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Theorem brwdomn0 9185
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 9182 . . . 4 Rel ≼*
21brrelex2i 5606 . . 3 (𝑋* 𝑌𝑌 ∈ V)
32a1i 11 . 2 (𝑋 ≠ ∅ → (𝑋* 𝑌𝑌 ∈ V))
4 fof 6633 . . . . . 6 (𝑧:𝑌onto𝑋𝑧:𝑌𝑋)
54fdmd 6556 . . . . 5 (𝑧:𝑌onto𝑋 → dom 𝑧 = 𝑌)
6 vex 3412 . . . . . 6 𝑧 ∈ V
76dmex 7689 . . . . 5 dom 𝑧 ∈ V
85, 7eqeltrrdi 2847 . . . 4 (𝑧:𝑌onto𝑋𝑌 ∈ V)
98exlimiv 1938 . . 3 (∃𝑧 𝑧:𝑌onto𝑋𝑌 ∈ V)
109a1i 11 . 2 (𝑋 ≠ ∅ → (∃𝑧 𝑧:𝑌onto𝑋𝑌 ∈ V))
11 brwdom 9183 . . . 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 220 . . . . 5 (𝑋 ≠ ∅ → (∃𝑧 𝑧:𝑌onto𝑋 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋)))
1514bicomd 226 . . . 4 (𝑋 ≠ ∅ → ((𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋) ↔ ∃𝑧 𝑧:𝑌onto𝑋))
1611, 15sylan9bbr 514 . . 3 ((𝑋 ≠ ∅ ∧ 𝑌 ∈ V) → (𝑋* 𝑌 ↔ ∃𝑧 𝑧:𝑌onto𝑋))
1716ex 416 . 2 (𝑋 ≠ ∅ → (𝑌 ∈ V → (𝑋* 𝑌 ↔ ∃𝑧 𝑧:𝑌onto𝑋)))
183, 10, 17pm5.21ndd 384 1 (𝑋 ≠ ∅ → (𝑋* 𝑌 ↔ ∃𝑧 𝑧:𝑌onto𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wo 847   = wceq 1543  wex 1787  wcel 2110  wne 2940  Vcvv 3408  c0 4237   class class class wbr 5053  dom cdm 5551  ontowfo 6378  * cwdom 9180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-xp 5557  df-rel 5558  df-cnv 5559  df-dm 5561  df-rn 5562  df-fn 6383  df-f 6384  df-fo 6386  df-wdom 9181
This theorem is referenced by:  brwdom2  9189  wdomtr  9191  wdompwdom  9194  canthwdom  9195  wdomfil  9675  fin1a2lem7  10020
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