Theorem brdomg 7851

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 6010	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–1-1→𝑦 ↔ 𝑓:𝐴–1-1→𝑦))
2	1	exbidv 1837	. . . 4 ⊢ (𝑥 = 𝐴 → (∃𝑓 𝑓:𝑥–1-1→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1→𝑦))
3		f1eq3 6011	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑓:𝐴–1-1→𝑦 ↔ 𝑓:𝐴–1-1→𝐵))
4	3	exbidv 1837	. . . 4 ⊢ (𝑦 = 𝐵 → (∃𝑓 𝑓:𝐴–1-1→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
5		df-dom 7843	. . . 4 ⊢ ≼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦}
6	2, 4, 5	brabg 4919	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝐶) → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
7	6	ex 449	. 2 ⊢ (𝐴 ∈ V → (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)))
8		reldom 7847	. . . . 5 ⊢ Rel ≼
9	8	brrelexi 5082	. . . 4 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ∈ V)
10		f1f 6014	. . . . . 6 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴⟶𝐵)
11		fdm 5964	. . . . . . 7 ⊢ (𝑓:𝐴⟶𝐵 → dom 𝑓 = 𝐴)
12		vex 3176	. . . . . . . 8 ⊢ 𝑓 ∈ V
13	12	dmex 6991	. . . . . . 7 ⊢ dom 𝑓 ∈ V
14	11, 13	syl6eqelr 2697	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → 𝐴 ∈ V)
15	10, 14	syl 17	. . . . 5 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝐴 ∈ V)
16	15	exlimiv 1845	. . . 4 ⊢ (∃𝑓 𝑓:𝐴–1-1→𝐵 → 𝐴 ∈ V)
17	9, 16	pm5.21ni 366	. . 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 175	1 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173   class class class wbr 4583  dom cdm 5038  ⟶wf 5800  –1-1→wf1 5801   ≼ cdom 7839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-fn 5807  df-f 5808  df-f1 5809  df-dom 7843

This theorem is referenced by:  brdomi  7852  brdom  7853  f1dom2g  7859  f1domg  7861  dom3d  7883  domdifsn  7928  fidomtri  8702  hashdom  13029  hashge3el3dif  13122  sizeusglecusg  26014  erdsze2lem1  30439  sizusglecusg  40679