Theorem domen 3284

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

Hypothesis

Ref	Expression
bren.1	⊢ B ∈ V

Assertion

Ref	Expression
domen	⊢ (A ≼ B ↔ ∃x(A ≈ x ∧ x ⊆ B))

Distinct variable group(s):   x,A   x,B

Proof of Theorem domen

Step	Hyp	Ref	Expression
1		visset 1350	. . . . 5 ⊢ f ∈ V
2	1	f11o 2821	. . . 4 ⊢ (f:A–1-1→B ↔ ∃x(f:A–1-1-onto→x ∧ x ⊆ B))
3	2	biex 733	. . 3 ⊢ (∃f f:A–1-1→B ↔ ∃f∃x(f:A–1-1-onto→x ∧ x ⊆ B))
4		excom 728	. . 3 ⊢ (∃f∃x(f:A–1-1-onto→x ∧ x ⊆ B) ↔ ∃x∃f(f:A–1-1-onto→x ∧ x ⊆ B))
5	3, 4	bitr 151	. 2 ⊢ (∃f f:A–1-1→B ↔ ∃x∃f(f:A–1-1-onto→x ∧ x ⊆ B))
6		bren.1	. . 3 ⊢ B ∈ V
7	6	brdom 3283	. 2 ⊢ (A ≼ B ↔ ∃f f:A–1-1→B)
8		visset 1350	. . . . . 6 ⊢ x ∈ V
9	8	bren 3282	. . . . 5 ⊢ (A ≈ x ↔ ∃f f:A–1-1-onto→x)
10	9	anbi1i 368	. . . 4 ⊢ ((A ≈ x ∧ x ⊆ B) ↔ (∃f f:A–1-1-onto→x ∧ x ⊆ B))
11		19.41v 963	. . . 4 ⊢ (∃f(f:A–1-1-onto→x ∧ x ⊆ B) ↔ (∃f f:A–1-1-onto→x ∧ x ⊆ B))
12	10, 11	bitr4 154	. . 3 ⊢ ((A ≈ x ∧ x ⊆ B) ↔ ∃f(f:A–1-1-onto→x ∧ x ⊆ B))
13	12	biex 733	. 2 ⊢ (∃x(A ≈ x ∧ x ⊆ B) ↔ ∃x∃f(f:A–1-1-onto→x ∧ x ⊆ B))
14	5, 7, 13	3bitr4 158	1 ⊢ (A ≼ B ↔ ∃x(A ≈ x ∧ x ⊆ B))

Syntax hints:   ↔ wb 127   ∧ wa 196  ∃wex 678   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487   class class class wbr 2054  –1-1→wf1 2419  –1-1-onto→wf1o 2421   ≈ cen 3271   ≼ cdom 3272

This theorem is referenced by:  domeng 3285  undom 3342  mapdom1 3387  mapdom2 3389  infcntss 3443  infxpidmlem10 4942  infxpidmlem12 4944

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-cnv 2426  df-dm 2428  df-rn 2429  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-en 3274  df-dom 3275

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