Theorem fvixp2 43830

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 8890	. . 3 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))
2	1	simp3bi 1148	. 2 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)
3	2	r19.21bi 3249	1 ⊢ ((𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 397   ∈ wcel 2107  ∀wral 3062  Vcvv 3475   Fn wfn 6534  ‘cfv 6539  Xcixp 8886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4527  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-br 5147  df-opab 5209  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-iota 6491  df-fun 6541  df-fn 6542  df-fv 6547  df-ixp 8887

This theorem is referenced by:  rrxsnicc  44950  ioorrnopnlem  44954  ioorrnopnxrlem  44956  hspdifhsp  45266  hoiqssbllem2  45273  iinhoiicclem  45323  iunhoiioolem  45325