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Theorem reprinrn 33231
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 6722 . . . . 5 (𝑐:(0..^𝑆)⟶(𝐴𝐵) ↔ (𝑐:(0..^𝑆)⟶𝐴𝑐:(0..^𝑆)⟶𝐵))
2 df-f 6500 . . . . . . 7 (𝑐:(0..^𝑆)⟶𝐵 ↔ (𝑐 Fn (0..^𝑆) ∧ ran 𝑐𝐵))
3 ffn 6668 . . . . . . . . . 10 (𝑐:(0..^𝑆)⟶𝐴𝑐 Fn (0..^𝑆))
43adantl 482 . . . . . . . . 9 ((𝜑𝑐:(0..^𝑆)⟶𝐴) → 𝑐 Fn (0..^𝑆))
54biantrurd 533 . . . . . . . 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 282 . . . . . 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 282 . . . 4 (𝜑 → (𝑐:(0..^𝑆)⟶(𝐴𝐵) ↔ (𝑐:(0..^𝑆)⟶𝐴 ∧ ran 𝑐𝐵)))
10 nnex 12159 . . . . . . . 8 ℕ ∈ V
1110a1i 11 . . . . . . 7 (𝜑 → ℕ ∈ V)
12 reprval.a . . . . . . 7 (𝜑𝐴 ⊆ ℕ)
1311, 12ssexd 5281 . . . . . 6 (𝜑𝐴 ∈ V)
14 inex1g 5276 . . . . . 6 (𝐴 ∈ V → (𝐴𝐵) ∈ V)
1513, 14syl 17 . . . . 5 (𝜑 → (𝐴𝐵) ∈ V)
16 ovex 7390 . . . . 5 (0..^𝑆) ∈ V
17 elmapg 8778 . . . . 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 8778 . . . . . 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 630 . . . 4 (𝜑 → ((𝑐 ∈ (𝐴m (0..^𝑆)) ∧ ran 𝑐𝐵) ↔ (𝑐:(0..^𝑆)⟶𝐴 ∧ ran 𝑐𝐵)))
229, 18, 213bitr4d 310 . . 3 (𝜑 → (𝑐 ∈ ((𝐴𝐵) ↑m (0..^𝑆)) ↔ (𝑐 ∈ (𝐴m (0..^𝑆)) ∧ ran 𝑐𝐵)))
2322anbi1d 630 . 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 4188 . . . . . 6 (𝐴𝐵) ⊆ 𝐴
2524, 12sstrid 3955 . . . . 5 (𝜑 → (𝐴𝐵) ⊆ ℕ)
26 reprval.m . . . . 5 (𝜑𝑀 ∈ ℤ)
27 reprval.s . . . . 5 (𝜑𝑆 ∈ ℕ0)
2825, 26, 27reprval 33223 . . . 4 (𝜑 → ((𝐴𝐵)(repr‘𝑆)𝑀) = {𝑐 ∈ ((𝐴𝐵) ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀})
2928eleq2d 2823 . . 3 (𝜑 → (𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀) ↔ 𝑐 ∈ {𝑐 ∈ ((𝐴𝐵) ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀}))
30 rabid 3427 . . 3 (𝑐 ∈ {𝑐 ∈ ((𝐴𝐵) ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀} ↔ (𝑐 ∈ ((𝐴𝐵) ↑m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀))
3129, 30bitrdi 286 . 2 (𝜑 → (𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀) ↔ (𝑐 ∈ ((𝐴𝐵) ↑m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀)))
3212, 26, 27reprval 33223 . . . . . 6 (𝜑 → (𝐴(repr‘𝑆)𝑀) = {𝑐 ∈ (𝐴m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀})
3332eleq2d 2823 . . . . 5 (𝜑 → (𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ↔ 𝑐 ∈ {𝑐 ∈ (𝐴m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀}))
34 rabid 3427 . . . . 5 (𝑐 ∈ {𝑐 ∈ (𝐴m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀} ↔ (𝑐 ∈ (𝐴m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀))
3533, 34bitrdi 286 . . . 4 (𝜑 → (𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ↔ (𝑐 ∈ (𝐴m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀)))
3635anbi1d 630 . . 3 (𝜑 → ((𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ran 𝑐𝐵) ↔ ((𝑐 ∈ (𝐴m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀) ∧ ran 𝑐𝐵)))
37 an32 644 . . 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 286 . 2 (𝜑 → ((𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ran 𝑐𝐵) ↔ ((𝑐 ∈ (𝐴m (0..^𝑆)) ∧ ran 𝑐𝐵) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀)))
3923, 31, 383bitr4d 310 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 396   = wceq 1541  wcel 2106  {crab 3407  Vcvv 3445  cin 3909  wss 3910  ran crn 5634   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  m cmap 8765  0cc0 11051  cn 12153  0cn0 12413  cz 12499  ..^cfzo 13567  Σcsu 15570  reprcrepr 33221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-addcl 11111
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-map 8767  df-neg 11388  df-nn 12154  df-z 12500  df-seq 13907  df-sum 15571  df-repr 33222
This theorem is referenced by:  hashreprin  33233
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