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Theorem fvixp2 41756
Description: Projection of a factor of an indexed Cartesian product. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
Assertion
Ref Expression
fvixp2 ((𝐹X𝑥𝐴 𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem fvixp2
StepHypRef Expression
1 elixp2 8461 . . 3 (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
21simp3bi 1144 . 2 (𝐹X𝑥𝐴 𝐵 → ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)
32r19.21bi 3203 1 ((𝐹X𝑥𝐴 𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2115  wral 3133  Vcvv 3480   Fn wfn 6338  cfv 6343  Xcixp 8457
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-iota 6302  df-fun 6345  df-fn 6346  df-fv 6351  df-ixp 8458
This theorem is referenced by:  rrxsnicc  42872  ioorrnopnlem  42876  ioorrnopnxrlem  42878  hspdifhsp  43185  hoiqssbllem2  43192  iinhoiicclem  43242  iunhoiioolem  43244
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