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Theorem dprdw 19061
Description: The property of being a finitely supported function in the family 𝑆. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 11-Jul-2019.)
Hypotheses
Ref Expression
dprdff.w 𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }
dprdff.1 (𝜑𝐺dom DProd 𝑆)
dprdff.2 (𝜑 → dom 𝑆 = 𝐼)
Assertion
Ref Expression
dprdw (𝜑 → (𝐹𝑊 ↔ (𝐹 Fn 𝐼 ∧ ∀𝑥𝐼 (𝐹𝑥) ∈ (𝑆𝑥) ∧ 𝐹 finSupp 0 )))
Distinct variable groups:   𝑥,,𝐹   𝑥,𝐺   ,𝑖,𝐼,𝑥   0 ,   𝜑,𝑥   𝑆,,𝑖,𝑥
Allowed substitution hints:   𝜑(,𝑖)   𝐹(𝑖)   𝐺(,𝑖)   𝑊(𝑥,,𝑖)   0 (𝑥,𝑖)

Proof of Theorem dprdw
StepHypRef Expression
1 elex 3510 . . . . 5 (𝐹X𝑖𝐼 (𝑆𝑖) → 𝐹 ∈ V)
21a1i 11 . . . 4 (𝜑 → (𝐹X𝑖𝐼 (𝑆𝑖) → 𝐹 ∈ V))
3 dprdff.1 . . . . . . 7 (𝜑𝐺dom DProd 𝑆)
4 dprdff.2 . . . . . . 7 (𝜑 → dom 𝑆 = 𝐼)
53, 4dprddomcld 19052 . . . . . 6 (𝜑𝐼 ∈ V)
6 fnex 6971 . . . . . . 7 ((𝐹 Fn 𝐼𝐼 ∈ V) → 𝐹 ∈ V)
76expcom 414 . . . . . 6 (𝐼 ∈ V → (𝐹 Fn 𝐼𝐹 ∈ V))
85, 7syl 17 . . . . 5 (𝜑 → (𝐹 Fn 𝐼𝐹 ∈ V))
98adantrd 492 . . . 4 (𝜑 → ((𝐹 Fn 𝐼 ∧ ∀𝑥𝐼 (𝐹𝑥) ∈ (𝑆𝑥)) → 𝐹 ∈ V))
10 fveq2 6663 . . . . . . . . 9 (𝑖 = 𝑥 → (𝑆𝑖) = (𝑆𝑥))
1110cbvixpv 8467 . . . . . . . 8 X𝑖𝐼 (𝑆𝑖) = X𝑥𝐼 (𝑆𝑥)
1211eleq2i 2901 . . . . . . 7 (𝐹X𝑖𝐼 (𝑆𝑖) ↔ 𝐹X𝑥𝐼 (𝑆𝑥))
13 elixp2 8453 . . . . . . 7 (𝐹X𝑥𝐼 (𝑆𝑥) ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐼 ∧ ∀𝑥𝐼 (𝐹𝑥) ∈ (𝑆𝑥)))
14 3anass 1087 . . . . . . 7 ((𝐹 ∈ V ∧ 𝐹 Fn 𝐼 ∧ ∀𝑥𝐼 (𝐹𝑥) ∈ (𝑆𝑥)) ↔ (𝐹 ∈ V ∧ (𝐹 Fn 𝐼 ∧ ∀𝑥𝐼 (𝐹𝑥) ∈ (𝑆𝑥))))
1512, 13, 143bitri 298 . . . . . 6 (𝐹X𝑖𝐼 (𝑆𝑖) ↔ (𝐹 ∈ V ∧ (𝐹 Fn 𝐼 ∧ ∀𝑥𝐼 (𝐹𝑥) ∈ (𝑆𝑥))))
1615baib 536 . . . . 5 (𝐹 ∈ V → (𝐹X𝑖𝐼 (𝑆𝑖) ↔ (𝐹 Fn 𝐼 ∧ ∀𝑥𝐼 (𝐹𝑥) ∈ (𝑆𝑥))))
1716a1i 11 . . . 4 (𝜑 → (𝐹 ∈ V → (𝐹X𝑖𝐼 (𝑆𝑖) ↔ (𝐹 Fn 𝐼 ∧ ∀𝑥𝐼 (𝐹𝑥) ∈ (𝑆𝑥)))))
182, 9, 17pm5.21ndd 381 . . 3 (𝜑 → (𝐹X𝑖𝐼 (𝑆𝑖) ↔ (𝐹 Fn 𝐼 ∧ ∀𝑥𝐼 (𝐹𝑥) ∈ (𝑆𝑥))))
1918anbi1d 629 . 2 (𝜑 → ((𝐹X𝑖𝐼 (𝑆𝑖) ∧ 𝐹 finSupp 0 ) ↔ ((𝐹 Fn 𝐼 ∧ ∀𝑥𝐼 (𝐹𝑥) ∈ (𝑆𝑥)) ∧ 𝐹 finSupp 0 )))
20 breq1 5060 . . 3 ( = 𝐹 → ( finSupp 0𝐹 finSupp 0 ))
21 dprdff.w . . 3 𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }
2220, 21elrab2 3680 . 2 (𝐹𝑊 ↔ (𝐹X𝑖𝐼 (𝑆𝑖) ∧ 𝐹 finSupp 0 ))
23 df-3an 1081 . 2 ((𝐹 Fn 𝐼 ∧ ∀𝑥𝐼 (𝐹𝑥) ∈ (𝑆𝑥) ∧ 𝐹 finSupp 0 ) ↔ ((𝐹 Fn 𝐼 ∧ ∀𝑥𝐼 (𝐹𝑥) ∈ (𝑆𝑥)) ∧ 𝐹 finSupp 0 ))
2419, 22, 233bitr4g 315 1 (𝜑 → (𝐹𝑊 ↔ (𝐹 Fn 𝐼 ∧ ∀𝑥𝐼 (𝐹𝑥) ∈ (𝑆𝑥) ∧ 𝐹 finSupp 0 )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  {crab 3139  Vcvv 3492   class class class wbr 5057  dom cdm 5548   Fn wfn 6343  cfv 6348  Xcixp 8449   finSupp cfsupp 8821   DProd cdprd 19044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-oprab 7149  df-mpo 7150  df-ixp 8450  df-dprd 19046
This theorem is referenced by:  dprdff  19063  dprdfcl  19064  dprdffsupp  19065  dprdsubg  19075
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