Theorem domssr 7054

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 7023	. . 3 |- ( A ~<_ B -> E. f f : A -1-1-> B )
2	1	3ad2ant3 1051	. 2 |- ( ( C e. V /\ B C_ C /\ A ~<_ B ) -> E. f f : A -1-1-> B )
3		simp2 1029	. . 3 |- ( ( C e. V /\ B C_ C /\ A ~<_ B ) -> B C_ C )
4		reldom 7017	. . . . 5 |- Rel ~<_
5	4	brrelex1i 4813	. . . 4 |- ( A ~<_ B -> A e. _V )
6	5	3ad2ant3 1051	. . 3 |- ( ( C e. V /\ B C_ C /\ A ~<_ B ) -> A e. _V )
7		simp1 1028	. . 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 5599	. . . . 5 |- ( ( f : A -1-1-> B /\ B C_ C ) -> f : A -1-1-> C )
10		vex 2824	. . . . . . 7 |- f e. _V
11		f1dom4g 7029	. . . . . . 7 |- ( ( ( f e. _V /\ A e. _V /\ C e. V ) /\ f : A -1-1-> C ) -> A ~<_ C )
12	10, 11	mp3anl1 1372	. . . . . 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 1651	. 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 1009   E.wex 1545   e. wcel 2209   _Vcvv 2821   C_ wss 3220   class class class wbr 4125   -1-1->wf1 5369   ~<_ cdom 7011

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-dom 7014

This theorem is referenced by:  rex2dom  7100