Theorem domen 8946

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 8945	. 2 ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)
3		vex 3461	. . . . . 6 ⊢ 𝑓 ∈ V
4	3	f11o 7932	. . . . 5 ⊢ (𝑓:𝐴–1-1→𝐵 ↔ ∃𝑥(𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵))
5	4	exbii 1871	. . . 4 ⊢ (∃𝑓 𝑓:𝐴–1-1→𝐵 ↔ ∃𝑓∃𝑥(𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵))
6		excom 2199	. . . 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 278	. . 3 ⊢ (∃𝑓 𝑓:𝐴–1-1→𝐵 ↔ ∃𝑥∃𝑓(𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵))
8		bren 8941	. . . . . 6 ⊢ (𝐴 ≈ 𝑥 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝑥)
9	8	anbi1i 635	. . . . 5 ⊢ ((𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ (∃𝑓 𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵))
10		19.41v 1972	. . . . 5 ⊢ (∃𝑓(𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ (∃𝑓 𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵))
11	9, 10	bitr4i 281	. . . 4 ⊢ ((𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ ∃𝑓(𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵))
12	11	exbii 1871	. . 3 ⊢ (∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ ∃𝑥∃𝑓(𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵))
13	7, 12	bitr4i 281	. 2 ⊢ (∃𝑓 𝑓:𝐴–1-1→𝐵 ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵))
14	2, 13	bitri 278	1 ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵))

Syntax hints:   ↔ wb 209   ∧ wa 400  ∃wex 1802   ∈ wcel 2145  Vcvv 3457   ⊆ wss 3907   class class class wbr 5104  –1-1→wf1 6522  –1-1-onto→wf1o 6524   ≈ cen 8928   ≼ cdom 8929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-11 2194  ax-ext 2737  ax-sep 5250  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5105  df-opab 5167  df-xp 5657  df-rel 5658  df-cnv 5659  df-dm 5661  df-rn 5662  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-en 8932  df-dom 8933

This theorem is referenced by:  domeng  8947  infcntss  9270  ramub2  17062  ram0  17070