Theorem brdomg 6914

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		reldom 6909	. . . 4 ⊢ Rel ≼
2	1	brrelex1i 4767	. . 3 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ∈ V)
3	2	a1i 9	. 2 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 → 𝐴 ∈ V))
4		f1f 5539	. . . . 5 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴⟶𝐵)
5		fdm 5485	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → dom 𝑓 = 𝐴)
6		vex 2803	. . . . . . 7 ⊢ 𝑓 ∈ V
7	6	dmex 4997	. . . . . 6 ⊢ dom 𝑓 ∈ V
8	5, 7	eqeltrrdi 2321	. . . . 5 ⊢ (𝑓:𝐴⟶𝐵 → 𝐴 ∈ V)
9	4, 8	syl 14	. . . 4 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝐴 ∈ V)
10	9	exlimiv 1644	. . 3 ⊢ (∃𝑓 𝑓:𝐴–1-1→𝐵 → 𝐴 ∈ V)
11	10	a1i 9	. 2 ⊢ (𝐵 ∈ 𝐶 → (∃𝑓 𝑓:𝐴–1-1→𝐵 → 𝐴 ∈ V))
12		f1eq2 5535	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–1-1→𝑦 ↔ 𝑓:𝐴–1-1→𝑦))
13	12	exbidv 1871	. . . 4 ⊢ (𝑥 = 𝐴 → (∃𝑓 𝑓:𝑥–1-1→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1→𝑦))
14		f1eq3 5536	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑓:𝐴–1-1→𝑦 ↔ 𝑓:𝐴–1-1→𝐵))
15	14	exbidv 1871	. . . 4 ⊢ (𝑦 = 𝐵 → (∃𝑓 𝑓:𝐴–1-1→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
16		df-dom 6906	. . . 4 ⊢ ≼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦}
17	13, 15, 16	brabg 4361	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝐶) → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
18	17	expcom 116	. 2 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ∈ V → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)))
19	3, 11, 18	pm5.21ndd 710	1 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1395  ∃wex 1538   ∈ wcel 2200  Vcvv 2800   class class class wbr 4086  dom cdm 4723  ⟶wf 5320  –1-1→wf1 5321   ≼ cdom 6903

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-xp 4729  df-rel 4730  df-cnv 4731  df-dm 4733  df-rn 4734  df-fn 5327  df-f 5328  df-f1 5329  df-dom 6906

This theorem is referenced by:  brdomi  6915  brdom  6916  f1dom2g  6924  f1domg  6926  dom3d  6942  dom1o  6997  phplem4dom  7043  djudom  7283  difinfsn  7290  djudoml  7424  djudomr  7425  nninfdc  13064