Theorem reprinrn 34631

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 6703	. . . . 5 ⊢ (𝑐:(0..^𝑆)⟶(𝐴 ∩ 𝐵) ↔ (𝑐:(0..^𝑆)⟶𝐴 ∧ 𝑐:(0..^𝑆)⟶𝐵))
2		df-f 6485	. . . . . . 7 ⊢ (𝑐:(0..^𝑆)⟶𝐵 ↔ (𝑐 Fn (0..^𝑆) ∧ ran 𝑐 ⊆ 𝐵))
3		ffn 6651	. . . . . . . . . 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 12131	. . . . . . . 8 ⊢ ℕ ∈ V
11	10	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℕ ∈ V)
12		reprval.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℕ)
13	11, 12	ssexd 5260	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V)
14		inex1g 5255	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ∩ 𝐵) ∈ V)
15	13, 14	syl 17	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ V)
16		ovex 7379	. . . . 5 ⊢ (0..^𝑆) ∈ V
17		elmapg 8763	. . . . 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 8763	. . . . . 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 4184	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
25	24, 12	sstrid 3941	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ ℕ)
26		reprval.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
27		reprval.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ ℕ0)
28	25, 26, 27	reprval 34623	. . . 4 ⊢ (𝜑 → ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀) = {𝑐 ∈ ((𝐴 ∩ 𝐵) ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀})
29	28	eleq2d 2817	. . 3 ⊢ (𝜑 → (𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀) ↔ 𝑐 ∈ {𝑐 ∈ ((𝐴 ∩ 𝐵) ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀}))
30		rabid 3416	. . 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 34623	. . . . . 6 ⊢ (𝜑 → (𝐴(repr‘𝑆)𝑀) = {𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀})
33	32	eleq2d 2817	. . . . 5 ⊢ (𝜑 → (𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ↔ 𝑐 ∈ {𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀}))
34		rabid 3416	. . . . 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 1541   ∈ wcel 2111  {crab 3395  Vcvv 3436   ∩ cin 3896   ⊆ wss 3897  ran crn 5615   Fn wfn 6476  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346   ↑_m cmap 8750  0cc0 11006  ℕcn 12125  ℕ₀cn0 12381  ℤcz 12468  ..^cfzo 13554  Σcsu 15593  reprcrepr 34621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-addcl 11066

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-map 8752  df-neg 11347  df-nn 12126  df-z 12469  df-seq 13909  df-sum 15594  df-repr 34622

This theorem is referenced by:  hashreprin  34633