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Theorem fvixp2 39703
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 7954 . . 3 (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
21simp3bi 1098 . 2 (𝐹X𝑥𝐴 𝐵 → ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)
32r19.21bi 2961 1 ((𝐹X𝑥𝐴 𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2030  wral 2941  Vcvv 3231   Fn wfn 5921  cfv 5926  Xcixp 7950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934  df-ixp 7951
This theorem is referenced by:  rrxsnicc  40838  ioorrnopnlem  40842  ioorrnopnxrlem  40844  hspdifhsp  41151  hoiqssbllem2  41158  iinhoiicclem  41208  iunhoiioolem  41210
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