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Theorem reprinrn 32171
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 6559 . . . . 5 (𝑐:(0..^𝑆)⟶(𝐴𝐵) ↔ (𝑐:(0..^𝑆)⟶𝐴𝑐:(0..^𝑆)⟶𝐵))
2 df-f 6344 . . . . . . 7 (𝑐:(0..^𝑆)⟶𝐵 ↔ (𝑐 Fn (0..^𝑆) ∧ ran 𝑐𝐵))
3 ffn 6505 . . . . . . . . . 10 (𝑐:(0..^𝑆)⟶𝐴𝑐 Fn (0..^𝑆))
43adantl 485 . . . . . . . . 9 ((𝜑𝑐:(0..^𝑆)⟶𝐴) → 𝑐 Fn (0..^𝑆))
54biantrurd 536 . . . . . . . 8 ((𝜑𝑐:(0..^𝑆)⟶𝐴) → (ran 𝑐𝐵 ↔ (𝑐 Fn (0..^𝑆) ∧ ran 𝑐𝐵)))
65bicomd 226 . . . . . . 7 ((𝜑𝑐:(0..^𝑆)⟶𝐴) → ((𝑐 Fn (0..^𝑆) ∧ ran 𝑐𝐵) ↔ ran 𝑐𝐵))
72, 6syl5bb 286 . . . . . 6 ((𝜑𝑐:(0..^𝑆)⟶𝐴) → (𝑐:(0..^𝑆)⟶𝐵 ↔ ran 𝑐𝐵))
87pm5.32da 582 . . . . 5 (𝜑 → ((𝑐:(0..^𝑆)⟶𝐴𝑐:(0..^𝑆)⟶𝐵) ↔ (𝑐:(0..^𝑆)⟶𝐴 ∧ ran 𝑐𝐵)))
91, 8syl5bb 286 . . . 4 (𝜑 → (𝑐:(0..^𝑆)⟶(𝐴𝐵) ↔ (𝑐:(0..^𝑆)⟶𝐴 ∧ ran 𝑐𝐵)))
10 nnex 11725 . . . . . . . 8 ℕ ∈ V
1110a1i 11 . . . . . . 7 (𝜑 → ℕ ∈ V)
12 reprval.a . . . . . . 7 (𝜑𝐴 ⊆ ℕ)
1311, 12ssexd 5193 . . . . . 6 (𝜑𝐴 ∈ V)
14 inex1g 5188 . . . . . 6 (𝐴 ∈ V → (𝐴𝐵) ∈ V)
1513, 14syl 17 . . . . 5 (𝜑 → (𝐴𝐵) ∈ V)
16 ovex 7206 . . . . 5 (0..^𝑆) ∈ V
17 elmapg 8453 . . . . 5 (((𝐴𝐵) ∈ V ∧ (0..^𝑆) ∈ V) → (𝑐 ∈ ((𝐴𝐵) ↑m (0..^𝑆)) ↔ 𝑐:(0..^𝑆)⟶(𝐴𝐵)))
1815, 16, 17sylancl 589 . . . 4 (𝜑 → (𝑐 ∈ ((𝐴𝐵) ↑m (0..^𝑆)) ↔ 𝑐:(0..^𝑆)⟶(𝐴𝐵)))
19 elmapg 8453 . . . . . 6 ((𝐴 ∈ V ∧ (0..^𝑆) ∈ V) → (𝑐 ∈ (𝐴m (0..^𝑆)) ↔ 𝑐:(0..^𝑆)⟶𝐴))
2013, 16, 19sylancl 589 . . . . 5 (𝜑 → (𝑐 ∈ (𝐴m (0..^𝑆)) ↔ 𝑐:(0..^𝑆)⟶𝐴))
2120anbi1d 633 . . . 4 (𝜑 → ((𝑐 ∈ (𝐴m (0..^𝑆)) ∧ ran 𝑐𝐵) ↔ (𝑐:(0..^𝑆)⟶𝐴 ∧ ran 𝑐𝐵)))
229, 18, 213bitr4d 314 . . 3 (𝜑 → (𝑐 ∈ ((𝐴𝐵) ↑m (0..^𝑆)) ↔ (𝑐 ∈ (𝐴m (0..^𝑆)) ∧ ran 𝑐𝐵)))
2322anbi1d 633 . 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 4120 . . . . . 6 (𝐴𝐵) ⊆ 𝐴
2524, 12sstrid 3889 . . . . 5 (𝜑 → (𝐴𝐵) ⊆ ℕ)
26 reprval.m . . . . 5 (𝜑𝑀 ∈ ℤ)
27 reprval.s . . . . 5 (𝜑𝑆 ∈ ℕ0)
2825, 26, 27reprval 32163 . . . 4 (𝜑 → ((𝐴𝐵)(repr‘𝑆)𝑀) = {𝑐 ∈ ((𝐴𝐵) ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀})
2928eleq2d 2819 . . 3 (𝜑 → (𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀) ↔ 𝑐 ∈ {𝑐 ∈ ((𝐴𝐵) ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀}))
30 rabid 3282 . . 3 (𝑐 ∈ {𝑐 ∈ ((𝐴𝐵) ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀} ↔ (𝑐 ∈ ((𝐴𝐵) ↑m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀))
3129, 30bitrdi 290 . 2 (𝜑 → (𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀) ↔ (𝑐 ∈ ((𝐴𝐵) ↑m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀)))
3212, 26, 27reprval 32163 . . . . . 6 (𝜑 → (𝐴(repr‘𝑆)𝑀) = {𝑐 ∈ (𝐴m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀})
3332eleq2d 2819 . . . . 5 (𝜑 → (𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ↔ 𝑐 ∈ {𝑐 ∈ (𝐴m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀}))
34 rabid 3282 . . . . 5 (𝑐 ∈ {𝑐 ∈ (𝐴m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀} ↔ (𝑐 ∈ (𝐴m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀))
3533, 34bitrdi 290 . . . 4 (𝜑 → (𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ↔ (𝑐 ∈ (𝐴m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀)))
3635anbi1d 633 . . 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 290 . 2 (𝜑 → ((𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ran 𝑐𝐵) ↔ ((𝑐 ∈ (𝐴m (0..^𝑆)) ∧ ran 𝑐𝐵) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀)))
3923, 31, 383bitr4d 314 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 209  wa 399   = wceq 1542  wcel 2114  {crab 3058  Vcvv 3399  cin 3843  wss 3844  ran crn 5527   Fn wfn 6335  wf 6336  cfv 6340  (class class class)co 7173  m cmap 8440  0cc0 10618  cn 11719  0cn0 11979  cz 12065  ..^cfzo 13127  Σcsu 15138  reprcrepr 32161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297  ax-un 7482  ax-cnex 10674  ax-resscn 10675  ax-1cn 10676  ax-addcl 10678
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3401  df-sbc 3682  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-pss 3863  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-tp 4522  df-op 4524  df-uni 4798  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5484  df-we 5486  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7176  df-oprab 7177  df-mpo 7178  df-om 7603  df-1st 7717  df-2nd 7718  df-wrecs 7979  df-recs 8040  df-rdg 8078  df-map 8442  df-neg 10954  df-nn 11720  df-z 12066  df-seq 13464  df-sum 15139  df-repr 32162
This theorem is referenced by:  hashreprin  32173
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