Theorem elixpconstg 39266

Description: Membership in an infinite Cartesian product of a constant B. (Contributed by Glauco Siliprandi, 8-Apr-2021.)

Assertion

Ref	Expression
elixpconstg	|- ( F e. V -> ( F e. X_ x e. A B <-> F : A --> B ) )

Distinct variable groups:   x, A   x, B   x, F

Allowed substitution hint:   V( x)

Proof of Theorem elixpconstg

Step	Hyp	Ref	Expression
1		elixp2 7912	. . . . . 6 |- ( F e. X_ x e. A B <-> ( F e. _V /\ F Fn A /\ A. x e. A ( F ` x ) e. B ) )
2	1	simp2bi 1077	. . . . 5 |- ( F e. X_ x e. A B -> F Fn A )
3	1	simp3bi 1078	. . . . 5 |- ( F e. X_ x e. A B -> A. x e. A ( F ` x ) e. B )
4	2, 3	jca 554	. . . 4 |- ( F e. X_ x e. A B -> ( F Fn A /\ A. x e. A ( F ` x ) e. B ) )
5		ffnfv 6388	. . . 4 |- ( F : A --> B <-> ( F Fn A /\ A. x e. A ( F ` x ) e. B ) )
6	4, 5	sylibr 224	. . 3 |- ( F e. X_ x e. A B -> F : A --> B )
7	6	a1i 11	. 2 |- ( F e. V -> ( F e. X_ x e. A B -> F : A --> B ) )
8		elex 3212	. . . . . 6 |- ( F e. V -> F e. _V )
9	8	adantr 481	. . . . 5 |- ( ( F e. V /\ F : A --> B ) -> F e. _V )
10		ffn 6045	. . . . . 6 |- ( F : A --> B -> F Fn A )
11	10	adantl 482	. . . . 5 |- ( ( F e. V /\ F : A --> B ) -> F Fn A )
12	5	simprbi 480	. . . . . 6 |- ( F : A --> B -> A. x e. A ( F ` x ) e. B )
13	12	adantl 482	. . . . 5 |- ( ( F e. V /\ F : A --> B ) -> A. x e. A ( F ` x ) e. B )
14	9, 11, 13	3jca 1242	. . . 4 |- ( ( F e. V /\ F : A --> B ) -> ( F e. _V /\ F Fn A /\ A. x e. A ( F ` x ) e. B ) )
15	14, 1	sylibr 224	. . 3 |- ( ( F e. V /\ F : A --> B ) -> F e. X_ x e. A B )
16	15	ex 450	. 2 |- ( F e. V -> ( F : A --> B -> F e. X_ x e. A B ) )
17	7, 16	impbid 202	1 |- ( F e. V -> ( F e. X_ x e. A B <-> F : A --> B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   e. wcel 1990   A.wral 2912   _Vcvv 3200   Fn wfn 5883   -->wf 5884   ` 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  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-eu 2474  df-mo 2475  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-sbc 3436  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-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ixp 7909

This theorem is referenced by:  iinhoiicclem  40887