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Theorem fowdom 8718
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 3400 . 2 (𝐹𝑉𝐹 ∈ V)
2 foeq1 6327 . . . . . 6 (𝑧 = 𝐹 → (𝑧:𝑌onto𝑋𝐹:𝑌onto𝑋))
32spcegv 3482 . . . . 5 (𝐹 ∈ V → (𝐹:𝑌onto𝑋 → ∃𝑧 𝑧:𝑌onto𝑋))
43imp 396 . . . 4 ((𝐹 ∈ V ∧ 𝐹:𝑌onto𝑋) → ∃𝑧 𝑧:𝑌onto𝑋)
54olcd 901 . . 3 ((𝐹 ∈ V ∧ 𝐹:𝑌onto𝑋) → (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋))
6 fof 6331 . . . . 5 (𝐹:𝑌onto𝑋𝐹:𝑌𝑋)
7 dmfex 7359 . . . . 5 ((𝐹 ∈ V ∧ 𝐹:𝑌𝑋) → 𝑌 ∈ V)
86, 7sylan2 587 . . . 4 ((𝐹 ∈ V ∧ 𝐹:𝑌onto𝑋) → 𝑌 ∈ V)
9 brwdom 8714 . . . 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 249 . 2 ((𝐹 ∈ V ∧ 𝐹:𝑌onto𝑋) → 𝑋* 𝑌)
121, 11sylan 576 1 ((𝐹𝑉𝐹:𝑌onto𝑋) → 𝑋* 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  wo 874   = wceq 1653  wex 1875  wcel 2157  Vcvv 3385  c0 4115   class class class wbr 4843  wf 6097  ontowfo 6099  * cwdom 8704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-fun 6103  df-fn 6104  df-f 6105  df-fo 6107  df-wdom 8706
This theorem is referenced by:  wdomref  8719  wdomtr  8722  wdom2d  8727  wdomima2g  8733  harwdom  8737  ixpiunwdom  8738  isf32lem10  9472  fin1a2lem7  9516
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