Theorem fundm2domnop0 11158

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 13158. (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 7023	. . 3 |- ( 2o ~<_ dom G -> E. a e. dom G E. b e. dom G -. a = b )
2		elvv 4794	. . . . . . . 8 |- ( G e. ( _V X. _V ) <-> E. x E. y G = <. x , y >. )
3		difeq1 3320	. . . . . . . . . . . . 13 |- ( G = <. x , y >. -> ( G \ { (/) } ) = ( <. x , y >. \ { (/) } ) )
4	3	funeqd 5355	. . . . . . . . . . . 12 |- ( G = <. x , y >. -> ( Fun ( G \ { (/) } ) <-> Fun ( <. x , y >. \ { (/) } ) ) )
5		opwo0id 4347	. . . . . . . . . . . . . . 15 |- <. x , y >. = ( <. x , y >. \ { (/) } )
6	5	eqcomi 2235	. . . . . . . . . . . . . 14 |- ( <. x , y >. \ { (/) } ) = <. x , y >.
7	6	funeqi 5354	. . . . . . . . . . . . 13 |- ( Fun ( <. x , y >. \ { (/) } ) <-> Fun <. x , y >. )
8		dmeq 4937	. . . . . . . . . . . . . . . . 17 |- ( G = <. x , y >. -> dom G = dom <. x , y >. )
9	8	eleq2d 2301	. . . . . . . . . . . . . . . 16 |- ( G = <. x , y >. -> ( a e. dom G <-> a e. dom <. x , y >. ) )
10	8	eleq2d 2301	. . . . . . . . . . . . . . . 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 2231	. . . . . . . . . . . . . . . . . 18 |- <. x , y >. = <. x , y >.
13		vex 2806	. . . . . . . . . . . . . . . . . 18 |- x e. _V
14		vex 2806	. . . . . . . . . . . . . . . . . 18 |- y e. _V
15	12, 13, 14	funopdmsn 5842	. . . . . . . . . . . . . . . . 17 |- ( ( Fun <. x , y >. /\ a e. dom <. x , y >. /\ b e. dom <. x , y >. ) -> a = b )
16	15	3expb 1231	. . . . . . . . . . . . . . . 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 1945	. . . . . . . . 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 636	. . . . . 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 2657	. . 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 1398   E.wex 1541   e. wcel 2202   E.wrex 2512   _Vcvv 2803   \ cdif 3198   (/)c0 3496   {csn 3673   <.cop 3676   class class class wbr 4093   X. cxp 4729   dom cdm 4731   Fun wfun 5327   2oc2o 6619   ~<_ cdom 6951

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-suc 4474  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fv 5341  df-1o 6625  df-2o 6626  df-dom 6954

This theorem is referenced by:  fundm2domnop  11159  fun2dmnop0  11160  funvtxdm2domval  15953  funiedgdm2domval  15954