Theorem domssr 6868

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 6837	. . 3 |- ( A ~<_ B -> E. f f : A -1-1-> B )
2	1	3ad2ant3 1022	. 2 |- ( ( C e. V /\ B C_ C /\ A ~<_ B ) -> E. f f : A -1-1-> B )
3		simp2 1000	. . 3 |- ( ( C e. V /\ B C_ C /\ A ~<_ B ) -> B C_ C )
4		reldom 6831	. . . . 5 |- Rel ~<_
5	4	brrelex1i 4717	. . . 4 |- ( A ~<_ B -> A e. _V )
6	5	3ad2ant3 1022	. . 3 |- ( ( C e. V /\ B C_ C /\ A ~<_ B ) -> A e. _V )
7		simp1 999	. . 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 5486	. . . . 5 |- ( ( f : A -1-1-> B /\ B C_ C ) -> f : A -1-1-> C )
10		vex 2774	. . . . . . 7 |- f e. _V
11		f1dom4g 6843	. . . . . . 7 |- ( ( ( f e. _V /\ A e. _V /\ C e. V ) /\ f : A -1-1-> C ) -> A ~<_ C )
12	10, 11	mp3anl1 1343	. . . . . 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 1620	. 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 980   E.wex 1514   e. wcel 2175   _Vcvv 2771   C_ wss 3165   class class class wbr 4043   -1-1->wf1 5267   ~<_ cdom 6825

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-dom 6828

This theorem is referenced by:  rex2dom  6909