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Theorem fvixp2 38384
 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 7798 . . . . 5 (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
21biimpi 205 . . . 4 (𝐹X𝑥𝐴 𝐵 → (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
32simp3d 1068 . . 3 (𝐹X𝑥𝐴 𝐵 → ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)
43adantr 480 . 2 ((𝐹X𝑥𝐴 𝐵𝑥𝐴) → ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)
5 simpr 476 . 2 ((𝐹X𝑥𝐴 𝐵𝑥𝐴) → 𝑥𝐴)
6 rspa 2914 . 2 ((∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
74, 5, 6syl2anc 691 1 ((𝐹X𝑥𝐴 𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   Fn wfn 5799  ‘cfv 5804  Xcixp 7794 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-ixp 7795 This theorem is referenced by:  rrxsnicc  39196  ioorrnopnlem  39200  ioorrnopnxrlem  39202  hspdifhsp  39506  hoiqssbllem2  39513  iinhoiicclem  39564  iunhoiioolem  39566
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