Theorem domen 4379

Description: Dominance in terms of equinumerosity. Example 1 of [Enderton] p. 146.

Hypothesis

Ref	Expression
bren.1	|- B e. V

Assertion

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

Distinct variable groups:   x,A   x,B

Proof of Theorem domen

Step	Hyp	Ref	Expression
1		visset 1813	. . . . 5 |- f e. V
2	1	f11o 3712	. . . 4 |- (f:A-1-1->B <-> E.x(f:A-1-1-onto->x /\ x (_ B))
3	2	exbii 1051	. . 3 |- (E.f f:A-1-1->B <-> E.fE.x(f:A-1-1-onto->x /\ x (_ B))
4		excom 1046	. . 3 |- (E.fE.x(f:A-1-1-onto->x /\ x (_ B) <-> E.xE.f(f:A-1-1-onto->x /\ x (_ B))
5	3, 4	bitr 173	. 2 |- (E.f f:A-1-1->B <-> E.xE.f(f:A-1-1-onto->x /\ x (_ B))
6		bren.1	. . 3 |- B e. V
7	6	brdom 4378	. 2 |- (A ~<_ B <-> E.f f:A-1-1->B)
8		visset 1813	. . . . . 6 |- x e. V
9	8	bren 4377	. . . . 5 |- (A ~~ x <-> E.f f:A-1-1-onto->x)
10	9	anbi1i 481	. . . 4 |- ((A ~~ x /\ x (_ B) <-> (E.f f:A-1-1-onto->x /\ x (_ B))
11		19.41v 1305	. . . 4 |- (E.f(f:A-1-1-onto->x /\ x (_ B) <-> (E.f f:A-1-1-onto->x /\ x (_ B))
12	10, 11	bitr4 176	. . 3 |- ((A ~~ x /\ x (_ B) <-> E.f(f:A-1-1-onto->x /\ x (_ B))
13	12	exbii 1051	. 2 |- (E.x(A ~~ x /\ x (_ B) <-> E.xE.f(f:A-1-1-onto->x /\ x (_ B))
14	5, 7, 13	3bitr4 183	1 |- (A ~<_ B <-> E.x(A ~~ x /\ x (_ B))

Syntax hints:   <-> wb 146   /\ wa 223   e. wcel 958  E.wex 980  Vcvv 1811   (_ wss 2047   class class class wbr 2619  -1-1->wf1 3179  -1-1-onto->wf1o 3181   ~~ cen 4364   ~<_ cdom 4365

This theorem is referenced by:  domeng 4380  undom 4438  mapdom1 4492  mapdom2 4494  infcntss 4554  infxpidmlem10 7561  infxpidmlem12 7563

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-xp 3184  df-rel 3185  df-cnv 3186  df-dm 3188  df-rn 3189  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-en 4368  df-dom 4369

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