Theorem prsprel 41737

Description: The elements of a pair from the set of all unordered pairs over a given set V are elements of the set V. (Contributed by AV, 22-Nov-2021.)

Assertion

Ref	Expression
prsprel	|- ( ( { X , Y } e. (Pairs ` V ) /\ ( X e. U /\ Y e. W ) ) -> ( X e. V /\ Y e. V ) )

Proof of Theorem prsprel

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sprel 41734	. . 3 |- ( { X , Y } e. (Pairs ` V ) -> E. a e. V E. b e. V { X , Y } = { a , b } )
2		preq12bg 4386	. . . . . . 7 |- ( ( ( X e. U /\ Y e. W ) /\ ( a e. V /\ b e. V ) ) -> ( { X , Y } = { a , b } <-> ( ( X = a /\ Y = b ) \/ ( X = b /\ Y = a ) ) ) )
3		eleq1 2689	. . . . . . . . . . . . 13 |- ( a = X -> ( a e. V <-> X e. V ) )
4	3	eqcoms 2630	. . . . . . . . . . . 12 |- ( X = a -> ( a e. V <-> X e. V ) )
5		eleq1 2689	. . . . . . . . . . . . 13 |- ( b = Y -> ( b e. V <-> Y e. V ) )
6	5	eqcoms 2630	. . . . . . . . . . . 12 |- ( Y = b -> ( b e. V <-> Y e. V ) )
7	4, 6	bi2anan9 917	. . . . . . . . . . 11 |- ( ( X = a /\ Y = b ) -> ( ( a e. V /\ b e. V ) <-> ( X e. V /\ Y e. V ) ) )
8	7	biimpd 219	. . . . . . . . . 10 |- ( ( X = a /\ Y = b ) -> ( ( a e. V /\ b e. V ) -> ( X e. V /\ Y e. V ) ) )
9		eleq1 2689	. . . . . . . . . . . . . 14 |- ( b = X -> ( b e. V <-> X e. V ) )
10	9	eqcoms 2630	. . . . . . . . . . . . 13 |- ( X = b -> ( b e. V <-> X e. V ) )
11		eleq1 2689	. . . . . . . . . . . . . 14 |- ( a = Y -> ( a e. V <-> Y e. V ) )
12	11	eqcoms 2630	. . . . . . . . . . . . 13 |- ( Y = a -> ( a e. V <-> Y e. V ) )
13	10, 12	bi2anan9 917	. . . . . . . . . . . 12 |- ( ( X = b /\ Y = a ) -> ( ( b e. V /\ a e. V ) <-> ( X e. V /\ Y e. V ) ) )
14	13	biimpd 219	. . . . . . . . . . 11 |- ( ( X = b /\ Y = a ) -> ( ( b e. V /\ a e. V ) -> ( X e. V /\ Y e. V ) ) )
15	14	ancomsd 470	. . . . . . . . . 10 |- ( ( X = b /\ Y = a ) -> ( ( a e. V /\ b e. V ) -> ( X e. V /\ Y e. V ) ) )
16	8, 15	jaoi 394	. . . . . . . . 9 |- ( ( ( X = a /\ Y = b ) \/ ( X = b /\ Y = a ) ) -> ( ( a e. V /\ b e. V ) -> ( X e. V /\ Y e. V ) ) )
17	16	com12 32	. . . . . . . 8 |- ( ( a e. V /\ b e. V ) -> ( ( ( X = a /\ Y = b ) \/ ( X = b /\ Y = a ) ) -> ( X e. V /\ Y e. V ) ) )
18	17	adantl 482	. . . . . . 7 |- ( ( ( X e. U /\ Y e. W ) /\ ( a e. V /\ b e. V ) ) -> ( ( ( X = a /\ Y = b ) \/ ( X = b /\ Y = a ) ) -> ( X e. V /\ Y e. V ) ) )
19	2, 18	sylbid 230	. . . . . 6 |- ( ( ( X e. U /\ Y e. W ) /\ ( a e. V /\ b e. V ) ) -> ( { X , Y } = { a , b } -> ( X e. V /\ Y e. V ) ) )
20	19	expcom 451	. . . . 5 |- ( ( a e. V /\ b e. V ) -> ( ( X e. U /\ Y e. W ) -> ( { X , Y } = { a , b } -> ( X e. V /\ Y e. V ) ) ) )
21	20	com23 86	. . . 4 |- ( ( a e. V /\ b e. V ) -> ( { X , Y } = { a , b } -> ( ( X e. U /\ Y e. W ) -> ( X e. V /\ Y e. V ) ) ) )
22	21	rexlimivv 3036	. . 3 |- ( E. a e. V E. b e. V { X , Y } = { a , b } -> ( ( X e. U /\ Y e. W ) -> ( X e. V /\ Y e. V ) ) )
23	1, 22	syl 17	. 2 |- ( { X , Y } e. (Pairs ` V ) -> ( ( X e. U /\ Y e. W ) -> ( X e. V /\ Y e. V ) ) )
24	23	imp 445	1 |- ( ( { X , Y } e. (Pairs ` V ) /\ ( X e. U /\ Y e. W ) ) -> ( X e. V /\ Y e. V ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   {cpr 4179   ` cfv 5888  Pairscspr 41727

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-rep 4771  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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-spr 41728

This theorem is referenced by:  prsssprel  41738  sprsymrelfolem2  41743