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Mirrors > Home > MPE Home > Th. List > brwdomn0 | Structured version Visualization version GIF version |
Description: Weak dominance over nonempty sets. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.) |
Ref | Expression |
---|---|
brwdomn0 | ⊢ (𝑋 ≠ ∅ → (𝑋 ≼* 𝑌 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relwdom 9024 | . . . 4 ⊢ Rel ≼* | |
2 | 1 | brrelex2i 5604 | . . 3 ⊢ (𝑋 ≼* 𝑌 → 𝑌 ∈ V) |
3 | 2 | a1i 11 | . 2 ⊢ (𝑋 ≠ ∅ → (𝑋 ≼* 𝑌 → 𝑌 ∈ V)) |
4 | fof 6585 | . . . . . 6 ⊢ (𝑧:𝑌–onto→𝑋 → 𝑧:𝑌⟶𝑋) | |
5 | 4 | fdmd 6518 | . . . . 5 ⊢ (𝑧:𝑌–onto→𝑋 → dom 𝑧 = 𝑌) |
6 | vex 3498 | . . . . . 6 ⊢ 𝑧 ∈ V | |
7 | 6 | dmex 7610 | . . . . 5 ⊢ dom 𝑧 ∈ V |
8 | 5, 7 | eqeltrrdi 2922 | . . . 4 ⊢ (𝑧:𝑌–onto→𝑋 → 𝑌 ∈ V) |
9 | 8 | exlimiv 1927 | . . 3 ⊢ (∃𝑧 𝑧:𝑌–onto→𝑋 → 𝑌 ∈ V) |
10 | 9 | a1i 11 | . 2 ⊢ (𝑋 ≠ ∅ → (∃𝑧 𝑧:𝑌–onto→𝑋 → 𝑌 ∈ V)) |
11 | brwdom 9025 | . . . 4 ⊢ (𝑌 ∈ V → (𝑋 ≼* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋))) | |
12 | df-ne 3017 | . . . . . 6 ⊢ (𝑋 ≠ ∅ ↔ ¬ 𝑋 = ∅) | |
13 | biorf 933 | . . . . . 6 ⊢ (¬ 𝑋 = ∅ → (∃𝑧 𝑧:𝑌–onto→𝑋 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋))) | |
14 | 12, 13 | sylbi 219 | . . . . 5 ⊢ (𝑋 ≠ ∅ → (∃𝑧 𝑧:𝑌–onto→𝑋 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋))) |
15 | 14 | bicomd 225 | . . . 4 ⊢ (𝑋 ≠ ∅ → ((𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋) ↔ ∃𝑧 𝑧:𝑌–onto→𝑋)) |
16 | 11, 15 | sylan9bbr 513 | . . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝑌 ∈ V) → (𝑋 ≼* 𝑌 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋)) |
17 | 16 | ex 415 | . 2 ⊢ (𝑋 ≠ ∅ → (𝑌 ∈ V → (𝑋 ≼* 𝑌 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋))) |
18 | 3, 10, 17 | pm5.21ndd 383 | 1 ⊢ (𝑋 ≠ ∅ → (𝑋 ≼* 𝑌 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∨ wo 843 = wceq 1533 ∃wex 1776 ∈ wcel 2110 ≠ wne 3016 Vcvv 3495 ∅c0 4291 class class class wbr 5059 dom cdm 5550 –onto→wfo 6348 ≼* cwdom 9015 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-xp 5556 df-rel 5557 df-cnv 5558 df-dm 5560 df-rn 5561 df-fn 6353 df-f 6354 df-fo 6356 df-wdom 9017 |
This theorem is referenced by: brwdom2 9031 wdomtr 9033 wdompwdom 9036 canthwdom 9037 wdomfil 9481 fin1a2lem7 9822 |
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