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Mirrors > Home > MPE Home > Th. List > brwdomi | Structured version Visualization version GIF version |
Description: Property of weak dominance, forward direction only. (Contributed by Mario Carneiro, 5-May-2015.) |
Ref | Expression |
---|---|
brwdomi | ⊢ (𝑋 ≼* 𝑌 → (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relwdom 8827 | . . . 4 ⊢ Rel ≼* | |
2 | 1 | brrelex2i 5460 | . . 3 ⊢ (𝑋 ≼* 𝑌 → 𝑌 ∈ V) |
3 | brwdom 8828 | . . 3 ⊢ (𝑌 ∈ V → (𝑋 ≼* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋))) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝑋 ≼* 𝑌 → (𝑋 ≼* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋))) |
5 | 4 | ibi 259 | 1 ⊢ (𝑋 ≼* 𝑌 → (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∨ wo 833 = wceq 1507 ∃wex 1742 ∈ wcel 2050 Vcvv 3415 ∅c0 4180 class class class wbr 4930 –onto→wfo 6188 ≼* cwdom 8818 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5061 ax-nul 5068 ax-pr 5187 ax-un 7281 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ral 3093 df-rex 3094 df-rab 3097 df-v 3417 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-nul 4181 df-if 4352 df-sn 4443 df-pr 4445 df-op 4449 df-uni 4714 df-br 4931 df-opab 4993 df-xp 5414 df-rel 5415 df-cnv 5416 df-dm 5418 df-rn 5419 df-fn 6193 df-fo 6196 df-wdom 8820 |
This theorem is referenced by: numwdom 9281 |
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