Theorem elmapsnd 39396

Description: Membership in a set exponentiated to a singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypotheses

Ref	Expression
elmapsnd.1	|- ( ph -> F Fn { A } )
elmapsnd.2	|- ( ph -> B e. V )
elmapsnd.3	|- ( ph -> ( F ` A ) e. B )

Assertion

Ref	Expression
elmapsnd	|- ( ph -> F e. ( B ^m { A } ) )

Proof of Theorem elmapsnd

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elmapsnd.1	. . . 4 |- ( ph -> F Fn { A } )
2		elsni 4194	. . . . . . . 8 |- ( x e. { A } -> x = A )
3	2	fveq2d 6195	. . . . . . 7 |- ( x e. { A } -> ( F ` x ) = ( F ` A ) )
4	3	adantl 482	. . . . . 6 |- ( ( ph /\ x e. { A } ) -> ( F ` x ) = ( F ` A ) )
5		elmapsnd.3	. . . . . . 7 |- ( ph -> ( F ` A ) e. B )
6	5	adantr 481	. . . . . 6 |- ( ( ph /\ x e. { A } ) -> ( F ` A ) e. B )
7	4, 6	eqeltrd 2701	. . . . 5 |- ( ( ph /\ x e. { A } ) -> ( F ` x ) e. B )
8	7	ralrimiva 2966	. . . 4 |- ( ph -> A. x e. { A } ( F ` x ) e. B )
9	1, 8	jca 554	. . 3 |- ( ph -> ( F Fn { A } /\ A. x e. { A } ( F ` x ) e. B ) )
10		ffnfv 6388	. . 3 |- ( F : { A } --> B <-> ( F Fn { A } /\ A. x e. { A } ( F ` x ) e. B ) )
11	9, 10	sylibr 224	. 2 |- ( ph -> F : { A } --> B )
12		elmapsnd.2	. . 3 |- ( ph -> B e. V )
13		snex 4908	. . . 4 |- { A } e. _V
14	13	a1i 11	. . 3 |- ( ph -> { A } e. _V )
15	12, 14	elmapd 7871	. 2 |- ( ph -> ( F e. ( B ^m { A } ) <-> F : { A } --> B ) )
16	11, 15	mpbird 247	1 |- ( ph -> F e. ( B ^m { A } ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   {csn 4177   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-pw 4160  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-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859

This theorem is referenced by:  ssmapsn  39408