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Theorem reprinrn 33928
Description: Representations with term in an intersection. (Contributed by Thierry Arnoux, 11-Dec-2021.)
Hypotheses
Ref Expression
reprval.a (πœ‘ β†’ 𝐴 βŠ† β„•)
reprval.m (πœ‘ β†’ 𝑀 ∈ β„€)
reprval.s (πœ‘ β†’ 𝑆 ∈ β„•0)
Assertion
Ref Expression
reprinrn (πœ‘ β†’ (𝑐 ∈ ((𝐴 ∩ 𝐡)(reprβ€˜π‘†)𝑀) ↔ (𝑐 ∈ (𝐴(reprβ€˜π‘†)𝑀) ∧ ran 𝑐 βŠ† 𝐡)))
Distinct variable groups:   𝐴,𝑐   𝑀,𝑐   𝑆,𝑐   πœ‘,𝑐   𝐡,𝑐

Proof of Theorem reprinrn
Dummy variable π‘Ž is distinct from all other variables.
StepHypRef Expression
1 fin 6770 . . . . 5 (𝑐:(0..^𝑆)⟢(𝐴 ∩ 𝐡) ↔ (𝑐:(0..^𝑆)⟢𝐴 ∧ 𝑐:(0..^𝑆)⟢𝐡))
2 df-f 6546 . . . . . . 7 (𝑐:(0..^𝑆)⟢𝐡 ↔ (𝑐 Fn (0..^𝑆) ∧ ran 𝑐 βŠ† 𝐡))
3 ffn 6716 . . . . . . . . . 10 (𝑐:(0..^𝑆)⟢𝐴 β†’ 𝑐 Fn (0..^𝑆))
43adantl 480 . . . . . . . . 9 ((πœ‘ ∧ 𝑐:(0..^𝑆)⟢𝐴) β†’ 𝑐 Fn (0..^𝑆))
54biantrurd 531 . . . . . . . 8 ((πœ‘ ∧ 𝑐:(0..^𝑆)⟢𝐴) β†’ (ran 𝑐 βŠ† 𝐡 ↔ (𝑐 Fn (0..^𝑆) ∧ ran 𝑐 βŠ† 𝐡)))
65bicomd 222 . . . . . . 7 ((πœ‘ ∧ 𝑐:(0..^𝑆)⟢𝐴) β†’ ((𝑐 Fn (0..^𝑆) ∧ ran 𝑐 βŠ† 𝐡) ↔ ran 𝑐 βŠ† 𝐡))
72, 6bitrid 282 . . . . . 6 ((πœ‘ ∧ 𝑐:(0..^𝑆)⟢𝐴) β†’ (𝑐:(0..^𝑆)⟢𝐡 ↔ ran 𝑐 βŠ† 𝐡))
87pm5.32da 577 . . . . 5 (πœ‘ β†’ ((𝑐:(0..^𝑆)⟢𝐴 ∧ 𝑐:(0..^𝑆)⟢𝐡) ↔ (𝑐:(0..^𝑆)⟢𝐴 ∧ ran 𝑐 βŠ† 𝐡)))
91, 8bitrid 282 . . . 4 (πœ‘ β†’ (𝑐:(0..^𝑆)⟢(𝐴 ∩ 𝐡) ↔ (𝑐:(0..^𝑆)⟢𝐴 ∧ ran 𝑐 βŠ† 𝐡)))
10 nnex 12222 . . . . . . . 8 β„• ∈ V
1110a1i 11 . . . . . . 7 (πœ‘ β†’ β„• ∈ V)
12 reprval.a . . . . . . 7 (πœ‘ β†’ 𝐴 βŠ† β„•)
1311, 12ssexd 5323 . . . . . 6 (πœ‘ β†’ 𝐴 ∈ V)
14 inex1g 5318 . . . . . 6 (𝐴 ∈ V β†’ (𝐴 ∩ 𝐡) ∈ V)
1513, 14syl 17 . . . . 5 (πœ‘ β†’ (𝐴 ∩ 𝐡) ∈ V)
16 ovex 7444 . . . . 5 (0..^𝑆) ∈ V
17 elmapg 8835 . . . . 5 (((𝐴 ∩ 𝐡) ∈ V ∧ (0..^𝑆) ∈ V) β†’ (𝑐 ∈ ((𝐴 ∩ 𝐡) ↑m (0..^𝑆)) ↔ 𝑐:(0..^𝑆)⟢(𝐴 ∩ 𝐡)))
1815, 16, 17sylancl 584 . . . 4 (πœ‘ β†’ (𝑐 ∈ ((𝐴 ∩ 𝐡) ↑m (0..^𝑆)) ↔ 𝑐:(0..^𝑆)⟢(𝐴 ∩ 𝐡)))
19 elmapg 8835 . . . . . 6 ((𝐴 ∈ V ∧ (0..^𝑆) ∈ V) β†’ (𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ↔ 𝑐:(0..^𝑆)⟢𝐴))
2013, 16, 19sylancl 584 . . . . 5 (πœ‘ β†’ (𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ↔ 𝑐:(0..^𝑆)⟢𝐴))
2120anbi1d 628 . . . 4 (πœ‘ β†’ ((𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∧ ran 𝑐 βŠ† 𝐡) ↔ (𝑐:(0..^𝑆)⟢𝐴 ∧ ran 𝑐 βŠ† 𝐡)))
229, 18, 213bitr4d 310 . . 3 (πœ‘ β†’ (𝑐 ∈ ((𝐴 ∩ 𝐡) ↑m (0..^𝑆)) ↔ (𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∧ ran 𝑐 βŠ† 𝐡)))
2322anbi1d 628 . 2 (πœ‘ β†’ ((𝑐 ∈ ((𝐴 ∩ 𝐡) ↑m (0..^𝑆)) ∧ Ξ£π‘Ž ∈ (0..^𝑆)(π‘β€˜π‘Ž) = 𝑀) ↔ ((𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∧ ran 𝑐 βŠ† 𝐡) ∧ Ξ£π‘Ž ∈ (0..^𝑆)(π‘β€˜π‘Ž) = 𝑀)))
24 inss1 4227 . . . . . 6 (𝐴 ∩ 𝐡) βŠ† 𝐴
2524, 12sstrid 3992 . . . . 5 (πœ‘ β†’ (𝐴 ∩ 𝐡) βŠ† β„•)
26 reprval.m . . . . 5 (πœ‘ β†’ 𝑀 ∈ β„€)
27 reprval.s . . . . 5 (πœ‘ β†’ 𝑆 ∈ β„•0)
2825, 26, 27reprval 33920 . . . 4 (πœ‘ β†’ ((𝐴 ∩ 𝐡)(reprβ€˜π‘†)𝑀) = {𝑐 ∈ ((𝐴 ∩ 𝐡) ↑m (0..^𝑆)) ∣ Ξ£π‘Ž ∈ (0..^𝑆)(π‘β€˜π‘Ž) = 𝑀})
2928eleq2d 2817 . . 3 (πœ‘ β†’ (𝑐 ∈ ((𝐴 ∩ 𝐡)(reprβ€˜π‘†)𝑀) ↔ 𝑐 ∈ {𝑐 ∈ ((𝐴 ∩ 𝐡) ↑m (0..^𝑆)) ∣ Ξ£π‘Ž ∈ (0..^𝑆)(π‘β€˜π‘Ž) = 𝑀}))
30 rabid 3450 . . 3 (𝑐 ∈ {𝑐 ∈ ((𝐴 ∩ 𝐡) ↑m (0..^𝑆)) ∣ Ξ£π‘Ž ∈ (0..^𝑆)(π‘β€˜π‘Ž) = 𝑀} ↔ (𝑐 ∈ ((𝐴 ∩ 𝐡) ↑m (0..^𝑆)) ∧ Ξ£π‘Ž ∈ (0..^𝑆)(π‘β€˜π‘Ž) = 𝑀))
3129, 30bitrdi 286 . 2 (πœ‘ β†’ (𝑐 ∈ ((𝐴 ∩ 𝐡)(reprβ€˜π‘†)𝑀) ↔ (𝑐 ∈ ((𝐴 ∩ 𝐡) ↑m (0..^𝑆)) ∧ Ξ£π‘Ž ∈ (0..^𝑆)(π‘β€˜π‘Ž) = 𝑀)))
3212, 26, 27reprval 33920 . . . . . 6 (πœ‘ β†’ (𝐴(reprβ€˜π‘†)𝑀) = {𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Ξ£π‘Ž ∈ (0..^𝑆)(π‘β€˜π‘Ž) = 𝑀})
3332eleq2d 2817 . . . . 5 (πœ‘ β†’ (𝑐 ∈ (𝐴(reprβ€˜π‘†)𝑀) ↔ 𝑐 ∈ {𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Ξ£π‘Ž ∈ (0..^𝑆)(π‘β€˜π‘Ž) = 𝑀}))
34 rabid 3450 . . . . 5 (𝑐 ∈ {𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Ξ£π‘Ž ∈ (0..^𝑆)(π‘β€˜π‘Ž) = 𝑀} ↔ (𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∧ Ξ£π‘Ž ∈ (0..^𝑆)(π‘β€˜π‘Ž) = 𝑀))
3533, 34bitrdi 286 . . . 4 (πœ‘ β†’ (𝑐 ∈ (𝐴(reprβ€˜π‘†)𝑀) ↔ (𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∧ Ξ£π‘Ž ∈ (0..^𝑆)(π‘β€˜π‘Ž) = 𝑀)))
3635anbi1d 628 . . 3 (πœ‘ β†’ ((𝑐 ∈ (𝐴(reprβ€˜π‘†)𝑀) ∧ ran 𝑐 βŠ† 𝐡) ↔ ((𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∧ Ξ£π‘Ž ∈ (0..^𝑆)(π‘β€˜π‘Ž) = 𝑀) ∧ ran 𝑐 βŠ† 𝐡)))
37 an32 642 . . 3 (((𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∧ Ξ£π‘Ž ∈ (0..^𝑆)(π‘β€˜π‘Ž) = 𝑀) ∧ ran 𝑐 βŠ† 𝐡) ↔ ((𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∧ ran 𝑐 βŠ† 𝐡) ∧ Ξ£π‘Ž ∈ (0..^𝑆)(π‘β€˜π‘Ž) = 𝑀))
3836, 37bitrdi 286 . 2 (πœ‘ β†’ ((𝑐 ∈ (𝐴(reprβ€˜π‘†)𝑀) ∧ ran 𝑐 βŠ† 𝐡) ↔ ((𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∧ ran 𝑐 βŠ† 𝐡) ∧ Ξ£π‘Ž ∈ (0..^𝑆)(π‘β€˜π‘Ž) = 𝑀)))
3923, 31, 383bitr4d 310 1 (πœ‘ β†’ (𝑐 ∈ ((𝐴 ∩ 𝐡)(reprβ€˜π‘†)𝑀) ↔ (𝑐 ∈ (𝐴(reprβ€˜π‘†)𝑀) ∧ ran 𝑐 βŠ† 𝐡)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  {crab 3430  Vcvv 3472   ∩ cin 3946   βŠ† wss 3947  ran crn 5676   Fn wfn 6537  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411   ↑m cmap 8822  0cc0 11112  β„•cn 12216  β„•0cn0 12476  β„€cz 12562  ..^cfzo 13631  Ξ£csu 15636  reprcrepr 33918
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-addcl 11172
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  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 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-map 8824  df-neg 11451  df-nn 12217  df-z 12563  df-seq 13971  df-sum 15637  df-repr 33919
This theorem is referenced by:  hashreprin  33930
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