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Theorem fowdom 9614
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 3482 . 2 (𝐹𝑉𝐹 ∈ V)
2 foeq1 6811 . . . . . 6 (𝑧 = 𝐹 → (𝑧:𝑌onto𝑋𝐹:𝑌onto𝑋))
32spcegv 3583 . . . . 5 (𝐹 ∈ V → (𝐹:𝑌onto𝑋 → ∃𝑧 𝑧:𝑌onto𝑋))
43imp 405 . . . 4 ((𝐹 ∈ V ∧ 𝐹:𝑌onto𝑋) → ∃𝑧 𝑧:𝑌onto𝑋)
54olcd 872 . . 3 ((𝐹 ∈ V ∧ 𝐹:𝑌onto𝑋) → (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋))
6 fof 6815 . . . . 5 (𝐹:𝑌onto𝑋𝐹:𝑌𝑋)
7 dmfex 7918 . . . . 5 ((𝐹 ∈ V ∧ 𝐹:𝑌𝑋) → 𝑌 ∈ V)
86, 7sylan2 591 . . . 4 ((𝐹 ∈ V ∧ 𝐹:𝑌onto𝑋) → 𝑌 ∈ V)
9 brwdom 9610 . . . 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 256 . 2 ((𝐹 ∈ V ∧ 𝐹:𝑌onto𝑋) → 𝑋* 𝑌)
121, 11sylan 578 1 ((𝐹𝑉𝐹:𝑌onto𝑋) → 𝑋* 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wo 845   = wceq 1534  wex 1774  wcel 2099  Vcvv 3462  c0 4325   class class class wbr 5153  wf 6550  ontowfo 6552  * cwdom 9607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-fun 6556  df-fn 6557  df-f 6558  df-fo 6560  df-wdom 9608
This theorem is referenced by:  wdomref  9615  wdomtr  9618  wdom2d  9623  wdomima2g  9629  ixpiunwdom  9633  harwdom  9634  isf32lem10  10405  fin1a2lem7  10449
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