Theorem domen 9001

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 9000	. 2 ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)
3		vex 3482	. . . . . 6 ⊢ 𝑓 ∈ V
4	3	f11o 7970	. . . . 5 ⊢ (𝑓:𝐴–1-1→𝐵 ↔ ∃𝑥(𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵))
5	4	exbii 1845	. . . 4 ⊢ (∃𝑓 𝑓:𝐴–1-1→𝐵 ↔ ∃𝑓∃𝑥(𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵))
6		excom 2160	. . . 4 ⊢ (∃𝑓∃𝑥(𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ ∃𝑥∃𝑓(𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵))
7	5, 6	bitri 275	. . 3 ⊢ (∃𝑓 𝑓:𝐴–1-1→𝐵 ↔ ∃𝑥∃𝑓(𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵))
8		bren 8994	. . . . . 6 ⊢ (𝐴 ≈ 𝑥 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝑥)
9	8	anbi1i 624	. . . . 5 ⊢ ((𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ (∃𝑓 𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵))
10		19.41v 1947	. . . . 5 ⊢ (∃𝑓(𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ (∃𝑓 𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵))
11	9, 10	bitr4i 278	. . . 4 ⊢ ((𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ ∃𝑓(𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵))
12	11	exbii 1845	. . 3 ⊢ (∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ ∃𝑥∃𝑓(𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵))
13	7, 12	bitr4i 278	. 2 ⊢ (∃𝑓 𝑓:𝐴–1-1→𝐵 ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵))
14	2, 13	bitri 275	1 ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵))

Syntax hints:   ↔ wb 206   ∧ wa 395  ∃wex 1776   ∈ wcel 2106  Vcvv 3478   ⊆ wss 3963   class class class wbr 5148  –1-1→wf1 6560  –1-1-onto→wf1o 6562   ≈ cen 8981   ≼ cdom 8982

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-en 8985  df-dom 8986

This theorem is referenced by:  domeng  9002  infcntss  9360  ramub2  17048  ram0  17056