Theorem fundm2domnop0 10988

Description: A function with a domain containing (at least) two different elements is not an ordered pair. This theorem (which requires that ( G \ { (/) } ) needs to be a function instead of G) is useful for proofs for extensible structures, see structn0fun 12787. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 7-Jun-2021.) (Proof shortened by AV, 15-Nov-2021.)

Assertion

Ref	Expression
fundm2domnop0	|- ( ( Fun ( G \ { (/) } ) /\ 2o ~<_ dom G ) -> -. G e. ( _V X. _V ) )

Proof of Theorem fundm2domnop0

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

Step	Hyp	Ref	Expression
1		2dom 6896	. . 3 |- ( 2o ~<_ dom G -> E. a e. dom G E. b e. dom G -. a = b )
2		elvv 4736	. . . . . . . 8 |- ( G e. ( _V X. _V ) <-> E. x E. y G = <. x , y >. )
3		difeq1 3283	. . . . . . . . . . . . 13 |- ( G = <. x , y >. -> ( G \ { (/) } ) = ( <. x , y >. \ { (/) } ) )
4	3	funeqd 5292	. . . . . . . . . . . 12 |- ( G = <. x , y >. -> ( Fun ( G \ { (/) } ) <-> Fun ( <. x , y >. \ { (/) } ) ) )
5		opwo0id 4292	. . . . . . . . . . . . . . 15 |- <. x , y >. = ( <. x , y >. \ { (/) } )
6	5	eqcomi 2208	. . . . . . . . . . . . . 14 |- ( <. x , y >. \ { (/) } ) = <. x , y >.
7	6	funeqi 5291	. . . . . . . . . . . . 13 |- ( Fun ( <. x , y >. \ { (/) } ) <-> Fun <. x , y >. )
8		dmeq 4877	. . . . . . . . . . . . . . . . 17 |- ( G = <. x , y >. -> dom G = dom <. x , y >. )
9	8	eleq2d 2274	. . . . . . . . . . . . . . . 16 |- ( G = <. x , y >. -> ( a e. dom G <-> a e. dom <. x , y >. ) )
10	8	eleq2d 2274	. . . . . . . . . . . . . . . 16 |- ( G = <. x , y >. -> ( b e. dom G <-> b e. dom <. x , y >. ) )
11	9, 10	anbi12d 473	. . . . . . . . . . . . . . 15 |- ( G = <. x , y >. -> ( ( a e. dom G /\ b e. dom G ) <-> ( a e. dom <. x , y >. /\ b e. dom <. x , y >. ) ) )
12		eqid 2204	. . . . . . . . . . . . . . . . . 18 |- <. x , y >. = <. x , y >.
13		vex 2774	. . . . . . . . . . . . . . . . . 18 |- x e. _V
14		vex 2774	. . . . . . . . . . . . . . . . . 18 |- y e. _V
15	12, 13, 14	funopdmsn 5763	. . . . . . . . . . . . . . . . 17 |- ( ( Fun <. x , y >. /\ a e. dom <. x , y >. /\ b e. dom <. x , y >. ) -> a = b )
16	15	3expb 1206	. . . . . . . . . . . . . . . 16 |- ( ( Fun <. x , y >. /\ ( a e. dom <. x , y >. /\ b e. dom <. x , y >. ) ) -> a = b )
17	16	expcom 116	. . . . . . . . . . . . . . 15 |- ( ( a e. dom <. x , y >. /\ b e. dom <. x , y >. ) -> ( Fun <. x , y >. -> a = b ) )
18	11, 17	biimtrdi 163	. . . . . . . . . . . . . 14 |- ( G = <. x , y >. -> ( ( a e. dom G /\ b e. dom G ) -> ( Fun <. x , y >. -> a = b ) ) )
19	18	com23 78	. . . . . . . . . . . . 13 |- ( G = <. x , y >. -> ( Fun <. x , y >. -> ( ( a e. dom G /\ b e. dom G ) -> a = b ) ) )
20	7, 19	biimtrid 152	. . . . . . . . . . . 12 |- ( G = <. x , y >. -> ( Fun ( <. x , y >. \ { (/) } ) -> ( ( a e. dom G /\ b e. dom G ) -> a = b ) ) )
21	4, 20	sylbid 150	. . . . . . . . . . 11 |- ( G = <. x , y >. -> ( Fun ( G \ { (/) } ) -> ( ( a e. dom G /\ b e. dom G ) -> a = b ) ) )
22	21	impcomd 255	. . . . . . . . . 10 |- ( G = <. x , y >. -> ( ( ( a e. dom G /\ b e. dom G ) /\ Fun ( G \ { (/) } ) ) -> a = b ) )
23	22	exlimivv 1919	. . . . . . . . 9 |- ( E. x E. y G = <. x , y >. -> ( ( ( a e. dom G /\ b e. dom G ) /\ Fun ( G \ { (/) } ) ) -> a = b ) )
24	23	com12 30	. . . . . . . 8 |- ( ( ( a e. dom G /\ b e. dom G ) /\ Fun ( G \ { (/) } ) ) -> ( E. x E. y G = <. x , y >. -> a = b ) )
25	2, 24	biimtrid 152	. . . . . . 7 |- ( ( ( a e. dom G /\ b e. dom G ) /\ Fun ( G \ { (/) } ) ) -> ( G e. ( _V X. _V ) -> a = b ) )
26	25	con3d 632	. . . . . 6 |- ( ( ( a e. dom G /\ b e. dom G ) /\ Fun ( G \ { (/) } ) ) -> ( -. a = b -> -. G e. ( _V X. _V ) ) )
27	26	ex 115	. . . . 5 |- ( ( a e. dom G /\ b e. dom G ) -> ( Fun ( G \ { (/) } ) -> ( -. a = b -> -. G e. ( _V X. _V ) ) ) )
28	27	com23 78	. . . 4 |- ( ( a e. dom G /\ b e. dom G ) -> ( -. a = b -> ( Fun ( G \ { (/) } ) -> -. G e. ( _V X. _V ) ) ) )
29	28	rexlimivv 2628	. . 3 |- ( E. a e. dom G E. b e. dom G -. a = b -> ( Fun ( G \ { (/) } ) -> -. G e. ( _V X. _V ) ) )
30	1, 29	syl 14	. 2 |- ( 2o ~<_ dom G -> ( Fun ( G \ { (/) } ) -> -. G e. ( _V X. _V ) ) )
31	30	impcom 125	1 |- ( ( Fun ( G \ { (/) } ) /\ 2o ~<_ dom G ) -> -. G e. ( _V X. _V ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1372   E.wex 1514   e. wcel 2175   E.wrex 2484   _Vcvv 2771   \ cdif 3162   (/)c0 3459   {csn 3632   <.cop 3635   class class class wbr 4043   X. cxp 4672   dom cdm 4674   Fun wfun 5264   2oc2o 6495   ~<_ cdom 6825

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-sbc 2998  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-id 4339  df-suc 4417  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fv 5278  df-1o 6501  df-2o 6502  df-dom 6828

This theorem is referenced by:  fundm2domnop  10989  fun2dmnop0  10990  funvtxdm2domval  15568  funiedgdm2domval  15569