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Theorem fowdom 8636
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 3364 . 2 (𝐹𝑉𝐹 ∈ V)
2 foeq1 6253 . . . . . 6 (𝑧 = 𝐹 → (𝑧:𝑌onto𝑋𝐹:𝑌onto𝑋))
32spcegv 3445 . . . . 5 (𝐹 ∈ V → (𝐹:𝑌onto𝑋 → ∃𝑧 𝑧:𝑌onto𝑋))
43imp 393 . . . 4 ((𝐹 ∈ V ∧ 𝐹:𝑌onto𝑋) → ∃𝑧 𝑧:𝑌onto𝑋)
54olcd 863 . . 3 ((𝐹 ∈ V ∧ 𝐹:𝑌onto𝑋) → (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋))
6 fof 6257 . . . . 5 (𝐹:𝑌onto𝑋𝐹:𝑌𝑋)
7 dmfex 7275 . . . . 5 ((𝐹 ∈ V ∧ 𝐹:𝑌𝑋) → 𝑌 ∈ V)
86, 7sylan2 580 . . . 4 ((𝐹 ∈ V ∧ 𝐹:𝑌onto𝑋) → 𝑌 ∈ V)
9 brwdom 8632 . . . 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 569 1 ((𝐹𝑉𝐹:𝑌onto𝑋) → 𝑋* 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wo 836   = wceq 1631  wex 1852  wcel 2145  Vcvv 3351  c0 4063   class class class wbr 4787  wf 6026  ontowfo 6028  * cwdom 8622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-fun 6032  df-fn 6033  df-f 6034  df-fo 6036  df-wdom 8624
This theorem is referenced by:  wdomref  8637  wdomtr  8640  wdom2d  8645  wdomima2g  8651  harwdom  8655  ixpiunwdom  8656  isf32lem10  9390  fin1a2lem7  9434
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