Theorem domen 9019

Description: Dominance in terms of equinumerosity. Example 1 of [Enderton] p. 146. (Contributed by NM, 15-Jun-1998.)

Hypothesis

Ref	Expression
bren.1	⊢ 𝐵 ∈ V

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem domen

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bren.1	. . 3 ⊢ 𝐵 ∈ V
2	1	brdom 9018	. 2 ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴⟶1-1𝐵)
3		vex 3458	. . . . . 6 ⊢ 𝑓 ∈ V
4	3	f11o 7986	. . . . 5 ⊢ (𝑓:𝐴⟶1-1𝐵 ↔ ∃𝑥(𝑓:𝐴⟶1-1onto𝑥 ∧ 𝑥 ⊆ 𝐵))
5	4	exbii 1881	. . . 4 ⊢ (∃𝑓 𝑓:𝐴⟶1-1𝐵 ↔ ∃𝑓∃𝑥(𝑓:𝐴⟶1-1onto𝑥 ∧ 𝑥 ⊆ 𝐵))
6		excom 2202	. . . 4 ⊢ (∃𝑓∃𝑥(𝑓:𝐴⟶1-1onto𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ ∃𝑥∃𝑓(𝑓:𝐴⟶1-1onto𝑥 ∧ 𝑥 ⊆ 𝐵))
7	5, 6	bitri 278	. . 3 ⊢ (∃𝑓 𝑓:𝐴⟶1-1𝐵 ↔ ∃𝑥∃𝑓(𝑓:𝐴⟶1-1onto𝑥 ∧ 𝑥 ⊆ 𝐵))
8		bren 9014	. . . . . 6 ⊢ (𝐴 ≈ 𝑥 ↔ ∃𝑓 𝑓:𝐴⟶1-1onto𝑥)
9	8	anbi1i 636	. . . . 5 ⊢ ((𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ (∃𝑓 𝑓:𝐴⟶1-1onto𝑥 ∧ 𝑥 ⊆ 𝐵))
10		19.41v 1984	. . . . 5 ⊢ (∃𝑓(𝑓:𝐴⟶1-1onto𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ (∃𝑓 𝑓:𝐴⟶1-1onto𝑥 ∧ 𝑥 ⊆ 𝐵))
11	9, 10	bitr4i 281	. . . 4 ⊢ ((𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ ∃𝑓(𝑓:𝐴⟶1-1onto𝑥 ∧ 𝑥 ⊆ 𝐵))
12	11	exbii 1881	. . 3 ⊢ (∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ ∃𝑥∃𝑓(𝑓:𝐴⟶1-1onto𝑥 ∧ 𝑥 ⊆ 𝐵))
13	7, 12	bitr4i 281	. 2 ⊢ (∃𝑓 𝑓:𝐴⟶1-1𝐵 ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵))
14	2, 13	bitri 278	1 ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401  ∃wex 1812   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900   class class class wbr 5106  ⟶1-1wf1 6558  ⟶1-1ontowf1o 6560   ≈ cen 9001   ≼ cdom 9002

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-11 2197  ax-ext 2735  ax-sep 5253  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-xp 5662  df-rel 5663  df-cnv 5664  df-dm 5666  df-rn 5667  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-en 9005  df-dom 9006

This theorem is used by:  domeng  9020  infcntss  9345  ramub2  17224  ram0  17232