Theorem 1dom1el 15637

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 6808	. . 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 5463	. . . . . . 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 5698	. . . . 5 |- ( ( ( A ~<_ 1o /\ B e. A /\ C e. A ) /\ f : A -1-1-> 1o ) -> ( f ` B ) e. 1o )
7		el1o 6495	. . . . 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 5698	. . . . 5 |- ( ( ( A ~<_ 1o /\ B e. A /\ C e. A ) /\ f : A -1-1-> 1o ) -> ( f ` C ) e. 1o )
11		el1o 6495	. . . . 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 2232	. . 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 5816	. . . 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 1913	1 |- ( ( A ~<_ 1o /\ B e. A /\ C e. A ) -> B = C )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   E.wex 1506   e. wcel 2167   (/)c0 3450   class class class wbr 4033   -->wf 5254   -1-1->wf1 5255   ` cfv 5258   1oc1o 6467   ~<_ cdom 6798

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-id 4328  df-suc 4406  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fv 5266  df-1o 6474  df-dom 6801

This theorem is referenced by: (None)