Theorem 1dom1el 15483

Description: If a set is dominated by one, then any two of its elements are equal. (Contributed by Jim Kingdon, 23-Apr-2025.)

Assertion

Ref	Expression
1dom1el	|- ( ( A ~<_ 1o /\ B e. A /\ C e. A ) -> B = C )

Proof of Theorem 1dom1el

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		brdomi 6803	. . 3 |- ( A ~<_ 1o -> E. f f : A -1-1-> 1o )
2	1	3ad2ant1 1020	. 2 |- ( ( A ~<_ 1o /\ B e. A /\ C e. A ) -> E. f f : A -1-1-> 1o )
3		f1f 5459	. . . . . . 7 |- ( f : A -1-1-> 1o -> f : A --> 1o )
4	3	adantl 277	. . . . . 6 |- ( ( ( A ~<_ 1o /\ B e. A /\ C e. A ) /\ f : A -1-1-> 1o ) -> f : A --> 1o )
5		simpl2 1003	. . . . . 6 |- ( ( ( A ~<_ 1o /\ B e. A /\ C e. A ) /\ f : A -1-1-> 1o ) -> B e. A )
6	4, 5	ffvelcdmd 5694	. . . . 5 |- ( ( ( A ~<_ 1o /\ B e. A /\ C e. A ) /\ f : A -1-1-> 1o ) -> ( f ` B ) e. 1o )
7		el1o 6490	. . . . 5 |- ( ( f ` B ) e. 1o <-> ( f ` B ) = (/) )
8	6, 7	sylib 122	. . . 4 |- ( ( ( A ~<_ 1o /\ B e. A /\ C e. A ) /\ f : A -1-1-> 1o ) -> ( f ` B ) = (/) )
9		simpl3 1004	. . . . . 6 |- ( ( ( A ~<_ 1o /\ B e. A /\ C e. A ) /\ f : A -1-1-> 1o ) -> C e. A )
10	4, 9	ffvelcdmd 5694	. . . . 5 |- ( ( ( A ~<_ 1o /\ B e. A /\ C e. A ) /\ f : A -1-1-> 1o ) -> ( f ` C ) e. 1o )
11		el1o 6490	. . . . 5 |- ( ( f ` C ) e. 1o <-> ( f ` C ) = (/) )
12	10, 11	sylib 122	. . . 4 |- ( ( ( A ~<_ 1o /\ B e. A /\ C e. A ) /\ f : A -1-1-> 1o ) -> ( f ` C ) = (/) )
13	8, 12	eqtr4d 2229	. . 3 |- ( ( ( A ~<_ 1o /\ B e. A /\ C e. A ) /\ f : A -1-1-> 1o ) -> ( f ` B ) = ( f ` C ) )
14		simpr 110	. . . 4 |- ( ( ( A ~<_ 1o /\ B e. A /\ C e. A ) /\ f : A -1-1-> 1o ) -> f : A -1-1-> 1o )
15		f1veqaeq 5812	. . . 4 |- ( ( f : A -1-1-> 1o /\ ( B e. A /\ C e. A ) ) -> ( ( f ` B ) = ( f ` C ) -> B = C ) )
16	14, 5, 9, 15	syl12anc 1247	. . 3 |- ( ( ( A ~<_ 1o /\ B e. A /\ C e. A ) /\ f : A -1-1-> 1o ) -> ( ( f ` B ) = ( f ` C ) -> B = C ) )
17	13, 16	mpd 13	. 2 |- ( ( ( A ~<_ 1o /\ B e. A /\ C e. A ) /\ f : A -1-1-> 1o ) -> B = C )
18	2, 17	exlimddv 1910	1 |- ( ( A ~<_ 1o /\ B e. A /\ C e. A ) -> B = C )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   E.wex 1503   e. wcel 2164   (/)c0 3446   class class class wbr 4029   -->wf 5250   -1-1->wf1 5251   ` cfv 5254   1oc1o 6462   ~<_ cdom 6793

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-suc 4402  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fv 5262  df-1o 6469  df-dom 6796

This theorem is referenced by: (None)