Theorem imarnf1pr 41301

Description: The image of the range of a function F under a function E if F is a function of a pair into the domain of E. (Contributed by Alexander van der Vekens, 2-Feb-2018.)

Assertion

Ref	Expression
imarnf1pr	|- ( ( X e. V /\ Y e. W ) -> ( ( ( F : { X , Y } --> dom E /\ E : dom E --> R ) /\ ( ( E ` ( F ` X ) ) = A /\ ( E ` ( F ` Y ) ) = B ) ) -> ( E " ran F ) = { A , B } ) )

Proof of Theorem imarnf1pr

Step	Hyp	Ref	Expression
1		ffn 6045	. . . . . . . . 9 |- ( E : dom E --> R -> E Fn dom E )
2	1	adantl 482	. . . . . . . 8 |- ( ( F : { X , Y } --> dom E /\ E : dom E --> R ) -> E Fn dom E )
3	2	adantr 481	. . . . . . 7 |- ( ( ( F : { X , Y } --> dom E /\ E : dom E --> R ) /\ ( X e. V /\ Y e. W ) ) -> E Fn dom E )
4		simpll 790	. . . . . . . 8 |- ( ( ( F : { X , Y } --> dom E /\ E : dom E --> R ) /\ ( X e. V /\ Y e. W ) ) -> F : { X , Y } --> dom E )
5		prid1g 4295	. . . . . . . . . 10 |- ( X e. V -> X e. { X , Y } )
6	5	adantr 481	. . . . . . . . 9 |- ( ( X e. V /\ Y e. W ) -> X e. { X , Y } )
7	6	adantl 482	. . . . . . . 8 |- ( ( ( F : { X , Y } --> dom E /\ E : dom E --> R ) /\ ( X e. V /\ Y e. W ) ) -> X e. { X , Y } )
8	4, 7	ffvelrnd 6360	. . . . . . 7 |- ( ( ( F : { X , Y } --> dom E /\ E : dom E --> R ) /\ ( X e. V /\ Y e. W ) ) -> ( F ` X ) e. dom E )
9		prid2g 4296	. . . . . . . . 9 |- ( Y e. W -> Y e. { X , Y } )
10	9	ad2antll 765	. . . . . . . 8 |- ( ( ( F : { X , Y } --> dom E /\ E : dom E --> R ) /\ ( X e. V /\ Y e. W ) ) -> Y e. { X , Y } )
11	4, 10	ffvelrnd 6360	. . . . . . 7 |- ( ( ( F : { X , Y } --> dom E /\ E : dom E --> R ) /\ ( X e. V /\ Y e. W ) ) -> ( F ` Y ) e. dom E )
12		fnimapr 6262	. . . . . . 7 |- ( ( E Fn dom E /\ ( F ` X ) e. dom E /\ ( F ` Y ) e. dom E ) -> ( E " { ( F ` X ) , ( F ` Y ) } ) = { ( E ` ( F ` X ) ) , ( E ` ( F ` Y ) ) } )
13	3, 8, 11, 12	syl3anc 1326	. . . . . 6 |- ( ( ( F : { X , Y } --> dom E /\ E : dom E --> R ) /\ ( X e. V /\ Y e. W ) ) -> ( E " { ( F ` X ) , ( F ` Y ) } ) = { ( E ` ( F ` X ) ) , ( E ` ( F ` Y ) ) } )
14	13	ex 450	. . . . 5 |- ( ( F : { X , Y } --> dom E /\ E : dom E --> R ) -> ( ( X e. V /\ Y e. W ) -> ( E " { ( F ` X ) , ( F ` Y ) } ) = { ( E ` ( F ` X ) ) , ( E ` ( F ` Y ) ) } ) )
15	14	adantr 481	. . . 4 |- ( ( ( F : { X , Y } --> dom E /\ E : dom E --> R ) /\ ( ( E ` ( F ` X ) ) = A /\ ( E ` ( F ` Y ) ) = B ) ) -> ( ( X e. V /\ Y e. W ) -> ( E " { ( F ` X ) , ( F ` Y ) } ) = { ( E ` ( F ` X ) ) , ( E ` ( F ` Y ) ) } ) )
16	15	impcom 446	. . 3 |- ( ( ( X e. V /\ Y e. W ) /\ ( ( F : { X , Y } --> dom E /\ E : dom E --> R ) /\ ( ( E ` ( F ` X ) ) = A /\ ( E ` ( F ` Y ) ) = B ) ) ) -> ( E " { ( F ` X ) , ( F ` Y ) } ) = { ( E ` ( F ` X ) ) , ( E ` ( F ` Y ) ) } )
17		ffn 6045	. . . . . . . . 9 |- ( F : { X , Y } --> dom E -> F Fn { X , Y } )
18		rnfdmpr 41300	. . . . . . . . 9 |- ( ( X e. V /\ Y e. W ) -> ( F Fn { X , Y } -> ran F = { ( F ` X ) , ( F ` Y ) } ) )
19	17, 18	syl5com 31	. . . . . . . 8 |- ( F : { X , Y } --> dom E -> ( ( X e. V /\ Y e. W ) -> ran F = { ( F ` X ) , ( F ` Y ) } ) )
20	19	adantr 481	. . . . . . 7 |- ( ( F : { X , Y } --> dom E /\ E : dom E --> R ) -> ( ( X e. V /\ Y e. W ) -> ran F = { ( F ` X ) , ( F ` Y ) } ) )
21	20	adantr 481	. . . . . 6 |- ( ( ( F : { X , Y } --> dom E /\ E : dom E --> R ) /\ ( ( E ` ( F ` X ) ) = A /\ ( E ` ( F ` Y ) ) = B ) ) -> ( ( X e. V /\ Y e. W ) -> ran F = { ( F ` X ) , ( F ` Y ) } ) )
22	21	impcom 446	. . . . 5 |- ( ( ( X e. V /\ Y e. W ) /\ ( ( F : { X , Y } --> dom E /\ E : dom E --> R ) /\ ( ( E ` ( F ` X ) ) = A /\ ( E ` ( F ` Y ) ) = B ) ) ) -> ran F = { ( F ` X ) , ( F ` Y ) } )
23	22	eqcomd 2628	. . . 4 |- ( ( ( X e. V /\ Y e. W ) /\ ( ( F : { X , Y } --> dom E /\ E : dom E --> R ) /\ ( ( E ` ( F ` X ) ) = A /\ ( E ` ( F ` Y ) ) = B ) ) ) -> { ( F ` X ) , ( F ` Y ) } = ran F )
24	23	imaeq2d 5466	. . 3 |- ( ( ( X e. V /\ Y e. W ) /\ ( ( F : { X , Y } --> dom E /\ E : dom E --> R ) /\ ( ( E ` ( F ` X ) ) = A /\ ( E ` ( F ` Y ) ) = B ) ) ) -> ( E " { ( F ` X ) , ( F ` Y ) } ) = ( E " ran F ) )
25		preq12 4270	. . . 4 |- ( ( ( E ` ( F ` X ) ) = A /\ ( E ` ( F ` Y ) ) = B ) -> { ( E ` ( F ` X ) ) , ( E ` ( F ` Y ) ) } = { A , B } )
26	25	ad2antll 765	. . 3 |- ( ( ( X e. V /\ Y e. W ) /\ ( ( F : { X , Y } --> dom E /\ E : dom E --> R ) /\ ( ( E ` ( F ` X ) ) = A /\ ( E ` ( F ` Y ) ) = B ) ) ) -> { ( E ` ( F ` X ) ) , ( E ` ( F ` Y ) ) } = { A , B } )
27	16, 24, 26	3eqtr3d 2664	. 2 |- ( ( ( X e. V /\ Y e. W ) /\ ( ( F : { X , Y } --> dom E /\ E : dom E --> R ) /\ ( ( E ` ( F ` X ) ) = A /\ ( E ` ( F ` Y ) ) = B ) ) ) -> ( E " ran F ) = { A , B } )
28	27	ex 450	1 |- ( ( X e. V /\ Y e. W ) -> ( ( ( F : { X , Y } --> dom E /\ E : dom E --> R ) /\ ( ( E ` ( F ` X ) ) = A /\ ( E ` ( F ` Y ) ) = B ) ) -> ( E " ran F ) = { A , B } ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cpr 4179   dom cdm 5114   ran crn 5115   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888

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-ne 2795  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-iun 4522  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-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896

This theorem is referenced by: (None)