Theorem brdomg 8703

Description: Dominance relation. (Contributed by NM, 15-Jun-1998.)

Assertion

Ref	Expression
brdomg	⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))

Distinct variable groups:   𝐴,𝑓   𝐵,𝑓

Allowed substitution hint:   𝐶(𝑓)

Proof of Theorem brdomg

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1eq2 6650	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–1-1→𝑦 ↔ 𝑓:𝐴–1-1→𝑦))
2	1	exbidv 1925	. . . 4 ⊢ (𝑥 = 𝐴 → (∃𝑓 𝑓:𝑥–1-1→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1→𝑦))
3		f1eq3 6651	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑓:𝐴–1-1→𝑦 ↔ 𝑓:𝐴–1-1→𝐵))
4	3	exbidv 1925	. . . 4 ⊢ (𝑦 = 𝐵 → (∃𝑓 𝑓:𝐴–1-1→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
5		df-dom 8693	. . . 4 ⊢ ≼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦}
6	2, 4, 5	brabg 5445	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝐶) → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
7	6	ex 412	. 2 ⊢ (𝐴 ∈ V → (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)))
8		reldom 8697	. . . . 5 ⊢ Rel ≼
9	8	brrelex1i 5634	. . . 4 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ∈ V)
10		f1f 6654	. . . . . 6 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴⟶𝐵)
11		fdm 6593	. . . . . . 7 ⊢ (𝑓:𝐴⟶𝐵 → dom 𝑓 = 𝐴)
12		vex 3426	. . . . . . . 8 ⊢ 𝑓 ∈ V
13	12	dmex 7732	. . . . . . 7 ⊢ dom 𝑓 ∈ V
14	11, 13	eqeltrrdi 2848	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → 𝐴 ∈ V)
15	10, 14	syl 17	. . . . 5 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝐴 ∈ V)
16	15	exlimiv 1934	. . . 4 ⊢ (∃𝑓 𝑓:𝐴–1-1→𝐵 → 𝐴 ∈ V)
17	9, 16	pm5.21ni 378	. . 3 ⊢ (¬ 𝐴 ∈ V → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
18	17	a1d 25	. 2 ⊢ (¬ 𝐴 ∈ V → (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)))
19	7, 18	pm2.61i 182	1 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   = wceq 1539  ∃wex 1783   ∈ wcel 2108  Vcvv 3422   class class class wbr 5070  dom cdm 5580  ⟶wf 6414  –1-1→wf1 6415   ≼ cdom 8689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-cnv 5588  df-dm 5590  df-rn 5591  df-fn 6421  df-f 6422  df-f1 6423  df-dom 8693

This theorem is referenced by:  brdomi  8704  brdom  8705  f1dom3g  8710  f1dom2gOLD  8713  f1domg  8715  dom3d  8737  domdifsn  8795  domtrfi  8938  fidomtri  9682  hashdom  14022  hashge3el3dif  14128  sizusglecusg  27733  erdsze2lem1  33065