Theorem domssr 6950

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 6919	. . 3 |- ( A ~<_ B -> E. f f : A -1-1-> B )
2	1	3ad2ant3 1046	. 2 |- ( ( C e. V /\ B C_ C /\ A ~<_ B ) -> E. f f : A -1-1-> B )
3		simp2 1024	. . 3 |- ( ( C e. V /\ B C_ C /\ A ~<_ B ) -> B C_ C )
4		reldom 6913	. . . . 5 |- Rel ~<_
5	4	brrelex1i 4769	. . . 4 |- ( A ~<_ B -> A e. _V )
6	5	3ad2ant3 1046	. . 3 |- ( ( C e. V /\ B C_ C /\ A ~<_ B ) -> A e. _V )
7		simp1 1023	. . 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 5548	. . . . 5 |- ( ( f : A -1-1-> B /\ B C_ C ) -> f : A -1-1-> C )
10		vex 2805	. . . . . . 7 |- f e. _V
11		f1dom4g 6925	. . . . . . 7 |- ( ( ( f e. _V /\ A e. _V /\ C e. V ) /\ f : A -1-1-> C ) -> A ~<_ C )
12	10, 11	mp3anl1 1367	. . . . . 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 1646	. 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 1004   E.wex 1540   e. wcel 2202   _Vcvv 2802   C_ wss 3200   class class class wbr 4088   -1-1->wf1 5323   ~<_ cdom 6907

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-dom 6910

This theorem is referenced by:  rex2dom  6995