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Mirrors > Home > MPE Home > Th. List > fowdom | Structured version Visualization version GIF version |
Description: An onto function implies weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
Ref | Expression |
---|---|
fowdom | ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝑌–onto→𝑋) → 𝑋 ≼* 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3459 | . 2 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V) | |
2 | foeq1 6561 | . . . . . 6 ⊢ (𝑧 = 𝐹 → (𝑧:𝑌–onto→𝑋 ↔ 𝐹:𝑌–onto→𝑋)) | |
3 | 2 | spcegv 3545 | . . . . 5 ⊢ (𝐹 ∈ V → (𝐹:𝑌–onto→𝑋 → ∃𝑧 𝑧:𝑌–onto→𝑋)) |
4 | 3 | imp 410 | . . . 4 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝑌–onto→𝑋) → ∃𝑧 𝑧:𝑌–onto→𝑋) |
5 | 4 | olcd 871 | . . 3 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝑌–onto→𝑋) → (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋)) |
6 | fof 6565 | . . . . 5 ⊢ (𝐹:𝑌–onto→𝑋 → 𝐹:𝑌⟶𝑋) | |
7 | dmfex 7623 | . . . . 5 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝑌⟶𝑋) → 𝑌 ∈ V) | |
8 | 6, 7 | sylan2 595 | . . . 4 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝑌–onto→𝑋) → 𝑌 ∈ V) |
9 | brwdom 9015 | . . . 4 ⊢ (𝑌 ∈ V → (𝑋 ≼* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋))) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝑌–onto→𝑋) → (𝑋 ≼* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋))) |
11 | 5, 10 | mpbird 260 | . 2 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝑌–onto→𝑋) → 𝑋 ≼* 𝑌) |
12 | 1, 11 | sylan 583 | 1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝑌–onto→𝑋) → 𝑋 ≼* 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 = wceq 1538 ∃wex 1781 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 class class class wbr 5030 ⟶wf 6320 –onto→wfo 6322 ≼* cwdom 9012 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-fun 6326 df-fn 6327 df-f 6328 df-fo 6330 df-wdom 9013 |
This theorem is referenced by: wdomref 9020 wdomtr 9023 wdom2d 9028 wdomima2g 9034 ixpiunwdom 9038 harwdom 9039 isf32lem10 9773 fin1a2lem7 9817 |
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