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Theorem reprinrn 33568
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 6768 . . . . 5 (𝑐:(0..^𝑆)⟶(𝐴𝐵) ↔ (𝑐:(0..^𝑆)⟶𝐴𝑐:(0..^𝑆)⟶𝐵))
2 df-f 6544 . . . . . . 7 (𝑐:(0..^𝑆)⟶𝐵 ↔ (𝑐 Fn (0..^𝑆) ∧ ran 𝑐𝐵))
3 ffn 6714 . . . . . . . . . 10 (𝑐:(0..^𝑆)⟶𝐴𝑐 Fn (0..^𝑆))
43adantl 483 . . . . . . . . 9 ((𝜑𝑐:(0..^𝑆)⟶𝐴) → 𝑐 Fn (0..^𝑆))
54biantrurd 534 . . . . . . . 8 ((𝜑𝑐:(0..^𝑆)⟶𝐴) → (ran 𝑐𝐵 ↔ (𝑐 Fn (0..^𝑆) ∧ ran 𝑐𝐵)))
65bicomd 222 . . . . . . 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 580 . . . . 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 12214 . . . . . . . 8 ℕ ∈ V
1110a1i 11 . . . . . . 7 (𝜑 → ℕ ∈ V)
12 reprval.a . . . . . . 7 (𝜑𝐴 ⊆ ℕ)
1311, 12ssexd 5323 . . . . . 6 (𝜑𝐴 ∈ V)
14 inex1g 5318 . . . . . 6 (𝐴 ∈ V → (𝐴𝐵) ∈ V)
1513, 14syl 17 . . . . 5 (𝜑 → (𝐴𝐵) ∈ V)
16 ovex 7437 . . . . 5 (0..^𝑆) ∈ V
17 elmapg 8829 . . . . 5 (((𝐴𝐵) ∈ V ∧ (0..^𝑆) ∈ V) → (𝑐 ∈ ((𝐴𝐵) ↑m (0..^𝑆)) ↔ 𝑐:(0..^𝑆)⟶(𝐴𝐵)))
1815, 16, 17sylancl 587 . . . 4 (𝜑 → (𝑐 ∈ ((𝐴𝐵) ↑m (0..^𝑆)) ↔ 𝑐:(0..^𝑆)⟶(𝐴𝐵)))
19 elmapg 8829 . . . . . 6 ((𝐴 ∈ V ∧ (0..^𝑆) ∈ V) → (𝑐 ∈ (𝐴m (0..^𝑆)) ↔ 𝑐:(0..^𝑆)⟶𝐴))
2013, 16, 19sylancl 587 . . . . 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 4227 . . . . . 6 (𝐴𝐵) ⊆ 𝐴
2524, 12sstrid 3992 . . . . 5 (𝜑 → (𝐴𝐵) ⊆ ℕ)
26 reprval.m . . . . 5 (𝜑𝑀 ∈ ℤ)
27 reprval.s . . . . 5 (𝜑𝑆 ∈ ℕ0)
2825, 26, 27reprval 33560 . . . 4 (𝜑 → ((𝐴𝐵)(repr‘𝑆)𝑀) = {𝑐 ∈ ((𝐴𝐵) ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀})
2928eleq2d 2820 . . 3 (𝜑 → (𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀) ↔ 𝑐 ∈ {𝑐 ∈ ((𝐴𝐵) ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀}))
30 rabid 3453 . . 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 33560 . . . . . 6 (𝜑 → (𝐴(repr‘𝑆)𝑀) = {𝑐 ∈ (𝐴m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀})
3332eleq2d 2820 . . . . 5 (𝜑 → (𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ↔ 𝑐 ∈ {𝑐 ∈ (𝐴m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀}))
34 rabid 3453 . . . . 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 645 . . 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 205  wa 397   = wceq 1542  wcel 2107  {crab 3433  Vcvv 3475  cin 3946  wss 3947  ran crn 5676   Fn wfn 6535  wf 6536  cfv 6540  (class class class)co 7404  m cmap 8816  0cc0 11106  cn 12208  0cn0 12468  cz 12554  ..^cfzo 13623  Σcsu 15628  reprcrepr 33558
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-addcl 11166
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7851  df-1st 7970  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-map 8818  df-neg 11443  df-nn 12209  df-z 12555  df-seq 13963  df-sum 15629  df-repr 33559
This theorem is referenced by:  hashreprin  33570
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