Theorem fvixp2 44566

Description: Projection of a factor of an indexed Cartesian product. (Contributed by Glauco Siliprandi, 24-Dec-2020.)

Assertion

Ref	Expression
fvixp2	⊢ ((𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem fvixp2

Step	Hyp	Ref	Expression
1		elixp2 8913	. . 3 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))
2	1	simp3bi 1145	. 2 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)
3	2	r19.21bi 3244	1 ⊢ ((𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2099  ∀wral 3057  Vcvv 3470   Fn wfn 6537  ‘cfv 6542  Xcixp 8909

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-12 2167  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3058  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fn 6545  df-fv 6550  df-ixp 8910

This theorem is referenced by:  rrxsnicc  45682  ioorrnopnlem  45686  ioorrnopnxrlem  45688  hspdifhsp  45998  hoiqssbllem2  46005  iinhoiicclem  46055  iunhoiioolem  46057

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