MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  brdomgOLD Structured version   Visualization version   GIF version

Theorem brdomgOLD 8998
Description: Obsolete version of brdomg 8997 as of 29-Nov-2024. (Contributed by NM, 15-Jun-1998.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
brdomgOLD (𝐵𝐶 → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
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.
StepHypRef Expression
1 f1eq2 6800 . . . . 5 (𝑥 = 𝐴 → (𝑓:𝑥1-1𝑦𝑓:𝐴1-1𝑦))
21exbidv 1921 . . . 4 (𝑥 = 𝐴 → (∃𝑓 𝑓:𝑥1-1𝑦 ↔ ∃𝑓 𝑓:𝐴1-1𝑦))
3 f1eq3 6801 . . . . 5 (𝑦 = 𝐵 → (𝑓:𝐴1-1𝑦𝑓:𝐴1-1𝐵))
43exbidv 1921 . . . 4 (𝑦 = 𝐵 → (∃𝑓 𝑓:𝐴1-1𝑦 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
5 df-dom 8987 . . . 4 ≼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1𝑦}
62, 4, 5brabg 5544 . . 3 ((𝐴 ∈ V ∧ 𝐵𝐶) → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
76ex 412 . 2 (𝐴 ∈ V → (𝐵𝐶 → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵)))
8 reldom 8991 . . . . 5 Rel ≼
98brrelex1i 5741 . . . 4 (𝐴𝐵𝐴 ∈ V)
10 f1f 6804 . . . . . 6 (𝑓:𝐴1-1𝐵𝑓:𝐴𝐵)
11 fdm 6745 . . . . . . 7 (𝑓:𝐴𝐵 → dom 𝑓 = 𝐴)
12 vex 3484 . . . . . . . 8 𝑓 ∈ V
1312dmex 7931 . . . . . . 7 dom 𝑓 ∈ V
1411, 13eqeltrrdi 2850 . . . . . 6 (𝑓:𝐴𝐵𝐴 ∈ V)
1510, 14syl 17 . . . . 5 (𝑓:𝐴1-1𝐵𝐴 ∈ V)
1615exlimiv 1930 . . . 4 (∃𝑓 𝑓:𝐴1-1𝐵𝐴 ∈ V)
179, 16pm5.21ni 377 . . 3 𝐴 ∈ V → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
1817a1d 25 . 2 𝐴 ∈ V → (𝐵𝐶 → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵)))
197, 18pm2.61i 182 1 (𝐵𝐶 → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206   = wceq 1540  wex 1779  wcel 2108  Vcvv 3480   class class class wbr 5143  dom cdm 5685  wf 6557  1-1wf1 6558  cdom 8983
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 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-fn 6564  df-f 6565  df-f1 6566  df-dom 8987
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator