Theorem fun2dmnop0 11064

Description: A function with a domain containing (at least) two different elements is not an ordered pair. This stronger version of fun2dmnop 11065 (with the less restrictive requirement that ( G \ { (/) } ) needs to be a function instead of G) is useful for proofs for extensible structures, see structn0fun 13040. (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 1024	. . 3 |- ( ( ( Fun ( G \ { (/) } ) /\ A =/= B /\ { A , B } C_ dom G ) /\ G e. ( _V X. _V ) ) -> Fun ( G \ { (/) } ) )
2		dmexg 4987	. . . 4 |- ( G e. ( _V X. _V ) -> dom G e. _V )
3		simpl3 1026	. . . . . 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 3772	. . . . . . 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 3225	. . . . 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 3773	. . . . . . 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 3225	. . . . 5 |- ( ( ( Fun ( G \ { (/) } ) /\ A =/= B /\ { A , B } C_ dom G ) /\ G e. ( _V X. _V ) ) -> B e. dom G )
12		simpl2 1025	. . . . 5 |- ( ( ( Fun ( G \ { (/) } ) /\ A =/= B /\ { A , B } C_ dom G ) /\ G e. ( _V X. _V ) ) -> A =/= B )
13		neeq1 2413	. . . . . 6 |- ( a = A -> ( a =/= b <-> A =/= b ) )
14		neeq2 2414	. . . . . 6 |- ( b = B -> ( A =/= b <-> A =/= B ) )
15	13, 14	rspc2ev 2922	. . . . 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 1271	. . . 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 6969	. . . 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 596	. . 3 |- ( ( ( Fun ( G \ { (/) } ) /\ A =/= B /\ { A , B } C_ dom G ) /\ G e. ( _V X. _V ) ) -> 2o ~<_ dom G )
19		fundm2domnop0 11062	. . 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 639	1 |- ( ( Fun ( G \ { (/) } ) /\ A =/= B /\ { A , B } C_ dom G ) -> -. G e. ( _V X. _V ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   /\ w3a 1002   e. wcel 2200   =/= wne 2400   E.wrex 2509   _Vcvv 2799   \ cdif 3194   C_ wss 3197   (/)c0 3491   {csn 3666   {cpr 3667   class class class wbr 4082   X. cxp 4716   dom cdm 4718   Fun wfun 5311   2oc2o 6554   ~<_ cdom 6884

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-1o 6560  df-2o 6561  df-en 6886  df-dom 6887

This theorem is referenced by:  fun2dmnop  11065  funvtxdm2vald  15826  funiedgdm2vald  15827