Theorem fundm2domnop0 11300

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 13365. (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 7093	. . 3 |- ( 2o ~<_ dom G -> E. a e. dom G E. b e. dom G -. a = b )
2		elvv 4837	. . . . . . . 8 |- ( G e. ( _V X. _V ) <-> E. x E. y G = <. x , y >. )
3		difeq1 3340	. . . . . . . . . . . . 13 |- ( G = <. x , y >. -> ( G \ { (/) } ) = ( <. x , y >. \ { (/) } ) )
4	3	funeqd 5399	. . . . . . . . . . . 12 |- ( G = <. x , y >. -> ( Fun ( G \ { (/) } ) <-> Fun ( <. x , y >. \ { (/) } ) ) )
5		opwo0id 4389	. . . . . . . . . . . . . . 15 |- <. x , y >. = ( <. x , y >. \ { (/) } )
6	5	eqcomi 2242	. . . . . . . . . . . . . 14 |- ( <. x , y >. \ { (/) } ) = <. x , y >.
7	6	funeqi 5398	. . . . . . . . . . . . 13 |- ( Fun ( <. x , y >. \ { (/) } ) <-> Fun <. x , y >. )
8		dmeq 4981	. . . . . . . . . . . . . . . . 17 |- ( G = <. x , y >. -> dom G = dom <. x , y >. )
9	8	eleq2d 2308	. . . . . . . . . . . . . . . 16 |- ( G = <. x , y >. -> ( a e. dom G <-> a e. dom <. x , y >. ) )
10	8	eleq2d 2308	. . . . . . . . . . . . . . . 16 |- ( G = <. x , y >. -> ( b e. dom G <-> b e. dom <. x , y >. ) )
11	9, 10	anbi12d 477	. . . . . . . . . . . . . . 15 |- ( G = <. x , y >. -> ( ( a e. dom G /\ b e. dom G ) <-> ( a e. dom <. x , y >. /\ b e. dom <. x , y >. ) ) )
12		eqid 2238	. . . . . . . . . . . . . . . . . 18 |- <. x , y >. = <. x , y >.
13		vex 2824	. . . . . . . . . . . . . . . . . 18 |- x e. _V
14		vex 2824	. . . . . . . . . . . . . . . . . 18 |- y e. _V
15	12, 13, 14	funopdmsn 5895	. . . . . . . . . . . . . . . . 17 |- ( ( Fun <. x , y >. /\ a e. dom <. x , y >. /\ b e. dom <. x , y >. ) -> a = b )
16	15	3expb 1235	. . . . . . . . . . . . . . . 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 1952	. . . . . . . . 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 640	. . . . . 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 2674	. . 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 ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1402   E.wex 1545   e. wcel 2209   E.wrex 2529   _Vcvv 2821   \ cdif 3217   (/)c0 3520   {csn 3709   <.cop 3712   class class class wbr 4130   X. cxp 4772   dom cdm 4774   Fun wfun 5371   2oc2o 6681   ~<_ cdom 7021

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-suc 4516  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fv 5385  df-1o 6687  df-2o 6688  df-dom 7024

This theorem is used by:  fundm2domnop  11301  fun2dmnop0  11302  funvtxdm2domval  16270  funiedgdm2domval  16271