Theorem fun2dmnop0 11110

Description: A function with a domain containing (at least) two different elements is not an ordered pair. This stronger version of fun2dmnop 11111 (with the less restrictive requirement that ( G \ { (/) } ) needs to be a function instead of G) is useful for proofs for extensible structures, see structn0fun 13094. (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 1026	. . 3 |- ( ( ( Fun ( G \ { (/) } ) /\ A =/= B /\ { A , B } C_ dom G ) /\ G e. ( _V X. _V ) ) -> Fun ( G \ { (/) } ) )
2		dmexg 4996	. . . 4 |- ( G e. ( _V X. _V ) -> dom G e. _V )
3		simpl3 1028	. . . . . 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 3777	. . . . . . 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 3228	. . . . 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 3778	. . . . . . 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 3228	. . . . 5 |- ( ( ( Fun ( G \ { (/) } ) /\ A =/= B /\ { A , B } C_ dom G ) /\ G e. ( _V X. _V ) ) -> B e. dom G )
12		simpl2 1027	. . . . 5 |- ( ( ( Fun ( G \ { (/) } ) /\ A =/= B /\ { A , B } C_ dom G ) /\ G e. ( _V X. _V ) ) -> A =/= B )
13		neeq1 2415	. . . . . 6 |- ( a = A -> ( a =/= b <-> A =/= b ) )
14		neeq2 2416	. . . . . 6 |- ( b = B -> ( A =/= b <-> A =/= B ) )
15	13, 14	rspc2ev 2925	. . . . 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 1273	. . . 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 6995	. . . 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 11108	. . 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 1004   e. wcel 2202   =/= wne 2402   E.wrex 2511   _Vcvv 2802   \ cdif 3197   C_ wss 3200   (/)c0 3494   {csn 3669   {cpr 3670   class class class wbr 4088   X. cxp 4723   dom cdm 4725   Fun wfun 5320   2oc2o 6575   ~<_ cdom 6907

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-1o 6581  df-2o 6582  df-en 6909  df-dom 6910

This theorem is referenced by:  fun2dmnop  11111  funvtxdm2vald  15881  funiedgdm2vald  15882