Theorem fvixp2 39389

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

Assertion

Ref	Expression
fvixp2	|- ( ( F e. X_ x e. A B /\ x e. A ) -> ( F ` x ) e. B )

Distinct variable groups:   x, A   x, F

Allowed substitution hint:   B( x)

Proof of Theorem fvixp2

Step	Hyp	Ref	Expression
1		elixp2 7912	. . 3 |- ( F e. X_ x e. A B <-> ( F e. _V /\ F Fn A /\ A. x e. A ( F ` x ) e. B ) )
2	1	simp3bi 1078	. 2 |- ( F e. X_ x e. A B -> A. x e. A ( F ` x ) e. B )
3	2	r19.21bi 2932	1 |- ( ( F e. X_ x e. A B /\ x e. A ) -> ( F ` x ) e. B )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   A.wral 2912   _Vcvv 3200   Fn wfn 5883   ` cfv 5888   X_cixp 7908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-ixp 7909

This theorem is referenced by:  rrxsnicc  40520  ioorrnopnlem  40524  ioorrnopnxrlem  40526  hspdifhsp  40830  hoiqssbllem2  40837  iinhoiicclem  40887  iunhoiioolem  40889