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Theorem reprinrn 34612
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 6789 . . . . 5 (𝑐:(0..^𝑆)⟶(𝐴𝐵) ↔ (𝑐:(0..^𝑆)⟶𝐴𝑐:(0..^𝑆)⟶𝐵))
2 df-f 6567 . . . . . . 7 (𝑐:(0..^𝑆)⟶𝐵 ↔ (𝑐 Fn (0..^𝑆) ∧ ran 𝑐𝐵))
3 ffn 6737 . . . . . . . . . 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 12270 . . . . . . . 8 ℕ ∈ V
1110a1i 11 . . . . . . 7 (𝜑 → ℕ ∈ V)
12 reprval.a . . . . . . 7 (𝜑𝐴 ⊆ ℕ)
1311, 12ssexd 5330 . . . . . 6 (𝜑𝐴 ∈ V)
14 inex1g 5325 . . . . . 6 (𝐴 ∈ V → (𝐴𝐵) ∈ V)
1513, 14syl 17 . . . . 5 (𝜑 → (𝐴𝐵) ∈ V)
16 ovex 7464 . . . . 5 (0..^𝑆) ∈ V
17 elmapg 8878 . . . . 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 8878 . . . . . 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 4245 . . . . . 6 (𝐴𝐵) ⊆ 𝐴
2524, 12sstrid 4007 . . . . 5 (𝜑 → (𝐴𝐵) ⊆ ℕ)
26 reprval.m . . . . 5 (𝜑𝑀 ∈ ℤ)
27 reprval.s . . . . 5 (𝜑𝑆 ∈ ℕ0)
2825, 26, 27reprval 34604 . . . 4 (𝜑 → ((𝐴𝐵)(repr‘𝑆)𝑀) = {𝑐 ∈ ((𝐴𝐵) ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀})
2928eleq2d 2825 . . 3 (𝜑 → (𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀) ↔ 𝑐 ∈ {𝑐 ∈ ((𝐴𝐵) ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀}))
30 rabid 3455 . . 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 34604 . . . . . 6 (𝜑 → (𝐴(repr‘𝑆)𝑀) = {𝑐 ∈ (𝐴m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀})
3332eleq2d 2825 . . . . 5 (𝜑 → (𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ↔ 𝑐 ∈ {𝑐 ∈ (𝐴m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀}))
34 rabid 3455 . . . . 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 1537  wcel 2106  {crab 3433  Vcvv 3478  cin 3962  wss 3963  ran crn 5690   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  m cmap 8865  0cc0 11153  cn 12264  0cn0 12524  cz 12611  ..^cfzo 13691  Σcsu 15719  reprcrepr 34602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-addcl 11213
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-map 8867  df-neg 11493  df-nn 12265  df-z 12612  df-seq 14040  df-sum 15720  df-repr 34603
This theorem is referenced by:  hashreprin  34614
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