Theorem fun2dmnop0 10990

Description: A function with a domain containing (at least) two different elements is not an ordered pair. This stronger version of fun2dmnop 10991 (with the less restrictive requirement that ( G \ { (/) } ) needs to be a function instead of G) is useful for proofs for extensible structures, see structn0fun 12787. (Contributed by AV, 21-Sep-2020.) (Revised by AV, 7-Jun-2021.)

Hypotheses

Ref	Expression
fun2dmnop.a	|- A e. _V
fun2dmnop.b	|- B e. _V

Assertion

Ref	Expression
fun2dmnop0	|- ( ( Fun ( G \ { (/) } ) /\ A =/= B /\ { A , B } C_ dom G ) -> -. G e. ( _V X. _V ) )

Proof of Theorem fun2dmnop0

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

Step	Hyp	Ref	Expression
1		simpl1 1002	. . 3 |- ( ( ( Fun ( G \ { (/) } ) /\ A =/= B /\ { A , B } C_ dom G ) /\ G e. ( _V X. _V ) ) -> Fun ( G \ { (/) } ) )
2		dmexg 4941	. . . 4 |- ( G e. ( _V X. _V ) -> dom G e. _V )
3		simpl3 1004	. . . . . 6 |- ( ( ( Fun ( G \ { (/) } ) /\ A =/= B /\ { A , B } C_ dom G ) /\ G e. ( _V X. _V ) ) -> { A , B } C_ dom G )
4		fun2dmnop.a	. . . . . . . 8 |- A e. _V
5	4	prid1 3738	. . . . . . 7 |- A e. { A , B }
6	5	a1i 9	. . . . . 6 |- ( ( ( Fun ( G \ { (/) } ) /\ A =/= B /\ { A , B } C_ dom G ) /\ G e. ( _V X. _V ) ) -> A e. { A , B } )
7	3, 6	sseldd 3193	. . . . 5 |- ( ( ( Fun ( G \ { (/) } ) /\ A =/= B /\ { A , B } C_ dom G ) /\ G e. ( _V X. _V ) ) -> A e. dom G )
8		fun2dmnop.b	. . . . . . . 8 |- B e. _V
9	8	prid2 3739	. . . . . . 7 |- B e. { A , B }
10	9	a1i 9	. . . . . 6 |- ( ( ( Fun ( G \ { (/) } ) /\ A =/= B /\ { A , B } C_ dom G ) /\ G e. ( _V X. _V ) ) -> B e. { A , B } )
11	3, 10	sseldd 3193	. . . . 5 |- ( ( ( Fun ( G \ { (/) } ) /\ A =/= B /\ { A , B } C_ dom G ) /\ G e. ( _V X. _V ) ) -> B e. dom G )
12		simpl2 1003	. . . . 5 |- ( ( ( Fun ( G \ { (/) } ) /\ A =/= B /\ { A , B } C_ dom G ) /\ G e. ( _V X. _V ) ) -> A =/= B )
13		neeq1 2388	. . . . . 6 |- ( a = A -> ( a =/= b <-> A =/= b ) )
14		neeq2 2389	. . . . . 6 |- ( b = B -> ( A =/= b <-> A =/= B ) )
15	13, 14	rspc2ev 2891	. . . . 5 |- ( ( A e. dom G /\ B e. dom G /\ A =/= B ) -> E. a e. dom G E. b e. dom G a =/= b )
16	7, 11, 12, 15	syl3anc 1249	. . . 4 |- ( ( ( Fun ( G \ { (/) } ) /\ A =/= B /\ { A , B } C_ dom G ) /\ G e. ( _V X. _V ) ) -> E. a e. dom G E. b e. dom G a =/= b )
17		rex2dom 6909	. . . 4 |- ( ( dom G e. _V /\ E. a e. dom G E. b e. dom G a =/= b ) -> 2o ~<_ dom G )
18	2, 16, 17	syl2an2 594	. . 3 |- ( ( ( Fun ( G \ { (/) } ) /\ A =/= B /\ { A , B } C_ dom G ) /\ G e. ( _V X. _V ) ) -> 2o ~<_ dom G )
19		fundm2domnop0 10988	. . 3 |- ( ( Fun ( G \ { (/) } ) /\ 2o ~<_ dom G ) -> -. G e. ( _V X. _V ) )
20	1, 18, 19	syl2anc 411	. 2 |- ( ( ( Fun ( G \ { (/) } ) /\ A =/= B /\ { A , B } C_ dom G ) /\ G e. ( _V X. _V ) ) -> -. G e. ( _V X. _V ) )
21	20	pm2.01da 637	1 |- ( ( Fun ( G \ { (/) } ) /\ A =/= B /\ { A , B } C_ dom G ) -> -. G e. ( _V X. _V ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   /\ w3a 980   e. wcel 2175   =/= wne 2375   E.wrex 2484   _Vcvv 2771   \ cdif 3162   C_ wss 3165   (/)c0 3459   {csn 3632   {cpr 3633   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-tr 4142  df-id 4339  df-iord 4412  df-on 4414  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-fo 5276  df-f1o 5277  df-fv 5278  df-1o 6501  df-2o 6502  df-en 6827  df-dom 6828

This theorem is referenced by:  fun2dmnop  10991  funvtxdm2vald  15570  funiedgdm2vald  15571