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Theorem fowdom 8436
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 3202 . 2 (𝐹𝑉𝐹 ∈ V)
2 foeq1 6078 . . . . . 6 (𝑧 = 𝐹 → (𝑧:𝑌onto𝑋𝐹:𝑌onto𝑋))
32spcegv 3284 . . . . 5 (𝐹 ∈ V → (𝐹:𝑌onto𝑋 → ∃𝑧 𝑧:𝑌onto𝑋))
43imp 445 . . . 4 ((𝐹 ∈ V ∧ 𝐹:𝑌onto𝑋) → ∃𝑧 𝑧:𝑌onto𝑋)
54olcd 408 . . 3 ((𝐹 ∈ V ∧ 𝐹:𝑌onto𝑋) → (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋))
6 fof 6082 . . . . 5 (𝐹:𝑌onto𝑋𝐹:𝑌𝑋)
7 dmfex 7086 . . . . 5 ((𝐹 ∈ V ∧ 𝐹:𝑌𝑋) → 𝑌 ∈ V)
86, 7sylan2 491 . . . 4 ((𝐹 ∈ V ∧ 𝐹:𝑌onto𝑋) → 𝑌 ∈ V)
9 brwdom 8432 . . . 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 247 . 2 ((𝐹 ∈ V ∧ 𝐹:𝑌onto𝑋) → 𝑋* 𝑌)
121, 11sylan 488 1 ((𝐹𝑉𝐹:𝑌onto𝑋) → 𝑋* 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wex 1701  wcel 1987  Vcvv 3190  c0 3897   class class class wbr 4623  wf 5853  ontowfo 5855  * cwdom 8422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-fun 5859  df-fn 5860  df-f 5861  df-fo 5863  df-wdom 8424
This theorem is referenced by:  wdomref  8437  wdomtr  8440  wdom2d  8445  wdomima2g  8451  harwdom  8455  ixpiunwdom  8456  isf32lem10  9144  fin1a2lem7  9188
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