reprinrn 34609

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 6740	. . . . 5 ⊢ (𝑐:(0..^𝑆)⟶(𝐴 ∩ 𝐵) ↔ (𝑐:(0..^𝑆)⟶𝐴 ∧ 𝑐:(0..^𝑆)⟶𝐵))
2		df-f 6515	. . . . . . 7 ⊢ (𝑐:(0..^𝑆)⟶𝐵 ↔ (𝑐 Fn (0..^𝑆) ∧ ran 𝑐 ⊆ 𝐵))
3		ffn 6688	. . . . . . . . . 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 12192	. . . . . . . 8 ⊢ ℕ ∈ V
11	10	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℕ ∈ V)
12		reprval.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℕ)
13	11, 12	ssexd 5279	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V)
14		inex1g 5274	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ∩ 𝐵) ∈ V)
15	13, 14	syl 17	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ V)
16		ovex 7420	. . . . 5 ⊢ (0..^𝑆) ∈ V
17		elmapg 8812	. . . . 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 8812	. . . . . 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 4200	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
25	24, 12	sstrid 3958	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ ℕ)
26		reprval.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
27		reprval.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ ℕ0)
28	25, 26, 27	reprval 34601	. . . 4 ⊢ (𝜑 → ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀) = {𝑐 ∈ ((𝐴 ∩ 𝐵) ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀})
29	28	eleq2d 2814	. . 3 ⊢ (𝜑 → (𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀) ↔ 𝑐 ∈ {𝑐 ∈ ((𝐴 ∩ 𝐵) ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀}))
30		rabid 3427	. . 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 34601	. . . . . 6 ⊢ (𝜑 → (𝐴(repr‘𝑆)𝑀) = {𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀})
33	32	eleq2d 2814	. . . . 5 ⊢ (𝜑 → (𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ↔ 𝑐 ∈ {𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀}))
34		rabid 3427	. . . . 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 2109  {crab 3405  Vcvv 3447   ∩ cin 3913   ⊆ wss 3914  ran crn 5639   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387   ↑_m cmap 8799  0cc0 11068  ℕcn 12186  ℕ₀cn0 12442  ℤcz 12529  ..^cfzo 13615  Σcsu 15652  reprcrepr 34599

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-addcl 11128

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-map 8801  df-neg 11408  df-nn 12187  df-z 12530  df-seq 13967  df-sum 15653  df-repr 34600

This theorem is referenced by:  hashreprin  34611