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Theorem reprinrn 34633
Description: Representations with term in an intersection. (Contributed by Thierry Arnoux, 11-Dec-2021.)
Hypotheses
Ref Expression
reprval.a (𝜑𝐴 ⊆ ℕ)
reprval.m (𝜑𝑀 ∈ ℤ)
reprval.s (𝜑𝑆 ∈ ℕ0)
Assertion
Ref Expression
reprinrn (𝜑 → (𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀) ↔ (𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ran 𝑐𝐵)))
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.
StepHypRef Expression
1 fin 6788 . . . . 5 (𝑐:(0..^𝑆)⟶(𝐴𝐵) ↔ (𝑐:(0..^𝑆)⟶𝐴𝑐:(0..^𝑆)⟶𝐵))
2 df-f 6565 . . . . . . 7 (𝑐:(0..^𝑆)⟶𝐵 ↔ (𝑐 Fn (0..^𝑆) ∧ ran 𝑐𝐵))
3 ffn 6736 . . . . . . . . . 10 (𝑐:(0..^𝑆)⟶𝐴𝑐 Fn (0..^𝑆))
43adantl 481 . . . . . . . . 9 ((𝜑𝑐:(0..^𝑆)⟶𝐴) → 𝑐 Fn (0..^𝑆))
54biantrurd 532 . . . . . . . 8 ((𝜑𝑐:(0..^𝑆)⟶𝐴) → (ran 𝑐𝐵 ↔ (𝑐 Fn (0..^𝑆) ∧ ran 𝑐𝐵)))
65bicomd 223 . . . . . . 7 ((𝜑𝑐:(0..^𝑆)⟶𝐴) → ((𝑐 Fn (0..^𝑆) ∧ ran 𝑐𝐵) ↔ ran 𝑐𝐵))
72, 6bitrid 283 . . . . . 6 ((𝜑𝑐:(0..^𝑆)⟶𝐴) → (𝑐:(0..^𝑆)⟶𝐵 ↔ ran 𝑐𝐵))
87pm5.32da 579 . . . . 5 (𝜑 → ((𝑐:(0..^𝑆)⟶𝐴𝑐:(0..^𝑆)⟶𝐵) ↔ (𝑐:(0..^𝑆)⟶𝐴 ∧ ran 𝑐𝐵)))
91, 8bitrid 283 . . . 4 (𝜑 → (𝑐:(0..^𝑆)⟶(𝐴𝐵) ↔ (𝑐:(0..^𝑆)⟶𝐴 ∧ ran 𝑐𝐵)))
10 nnex 12272 . . . . . . . 8 ℕ ∈ V
1110a1i 11 . . . . . . 7 (𝜑 → ℕ ∈ V)
12 reprval.a . . . . . . 7 (𝜑𝐴 ⊆ ℕ)
1311, 12ssexd 5324 . . . . . 6 (𝜑𝐴 ∈ V)
14 inex1g 5319 . . . . . 6 (𝐴 ∈ V → (𝐴𝐵) ∈ V)
1513, 14syl 17 . . . . 5 (𝜑 → (𝐴𝐵) ∈ V)
16 ovex 7464 . . . . 5 (0..^𝑆) ∈ V
17 elmapg 8879 . . . . 5 (((𝐴𝐵) ∈ V ∧ (0..^𝑆) ∈ V) → (𝑐 ∈ ((𝐴𝐵) ↑m (0..^𝑆)) ↔ 𝑐:(0..^𝑆)⟶(𝐴𝐵)))
1815, 16, 17sylancl 586 . . . 4 (𝜑 → (𝑐 ∈ ((𝐴𝐵) ↑m (0..^𝑆)) ↔ 𝑐:(0..^𝑆)⟶(𝐴𝐵)))
19 elmapg 8879 . . . . . 6 ((𝐴 ∈ V ∧ (0..^𝑆) ∈ V) → (𝑐 ∈ (𝐴m (0..^𝑆)) ↔ 𝑐:(0..^𝑆)⟶𝐴))
2013, 16, 19sylancl 586 . . . . 5 (𝜑 → (𝑐 ∈ (𝐴m (0..^𝑆)) ↔ 𝑐:(0..^𝑆)⟶𝐴))
2120anbi1d 631 . . . 4 (𝜑 → ((𝑐 ∈ (𝐴m (0..^𝑆)) ∧ ran 𝑐𝐵) ↔ (𝑐:(0..^𝑆)⟶𝐴 ∧ ran 𝑐𝐵)))
229, 18, 213bitr4d 311 . . 3 (𝜑 → (𝑐 ∈ ((𝐴𝐵) ↑m (0..^𝑆)) ↔ (𝑐 ∈ (𝐴m (0..^𝑆)) ∧ ran 𝑐𝐵)))
2322anbi1d 631 . 2 (𝜑 → ((𝑐 ∈ ((𝐴𝐵) ↑m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀) ↔ ((𝑐 ∈ (𝐴m (0..^𝑆)) ∧ ran 𝑐𝐵) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀)))
24 inss1 4237 . . . . . 6 (𝐴𝐵) ⊆ 𝐴
2524, 12sstrid 3995 . . . . 5 (𝜑 → (𝐴𝐵) ⊆ ℕ)
26 reprval.m . . . . 5 (𝜑𝑀 ∈ ℤ)
27 reprval.s . . . . 5 (𝜑𝑆 ∈ ℕ0)
2825, 26, 27reprval 34625 . . . 4 (𝜑 → ((𝐴𝐵)(repr‘𝑆)𝑀) = {𝑐 ∈ ((𝐴𝐵) ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀})
2928eleq2d 2827 . . 3 (𝜑 → (𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀) ↔ 𝑐 ∈ {𝑐 ∈ ((𝐴𝐵) ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀}))
30 rabid 3458 . . 3 (𝑐 ∈ {𝑐 ∈ ((𝐴𝐵) ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀} ↔ (𝑐 ∈ ((𝐴𝐵) ↑m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀))
3129, 30bitrdi 287 . 2 (𝜑 → (𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀) ↔ (𝑐 ∈ ((𝐴𝐵) ↑m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀)))
3212, 26, 27reprval 34625 . . . . . 6 (𝜑 → (𝐴(repr‘𝑆)𝑀) = {𝑐 ∈ (𝐴m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀})
3332eleq2d 2827 . . . . 5 (𝜑 → (𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ↔ 𝑐 ∈ {𝑐 ∈ (𝐴m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀}))
34 rabid 3458 . . . . 5 (𝑐 ∈ {𝑐 ∈ (𝐴m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀} ↔ (𝑐 ∈ (𝐴m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀))
3533, 34bitrdi 287 . . . 4 (𝜑 → (𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ↔ (𝑐 ∈ (𝐴m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀)))
3635anbi1d 631 . . 3 (𝜑 → ((𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ran 𝑐𝐵) ↔ ((𝑐 ∈ (𝐴m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀) ∧ ran 𝑐𝐵)))
37 an32 646 . . 3 (((𝑐 ∈ (𝐴m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀) ∧ ran 𝑐𝐵) ↔ ((𝑐 ∈ (𝐴m (0..^𝑆)) ∧ ran 𝑐𝐵) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀))
3836, 37bitrdi 287 . 2 (𝜑 → ((𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ran 𝑐𝐵) ↔ ((𝑐 ∈ (𝐴m (0..^𝑆)) ∧ ran 𝑐𝐵) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀)))
3923, 31, 383bitr4d 311 1 (𝜑 → (𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀) ↔ (𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ran 𝑐𝐵)))
Colors of variables: wff setvar class
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  cn 12266  0cn0 12526  cz 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
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