Theorem sprsymrelfvlem 41740

Description: Lemma for sprsymrelf 41745 and sprsymrelfv 41744. (Contributed by AV, 19-Nov-2021.)

Assertion

Ref	Expression
sprsymrelfvlem	|- ( P C_ (Pairs ` V ) -> { <. x , y >. | E. c e. P c = { x , y } } e. ~P ( V X. V ) )

Distinct variable groups:   P, c, x, y   V, c, x, y

Proof of Theorem sprsymrelfvlem

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 473	. . . . 5 |- ( ( V e. _V /\ P C_ (Pairs ` V ) ) -> V e. _V )
2		eleq1 2689	. . . . . . . . . . . 12 |- ( c = { x , y } -> ( c e. P <-> { x , y } e. P ) )
3		vex 3203	. . . . . . . . . . . . 13 |- x e. _V
4		vex 3203	. . . . . . . . . . . . 13 |- y e. _V
5		prsssprel 41738	. . . . . . . . . . . . . . 15 |- ( ( P C_ (Pairs ` V ) /\ { x , y } e. P /\ ( x e. _V /\ y e. _V ) ) -> ( x e. V /\ y e. V ) )
6	5	3exp 1264	. . . . . . . . . . . . . 14 |- ( P C_ (Pairs ` V ) -> ( { x , y } e. P -> ( ( x e. _V /\ y e. _V ) -> ( x e. V /\ y e. V ) ) ) )
7	6	com13 88	. . . . . . . . . . . . 13 |- ( ( x e. _V /\ y e. _V ) -> ( { x , y } e. P -> ( P C_ (Pairs ` V ) -> ( x e. V /\ y e. V ) ) ) )
8	3, 4, 7	mp2an 708	. . . . . . . . . . . 12 |- ( { x , y } e. P -> ( P C_ (Pairs ` V ) -> ( x e. V /\ y e. V ) ) )
9	2, 8	syl6bi 243	. . . . . . . . . . 11 |- ( c = { x , y } -> ( c e. P -> ( P C_ (Pairs ` V ) -> ( x e. V /\ y e. V ) ) ) )
10	9	com12 32	. . . . . . . . . 10 |- ( c e. P -> ( c = { x , y } -> ( P C_ (Pairs ` V ) -> ( x e. V /\ y e. V ) ) ) )
11	10	rexlimiv 3027	. . . . . . . . 9 |- ( E. c e. P c = { x , y } -> ( P C_ (Pairs ` V ) -> ( x e. V /\ y e. V ) ) )
12	11	com12 32	. . . . . . . 8 |- ( P C_ (Pairs ` V ) -> ( E. c e. P c = { x , y } -> ( x e. V /\ y e. V ) ) )
13	12	adantl 482	. . . . . . 7 |- ( ( V e. _V /\ P C_ (Pairs ` V ) ) -> ( E. c e. P c = { x , y } -> ( x e. V /\ y e. V ) ) )
14	13	imp 445	. . . . . 6 |- ( ( ( V e. _V /\ P C_ (Pairs ` V ) ) /\ E. c e. P c = { x , y } ) -> ( x e. V /\ y e. V ) )
15	14	simpld 475	. . . . 5 |- ( ( ( V e. _V /\ P C_ (Pairs ` V ) ) /\ E. c e. P c = { x , y } ) -> x e. V )
16	14	simprd 479	. . . . 5 |- ( ( ( V e. _V /\ P C_ (Pairs ` V ) ) /\ E. c e. P c = { x , y } ) -> y e. V )
17	1, 1, 15, 16	opabex2 7227	. . . 4 |- ( ( V e. _V /\ P C_ (Pairs ` V ) ) -> { <. x , y >. | E. c e. P c = { x , y } } e. _V )
18		elopab 4983	. . . . . . 7 |- ( p e. { <. x , y >. | E. c e. P c = { x , y } } <-> E. x E. y ( p = <. x , y >. /\ E. c e. P c = { x , y } ) )
19	11	adantl 482	. . . . . . . . . . . 12 |- ( ( p = <. x , y >. /\ E. c e. P c = { x , y } ) -> ( P C_ (Pairs ` V ) -> ( x e. V /\ y e. V ) ) )
20	19	adantld 483	. . . . . . . . . . 11 |- ( ( p = <. x , y >. /\ E. c e. P c = { x , y } ) -> ( ( V e. _V /\ P C_ (Pairs ` V ) ) -> ( x e. V /\ y e. V ) ) )
21	20	imp 445	. . . . . . . . . 10 |- ( ( ( p = <. x , y >. /\ E. c e. P c = { x , y } ) /\ ( V e. _V /\ P C_ (Pairs ` V ) ) ) -> ( x e. V /\ y e. V ) )
22		eleq1 2689	. . . . . . . . . . . 12 |- ( p = <. x , y >. -> ( p e. ( V X. V ) <-> <. x , y >. e. ( V X. V ) ) )
23	22	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( p = <. x , y >. /\ E. c e. P c = { x , y } ) /\ ( V e. _V /\ P C_ (Pairs ` V ) ) ) -> ( p e. ( V X. V ) <-> <. x , y >. e. ( V X. V ) ) )
24		opelxp 5146	. . . . . . . . . . 11 |- ( <. x , y >. e. ( V X. V ) <-> ( x e. V /\ y e. V ) )
25	23, 24	syl6bb 276	. . . . . . . . . 10 |- ( ( ( p = <. x , y >. /\ E. c e. P c = { x , y } ) /\ ( V e. _V /\ P C_ (Pairs ` V ) ) ) -> ( p e. ( V X. V ) <-> ( x e. V /\ y e. V ) ) )
26	21, 25	mpbird 247	. . . . . . . . 9 |- ( ( ( p = <. x , y >. /\ E. c e. P c = { x , y } ) /\ ( V e. _V /\ P C_ (Pairs ` V ) ) ) -> p e. ( V X. V ) )
27	26	ex 450	. . . . . . . 8 |- ( ( p = <. x , y >. /\ E. c e. P c = { x , y } ) -> ( ( V e. _V /\ P C_ (Pairs ` V ) ) -> p e. ( V X. V ) ) )
28	27	exlimivv 1860	. . . . . . 7 |- ( E. x E. y ( p = <. x , y >. /\ E. c e. P c = { x , y } ) -> ( ( V e. _V /\ P C_ (Pairs ` V ) ) -> p e. ( V X. V ) ) )
29	18, 28	sylbi 207	. . . . . 6 |- ( p e. { <. x , y >. | E. c e. P c = { x , y } } -> ( ( V e. _V /\ P C_ (Pairs ` V ) ) -> p e. ( V X. V ) ) )
30	29	com12 32	. . . . 5 |- ( ( V e. _V /\ P C_ (Pairs ` V ) ) -> ( p e. { <. x , y >. | E. c e. P c = { x , y } } -> p e. ( V X. V ) ) )
31	30	ssrdv 3609	. . . 4 |- ( ( V e. _V /\ P C_ (Pairs ` V ) ) -> { <. x , y >. | E. c e. P c = { x , y } } C_ ( V X. V ) )
32	17, 31	elpwd 4167	. . 3 |- ( ( V e. _V /\ P C_ (Pairs ` V ) ) -> { <. x , y >. | E. c e. P c = { x , y } } e. ~P ( V X. V ) )
33	32	ex 450	. 2 |- ( V e. _V -> ( P C_ (Pairs ` V ) -> { <. x , y >. | E. c e. P c = { x , y } } e. ~P ( V X. V ) ) )
34		fvprc 6185	. . . . 5 |- ( -. V e. _V -> (Pairs ` V ) = (/) )
35	34	sseq2d 3633	. . . 4 |- ( -. V e. _V -> ( P C_ (Pairs ` V ) <-> P C_ (/) ) )
36		ss0b 3973	. . . 4 |- ( P C_ (/) <-> P = (/) )
37	35, 36	syl6bb 276	. . 3 |- ( -. V e. _V -> ( P C_ (Pairs ` V ) <-> P = (/) ) )
38		rex0 3938	. . . . . . 7 |- -. E. c e. (/) c = { x , y }
39		rexeq 3139	. . . . . . 7 |- ( P = (/) -> ( E. c e. P c = { x , y } <-> E. c e. (/) c = { x , y } ) )
40	38, 39	mtbiri 317	. . . . . 6 |- ( P = (/) -> -. E. c e. P c = { x , y } )
41	40	alrimivv 1856	. . . . 5 |- ( P = (/) -> A. x A. y -. E. c e. P c = { x , y } )
42		opab0 5007	. . . . 5 |- ( { <. x , y >. | E. c e. P c = { x , y } } = (/) <-> A. x A. y -. E. c e. P c = { x , y } )
43	41, 42	sylibr 224	. . . 4 |- ( P = (/) -> { <. x , y >. | E. c e. P c = { x , y } } = (/) )
44		0elpw 4834	. . . 4 |- (/) e. ~P ( V X. V )
45	43, 44	syl6eqel 2709	. . 3 |- ( P = (/) -> { <. x , y >. | E. c e. P c = { x , y } } e. ~P ( V X. V ) )
46	37, 45	syl6bi 243	. 2 |- ( -. V e. _V -> ( P C_ (Pairs ` V ) -> { <. x , y >. | E. c e. P c = { x , y } } e. ~P ( V X. V ) ) )
47	33, 46	pm2.61i 176	1 |- ( P C_ (Pairs ` V ) -> { <. x , y >. | E. c e. P c = { x , y } } e. ~P ( V X. V ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {cpr 4179   <.cop 4183   {copab 4712   X. cxp 5112   ` 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:  sprsymrelfv  41744  sprsymrelf  41745