Theorem domssr 6927

Description: If C is a superset of B and B dominates A, then C also dominates A. (Contributed by BTernaryTau, 7-Dec-2024.)

Assertion

Ref	Expression
domssr	|- ( ( C e. V /\ B C_ C /\ A ~<_ B ) -> A ~<_ C )

Proof of Theorem domssr

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		brdomi 6896	. . 3 |- ( A ~<_ B -> E. f f : A -1-1-> B )
2	1	3ad2ant3 1044	. 2 |- ( ( C e. V /\ B C_ C /\ A ~<_ B ) -> E. f f : A -1-1-> B )
3		simp2 1022	. . 3 |- ( ( C e. V /\ B C_ C /\ A ~<_ B ) -> B C_ C )
4		reldom 6890	. . . . 5 |- Rel ~<_
5	4	brrelex1i 4761	. . . 4 |- ( A ~<_ B -> A e. _V )
6	5	3ad2ant3 1044	. . 3 |- ( ( C e. V /\ B C_ C /\ A ~<_ B ) -> A e. _V )
7		simp1 1021	. . 3 |- ( ( C e. V /\ B C_ C /\ A ~<_ B ) -> C e. V )
8	3, 6, 7	jca32 310	. 2 |- ( ( C e. V /\ B C_ C /\ A ~<_ B ) -> ( B C_ C /\ ( A e. _V /\ C e. V ) ) )
9		f1ss 5536	. . . . 5 |- ( ( f : A -1-1-> B /\ B C_ C ) -> f : A -1-1-> C )
10		vex 2802	. . . . . . 7 |- f e. _V
11		f1dom4g 6902	. . . . . . 7 |- ( ( ( f e. _V /\ A e. _V /\ C e. V ) /\ f : A -1-1-> C ) -> A ~<_ C )
12	10, 11	mp3anl1 1365	. . . . . 6 |- ( ( ( A e. _V /\ C e. V ) /\ f : A -1-1-> C ) -> A ~<_ C )
13	12	ancoms 268	. . . . 5 |- ( ( f : A -1-1-> C /\ ( A e. _V /\ C e. V ) ) -> A ~<_ C )
14	9, 13	sylan 283	. . . 4 |- ( ( ( f : A -1-1-> B /\ B C_ C ) /\ ( A e. _V /\ C e. V ) ) -> A ~<_ C )
15	14	expl 378	. . 3 |- ( f : A -1-1-> B -> ( ( B C_ C /\ ( A e. _V /\ C e. V ) ) -> A ~<_ C ) )
16	15	exlimiv 1644	. 2 |- ( E. f f : A -1-1-> B -> ( ( B C_ C /\ ( A e. _V /\ C e. V ) ) -> A ~<_ C ) )
17	2, 8, 16	sylc 62	1 |- ( ( C e. V /\ B C_ C /\ A ~<_ B ) -> A ~<_ C )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   E.wex 1538   e. wcel 2200   _Vcvv 2799   C_ wss 3197   class class class wbr 4082   -1-1->wf1 5314   ~<_ cdom 6884

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-dom 6887

This theorem is referenced by:  rex2dom  6969