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Theorem fowdom 9609
Description: An onto function implies weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.)
Assertion
Ref Expression
fowdom ((𝐹𝑉𝐹:𝑌onto𝑋) → 𝑋* 𝑌)

Proof of Theorem fowdom
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 elex 3499 . 2 (𝐹𝑉𝐹 ∈ V)
2 foeq1 6817 . . . . . 6 (𝑧 = 𝐹 → (𝑧:𝑌onto𝑋𝐹:𝑌onto𝑋))
32spcegv 3597 . . . . 5 (𝐹 ∈ V → (𝐹:𝑌onto𝑋 → ∃𝑧 𝑧:𝑌onto𝑋))
43imp 406 . . . 4 ((𝐹 ∈ V ∧ 𝐹:𝑌onto𝑋) → ∃𝑧 𝑧:𝑌onto𝑋)
54olcd 874 . . 3 ((𝐹 ∈ V ∧ 𝐹:𝑌onto𝑋) → (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋))
6 fof 6821 . . . . 5 (𝐹:𝑌onto𝑋𝐹:𝑌𝑋)
7 dmfex 7928 . . . . 5 ((𝐹 ∈ V ∧ 𝐹:𝑌𝑋) → 𝑌 ∈ V)
86, 7sylan2 593 . . . 4 ((𝐹 ∈ V ∧ 𝐹:𝑌onto𝑋) → 𝑌 ∈ V)
9 brwdom 9605 . . . 4 (𝑌 ∈ V → (𝑋* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋)))
108, 9syl 17 . . 3 ((𝐹 ∈ V ∧ 𝐹:𝑌onto𝑋) → (𝑋* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋)))
115, 10mpbird 257 . 2 ((𝐹 ∈ V ∧ 𝐹:𝑌onto𝑋) → 𝑋* 𝑌)
121, 11sylan 580 1 ((𝐹𝑉𝐹:𝑌onto𝑋) → 𝑋* 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847   = wceq 1537  wex 1776  wcel 2106  Vcvv 3478  c0 4339   class class class wbr 5148  wf 6559  ontowfo 6561  * cwdom 9602
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-co 5698  df-dm 5699  df-rn 5700  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-wdom 9603
This theorem is referenced by:  wdomref  9610  wdomtr  9613  wdom2d  9618  wdomima2g  9624  ixpiunwdom  9628  harwdom  9629  isf32lem10  10400  fin1a2lem7  10444
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