Theorem graop 25921

Description: Any representation of a graph G (especially as extensible structure G = { <. ( Base ` ndx ) , V >. , <. (.ef ` ndx ) , E >. }) is convertible in a representation of the graph as ordered pair. (Contributed by AV, 7-Oct-2020.)

Hypothesis

Ref	Expression
graop.h	|- H = <. (Vtx ` G ) , (iEdg ` G ) >.

Assertion

Ref	Expression
graop	|- ( (Vtx ` G ) = (Vtx ` H ) /\ (iEdg ` G ) = (iEdg ` H ) )

Proof of Theorem graop

Step	Hyp	Ref	Expression
1		graop.h	. . . 4 |- H = <. (Vtx ` G ) , (iEdg ` G ) >.
2	1	fveq2i 6194	. . 3 |- (Vtx ` H ) = (Vtx ` <. (Vtx ` G ) , (iEdg ` G ) >. )
3		fvex 6201	. . . 4 |- (Vtx ` G ) e. _V
4		fvex 6201	. . . 4 |- (iEdg ` G ) e. _V
5		opvtxfv 25884	. . . 4 |- ( ( (Vtx ` G ) e. _V /\ (iEdg ` G ) e. _V ) -> (Vtx ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = (Vtx ` G ) )
6	3, 4, 5	mp2an 708	. . 3 |- (Vtx ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = (Vtx ` G )
7	2, 6	eqtr2i 2645	. 2 |- (Vtx ` G ) = (Vtx ` H )
8	1	fveq2i 6194	. . 3 |- (iEdg ` H ) = (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. )
9		opiedgfv 25887	. . . 4 |- ( ( (Vtx ` G ) e. _V /\ (iEdg ` G ) e. _V ) -> (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = (iEdg ` G ) )
10	3, 4, 9	mp2an 708	. . 3 |- (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = (iEdg ` G )
11	8, 10	eqtr2i 2645	. 2 |- (iEdg ` G ) = (iEdg ` H )
12	7, 11	pm3.2i 471	1 |- ( (Vtx ` G ) = (Vtx ` H ) /\ (iEdg ` G ) = (iEdg ` H ) )

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   ` cfv 5888  Vtxcvtx 25874  iEdgciedg 25875

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-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-fv 5896  df-1st 7168  df-2nd 7169  df-vtx 25876  df-iedg 25877

This theorem is referenced by: (None)