Theorem brdomgOLD 8972

Description: Obsolete version of brdomg 8971 as of 29-Nov-2024. (Contributed by NM, 15-Jun-1998.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
brdomgOLD	⊢ (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 brdomgOLD

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

Step	Hyp	Ref	Expression
1		f1eq2 6770	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–1-1→𝑦 ↔ 𝑓:𝐴–1-1→𝑦))
2	1	exbidv 1921	. . . 4 ⊢ (𝑥 = 𝐴 → (∃𝑓 𝑓:𝑥–1-1→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1→𝑦))
3		f1eq3 6771	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑓:𝐴–1-1→𝑦 ↔ 𝑓:𝐴–1-1→𝐵))
4	3	exbidv 1921	. . . 4 ⊢ (𝑦 = 𝐵 → (∃𝑓 𝑓:𝐴–1-1→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
5		df-dom 8961	. . . 4 ⊢ ≼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦}
6	2, 4, 5	brabg 5514	. . 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 8965	. . . . 5 ⊢ Rel ≼
9	8	brrelex1i 5710	. . . 4 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ∈ V)
10		f1f 6774	. . . . . 6 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴⟶𝐵)
11		fdm 6715	. . . . . . 7 ⊢ (𝑓:𝐴⟶𝐵 → dom 𝑓 = 𝐴)
12		vex 3463	. . . . . . . 8 ⊢ 𝑓 ∈ V
13	12	dmex 7905	. . . . . . 7 ⊢ dom 𝑓 ∈ V
14	11, 13	eqeltrrdi 2843	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → 𝐴 ∈ V)
15	10, 14	syl 17	. . . . 5 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝐴 ∈ V)
16	15	exlimiv 1930	. . . 4 ⊢ (∃𝑓 𝑓:𝐴–1-1→𝐵 → 𝐴 ∈ V)
17	9, 16	pm5.21ni 377	. . 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 206   = wceq 1540  ∃wex 1779   ∈ wcel 2108  Vcvv 3459   class class class wbr 5119  dom cdm 5654  ⟶wf 6527  –1-1→wf1 6528   ≼ cdom 8957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-xp 5660  df-rel 5661  df-cnv 5662  df-dm 5664  df-rn 5665  df-fn 6534  df-f 6535  df-f1 6536  df-dom 8961

This theorem is referenced by: (None)