Theorem domeng 4386

Description: Dominance in terms of equinumerosity, with the sethood requirement expressed as an antecedent. Example 1 of [Enderton] p. 146.

Assertion

Ref	Expression
domeng	|- (B e. C -> (A ~<_ B <-> E.x(A ~~ x /\ x (_ B)))

Distinct variable groups:   x,A   x,B

Proof of Theorem domeng

Step	Hyp	Ref	Expression
1		breq2 2628	. 2 |- (y = B -> (A ~<_ y <-> A ~<_ B))
2		sseq2 2086	. . . 4 |- (y = B -> (x (_ y <-> x (_ B))
3	2	anbi2d 618	. . 3 |- (y = B -> ((A ~~ x /\ x (_ y) <-> (A ~~ x /\ x (_ B)))
4	3	exbidv 1281	. 2 |- (y = B -> (E.x(A ~~ x /\ x (_ y) <-> E.x(A ~~ x /\ x (_ B)))
5		visset 1816	. . 3 |- y e. V
6	5	domen 4385	. 2 |- (A ~<_ y <-> E.x(A ~~ x /\ x (_ y))
7	1, 4, 6	vtoclbg 1851	1 |- (B e. C -> (A ~<_ B <-> E.x(A ~~ x /\ x (_ B)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  E.wex 982   (_ wss 2050   class class class wbr 2624   ~~ cen 4370   ~<_ cdom 4371

This theorem is referenced by:  domfi 4549  domfiOLD 4550  isfinite2 4557  isfinite2OLD 4558

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-xp 3190  df-rel 3191  df-cnv 3192  df-dm 3194  df-rn 3195  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-en 4374  df-dom 4375

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