Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 9182 | . . . 4 ⊢ Rel ≼* | |
2 | 1 | brrelex2i 5606 | . . 3 ⊢ (𝑋 ≼* 𝑌 → 𝑌 ∈ V) |
3 | brwdom 9183 | . . 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 270 | 1 ⊢ (𝑋 ≼* 𝑌 → (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∨ wo 847 = wceq 1543 ∃wex 1787 ∈ wcel 2110 Vcvv 3408 ∅c0 4237 class class class wbr 5053 –onto→wfo 6378 ≼* cwdom 9180 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-xp 5557 df-rel 5558 df-cnv 5559 df-dm 5561 df-rn 5562 df-fn 6383 df-fo 6386 df-wdom 9181 |
This theorem is referenced by: numwdom 9673 |
Copyright terms: Public domain | W3C validator |