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Theorem brdomgOLD 8997
Description: Obsolete version of brdomg 8996 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 6801 . . . . 5 (𝑥 = 𝐴 → (𝑓:𝑥1-1𝑦𝑓:𝐴1-1𝑦))
21exbidv 1919 . . . 4 (𝑥 = 𝐴 → (∃𝑓 𝑓:𝑥1-1𝑦 ↔ ∃𝑓 𝑓:𝐴1-1𝑦))
3 f1eq3 6802 . . . . 5 (𝑦 = 𝐵 → (𝑓:𝐴1-1𝑦𝑓:𝐴1-1𝐵))
43exbidv 1919 . . . 4 (𝑦 = 𝐵 → (∃𝑓 𝑓:𝐴1-1𝑦 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
5 df-dom 8986 . . . 4 ≼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1𝑦}
62, 4, 5brabg 5549 . . 3 ((𝐴 ∈ V ∧ 𝐵𝐶) → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
76ex 412 . 2 (𝐴 ∈ V → (𝐵𝐶 → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵)))
8 reldom 8990 . . . . 5 Rel ≼
98brrelex1i 5745 . . . 4 (𝐴𝐵𝐴 ∈ V)
10 f1f 6805 . . . . . 6 (𝑓:𝐴1-1𝐵𝑓:𝐴𝐵)
11 fdm 6746 . . . . . . 7 (𝑓:𝐴𝐵 → dom 𝑓 = 𝐴)
12 vex 3482 . . . . . . . 8 𝑓 ∈ V
1312dmex 7932 . . . . . . 7 dom 𝑓 ∈ V
1411, 13eqeltrrdi 2848 . . . . . 6 (𝑓:𝐴𝐵𝐴 ∈ V)
1510, 14syl 17 . . . . 5 (𝑓:𝐴1-1𝐵𝐴 ∈ V)
1615exlimiv 1928 . . . 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 1537  wex 1776  wcel 2106  Vcvv 3478   class class class wbr 5148  dom cdm 5689  wf 6559  1-1wf1 6560  cdom 8982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-fn 6566  df-f 6567  df-f1 6568  df-dom 8986
This theorem is referenced by: (None)
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