Theorem fun2dmnop0 11160

Description: A function with a domain containing (at least) two different elements is not an ordered pair. This stronger version of fun2dmnop 11161 (with the less restrictive requirement that ( G \ { (/) } ) needs to be a function instead of G) is useful for proofs for extensible structures, see structn0fun 13158. (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 1027	. . 3 |- ( ( ( Fun ( G \ { (/) } ) /\ A =/= B /\ { A , B } C_ dom G ) /\ G e. ( _V X. _V ) ) -> Fun ( G \ { (/) } ) )
2		dmexg 5002	. . . 4 |- ( G e. ( _V X. _V ) -> dom G e. _V )
3		simpl3 1029	. . . . . 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 3781	. . . . . . 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 3229	. . . . 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 3782	. . . . . . 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 3229	. . . . 5 |- ( ( ( Fun ( G \ { (/) } ) /\ A =/= B /\ { A , B } C_ dom G ) /\ G e. ( _V X. _V ) ) -> B e. dom G )
12		simpl2 1028	. . . . 5 |- ( ( ( Fun ( G \ { (/) } ) /\ A =/= B /\ { A , B } C_ dom G ) /\ G e. ( _V X. _V ) ) -> A =/= B )
13		neeq1 2416	. . . . . 6 |- ( a = A -> ( a =/= b <-> A =/= b ) )
14		neeq2 2417	. . . . . 6 |- ( b = B -> ( A =/= b <-> A =/= B ) )
15	13, 14	rspc2ev 2926	. . . . 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 1274	. . . 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 7039	. . . 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 598	. . 3 |- ( ( ( Fun ( G \ { (/) } ) /\ A =/= B /\ { A , B } C_ dom G ) /\ G e. ( _V X. _V ) ) -> 2o ~<_ dom G )
19		fundm2domnop0 11158	. . 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 641	1 |- ( ( Fun ( G \ { (/) } ) /\ A =/= B /\ { A , B } C_ dom G ) -> -. G e. ( _V X. _V ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   /\ w3a 1005   e. wcel 2202   =/= wne 2403   E.wrex 2512   _Vcvv 2803   \ cdif 3198   C_ wss 3201   (/)c0 3496   {csn 3673   {cpr 3674   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-tr 4193  df-id 4396  df-iord 4469  df-on 4471  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-fo 5339  df-f1o 5340  df-fv 5341  df-1o 6625  df-2o 6626  df-en 6953  df-dom 6954

This theorem is referenced by:  fun2dmnop  11161  funvtxdm2vald  15955  funiedgdm2vald  15956