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Theorem brwdomn0 9609
Description: Weak dominance over nonempty sets. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
Assertion
Ref Expression
brwdomn0 (𝑋 ≠ ∅ → (𝑋* 𝑌 ↔ ∃𝑧 𝑧:𝑌onto𝑋))
Distinct variable groups:   𝑧,𝑋   𝑧,𝑌

Proof of Theorem brwdomn0
StepHypRef Expression
1 relwdom 9606 . . . 4 Rel ≼*
21brrelex2i 5742 . . 3 (𝑋* 𝑌𝑌 ∈ V)
32a1i 11 . 2 (𝑋 ≠ ∅ → (𝑋* 𝑌𝑌 ∈ V))
4 fof 6820 . . . . . 6 (𝑧:𝑌onto𝑋𝑧:𝑌𝑋)
54fdmd 6746 . . . . 5 (𝑧:𝑌onto𝑋 → dom 𝑧 = 𝑌)
6 vex 3484 . . . . . 6 𝑧 ∈ V
76dmex 7931 . . . . 5 dom 𝑧 ∈ V
85, 7eqeltrrdi 2850 . . . 4 (𝑧:𝑌onto𝑋𝑌 ∈ V)
98exlimiv 1930 . . 3 (∃𝑧 𝑧:𝑌onto𝑋𝑌 ∈ V)
109a1i 11 . 2 (𝑋 ≠ ∅ → (∃𝑧 𝑧:𝑌onto𝑋𝑌 ∈ V))
11 brwdom 9607 . . . 4 (𝑌 ∈ V → (𝑋* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋)))
12 df-ne 2941 . . . . . 6 (𝑋 ≠ ∅ ↔ ¬ 𝑋 = ∅)
13 biorf 937 . . . . . 6 𝑋 = ∅ → (∃𝑧 𝑧:𝑌onto𝑋 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋)))
1412, 13sylbi 217 . . . . 5 (𝑋 ≠ ∅ → (∃𝑧 𝑧:𝑌onto𝑋 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋)))
1514bicomd 223 . . . 4 (𝑋 ≠ ∅ → ((𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋) ↔ ∃𝑧 𝑧:𝑌onto𝑋))
1611, 15sylan9bbr 510 . . 3 ((𝑋 ≠ ∅ ∧ 𝑌 ∈ V) → (𝑋* 𝑌 ↔ ∃𝑧 𝑧:𝑌onto𝑋))
1716ex 412 . 2 (𝑋 ≠ ∅ → (𝑌 ∈ V → (𝑋* 𝑌 ↔ ∃𝑧 𝑧:𝑌onto𝑋)))
183, 10, 17pm5.21ndd 379 1 (𝑋 ≠ ∅ → (𝑋* 𝑌 ↔ ∃𝑧 𝑧:𝑌onto𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wo 848   = wceq 1540  wex 1779  wcel 2108  wne 2940  Vcvv 3480  c0 4333   class class class wbr 5143  dom cdm 5685  ontowfo 6559  * cwdom 9604
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-ne 2941  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-fo 6567  df-wdom 9605
This theorem is referenced by:  brwdom2  9613  wdomtr  9615  wdompwdom  9618  canthwdom  9619  wdomfil  10101  fin1a2lem7  10446
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