Theorem brdomg 7036

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 7031	. . . 4 ⊢ Rel ≼
2	1	brrelex1i 4820	. . 3 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ∈ V)
3	2	a1i 9	. 2 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 → 𝐴 ∈ V))
4		f1f 5601	. . . . 5 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴⟶𝐵)
5		fdm 5542	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → dom 𝑓 = 𝐴)
6		vex 2824	. . . . . . 7 ⊢ 𝑓 ∈ V
7	6	dmex 5051	. . . . . 6 ⊢ dom 𝑓 ∈ V
8	5, 7	eqeltrrdi 2330	. . . . 5 ⊢ (𝑓:𝐴⟶𝐵 → 𝐴 ∈ V)
9	4, 8	syl 14	. . . 4 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝐴 ∈ V)
10	9	exlimiv 1651	. . 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 5597	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–1-1→𝑦 ↔ 𝑓:𝐴–1-1→𝑦))
13	12	exbidv 1878	. . . 4 ⊢ (𝑥 = 𝐴 → (∃𝑓 𝑓:𝑥–1-1→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1→𝑦))
14		f1eq3 5598	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑓:𝐴–1-1→𝑦 ↔ 𝑓:𝐴–1-1→𝐵))
15	14	exbidv 1878	. . . 4 ⊢ (𝑦 = 𝐵 → (∃𝑓 𝑓:𝐴–1-1→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
16		df-dom 7028	. . . 4 ⊢ ≼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦}
17	13, 15, 16	brabg 4413	. . 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 717	1 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 105   = wceq 1402  ∃wex 1545   ∈ wcel 2209  Vcvv 2821   class class class wbr 4131  dom cdm 4776  ⟶wf 5376  –1-1→wf1 5377   ≼ cdom 7025

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-opab 4194  df-xp 4782  df-rel 4783  df-cnv 4784  df-dm 4786  df-rn 4787  df-fn 5383  df-f 5384  df-f1 5385  df-dom 7028

This theorem is used by:  brdomi  7037  brdom  7038  f1dom2g  7046  f1domg  7048  dom3d  7064  dom1o  7120  phplem4dom  7167  djudom  7439  difinfsn  7446  djudoml  7581  djudomr  7582  nninfdc  13402  birthdaylem3  16227