Theorem 1dom1el 7073

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 6999	. . 3 |- ( A ~<_ 1o -> E. f f : A -1-1-> 1o )
2	1	3ad2ant1 1045	. 2 |- ( ( A ~<_ 1o /\ B e. A /\ C e. A ) -> E. f f : A -1-1-> 1o )
3		f1f 5578	. . . . . . 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 1028	. . . . . 6 |- ( ( ( A ~<_ 1o /\ B e. A /\ C e. A ) /\ f : A -1-1-> 1o ) -> B e. A )
6	4, 5	ffvelcdmd 5818	. . . . 5 |- ( ( ( A ~<_ 1o /\ B e. A /\ C e. A ) /\ f : A -1-1-> 1o ) -> ( f ` B ) e. 1o )
7		el1o 6683	. . . . 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 1029	. . . . . 6 |- ( ( ( A ~<_ 1o /\ B e. A /\ C e. A ) /\ f : A -1-1-> 1o ) -> C e. A )
10	4, 9	ffvelcdmd 5818	. . . . 5 |- ( ( ( A ~<_ 1o /\ B e. A /\ C e. A ) /\ f : A -1-1-> 1o ) -> ( f ` C ) e. 1o )
11		el1o 6683	. . . . 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 2270	. . 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 5948	. . . 4 |- ( ( f : A -1-1-> 1o /\ ( B e. A /\ C e. A ) ) -> ( ( f ` B ) = ( f ` C ) -> B = C ) )
16	14, 5, 9, 15	syl12anc 1272	. . 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 1950	1 |- ( ( A ~<_ 1o /\ B e. A /\ C e. A ) -> B = C )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   E.wex 1541   e. wcel 2205   (/)c0 3512   class class class wbr 4114   -->wf 5353   -1-1->wf1 5354   ` cfv 5357   1oc1o 6653   ~<_ cdom 6987

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-suc 4497  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fv 5365  df-1o 6660  df-dom 6990

This theorem is referenced by:  modom  7074