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Theorem brwdomn0 9513
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 9510 . . . 4 Rel ≼*
21brrelex2i 5693 . . 3 (𝑋* 𝑌𝑌 ∈ V)
32a1i 11 . 2 (𝑋 ≠ ∅ → (𝑋* 𝑌𝑌 ∈ V))
4 fof 6760 . . . . . 6 (𝑧:𝑌onto𝑋𝑧:𝑌𝑋)
54fdmd 6683 . . . . 5 (𝑧:𝑌onto𝑋 → dom 𝑧 = 𝑌)
6 vex 3451 . . . . . 6 𝑧 ∈ V
76dmex 7852 . . . . 5 dom 𝑧 ∈ V
85, 7eqeltrrdi 2843 . . . 4 (𝑧:𝑌onto𝑋𝑌 ∈ V)
98exlimiv 1934 . . 3 (∃𝑧 𝑧:𝑌onto𝑋𝑌 ∈ V)
109a1i 11 . 2 (𝑋 ≠ ∅ → (∃𝑧 𝑧:𝑌onto𝑋𝑌 ∈ V))
11 brwdom 9511 . . . 4 (𝑌 ∈ V → (𝑋* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋)))
12 df-ne 2941 . . . . . 6 (𝑋 ≠ ∅ ↔ ¬ 𝑋 = ∅)
13 biorf 936 . . . . . 6 𝑋 = ∅ → (∃𝑧 𝑧:𝑌onto𝑋 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋)))
1412, 13sylbi 216 . . . . 5 (𝑋 ≠ ∅ → (∃𝑧 𝑧:𝑌onto𝑋 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋)))
1514bicomd 222 . . . 4 (𝑋 ≠ ∅ → ((𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋) ↔ ∃𝑧 𝑧:𝑌onto𝑋))
1611, 15sylan9bbr 512 . . 3 ((𝑋 ≠ ∅ ∧ 𝑌 ∈ V) → (𝑋* 𝑌 ↔ ∃𝑧 𝑧:𝑌onto𝑋))
1716ex 414 . 2 (𝑋 ≠ ∅ → (𝑌 ∈ V → (𝑋* 𝑌 ↔ ∃𝑧 𝑧:𝑌onto𝑋)))
183, 10, 17pm5.21ndd 381 1 (𝑋 ≠ ∅ → (𝑋* 𝑌 ↔ ∃𝑧 𝑧:𝑌onto𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wo 846   = wceq 1542  wex 1782  wcel 2107  wne 2940  Vcvv 3447  c0 4286   class class class wbr 5109  dom cdm 5637  ontowfo 6498  * cwdom 9508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-xp 5643  df-rel 5644  df-cnv 5645  df-dm 5647  df-rn 5648  df-fn 6503  df-f 6504  df-fo 6506  df-wdom 9509
This theorem is referenced by:  brwdom2  9517  wdomtr  9519  wdompwdom  9522  canthwdom  9523  wdomfil  10005  fin1a2lem7  10350
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