Theorem 1dom1el 15931

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 6838	. . 3 |- ( A ~<_ 1o -> E. f f : A -1-1-> 1o )
2	1	3ad2ant1 1021	. 2 |- ( ( A ~<_ 1o /\ B e. A /\ C e. A ) -> E. f f : A -1-1-> 1o )
3		f1f 5481	. . . . . . 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 1004	. . . . . 6 |- ( ( ( A ~<_ 1o /\ B e. A /\ C e. A ) /\ f : A -1-1-> 1o ) -> B e. A )
6	4, 5	ffvelcdmd 5716	. . . . 5 |- ( ( ( A ~<_ 1o /\ B e. A /\ C e. A ) /\ f : A -1-1-> 1o ) -> ( f ` B ) e. 1o )
7		el1o 6523	. . . . 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 1005	. . . . . 6 |- ( ( ( A ~<_ 1o /\ B e. A /\ C e. A ) /\ f : A -1-1-> 1o ) -> C e. A )
10	4, 9	ffvelcdmd 5716	. . . . 5 |- ( ( ( A ~<_ 1o /\ B e. A /\ C e. A ) /\ f : A -1-1-> 1o ) -> ( f ` C ) e. 1o )
11		el1o 6523	. . . . 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 2241	. . 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 5838	. . . 4 |- ( ( f : A -1-1-> 1o /\ ( B e. A /\ C e. A ) ) -> ( ( f ` B ) = ( f ` C ) -> B = C ) )
16	14, 5, 9, 15	syl12anc 1248	. . 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 1922	1 |- ( ( A ~<_ 1o /\ B e. A /\ C e. A ) -> B = C )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 981   = wceq 1373   E.wex 1515   e. wcel 2176   (/)c0 3460   class class class wbr 4044   -->wf 5267   -1-1->wf1 5268   ` cfv 5271   1oc1o 6495   ~<_ cdom 6826

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-id 4340  df-suc 4418  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fv 5279  df-1o 6502  df-dom 6829

This theorem is referenced by: (None)