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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 9558 | . . . 4 ⊢ Rel ≼* | |
2 | 1 | brrelex2i 5724 | . . 3 ⊢ (𝑋 ≼* 𝑌 → 𝑌 ∈ V) |
3 | 2 | a1i 11 | . 2 ⊢ (𝑋 ≠ ∅ → (𝑋 ≼* 𝑌 → 𝑌 ∈ V)) |
4 | fof 6796 | . . . . . 6 ⊢ (𝑧:𝑌–onto→𝑋 → 𝑧:𝑌⟶𝑋) | |
5 | 4 | fdmd 6719 | . . . . 5 ⊢ (𝑧:𝑌–onto→𝑋 → dom 𝑧 = 𝑌) |
6 | vex 3470 | . . . . . 6 ⊢ 𝑧 ∈ V | |
7 | 6 | dmex 7896 | . . . . 5 ⊢ dom 𝑧 ∈ V |
8 | 5, 7 | eqeltrrdi 2834 | . . . 4 ⊢ (𝑧:𝑌–onto→𝑋 → 𝑌 ∈ V) |
9 | 8 | exlimiv 1925 | . . 3 ⊢ (∃𝑧 𝑧:𝑌–onto→𝑋 → 𝑌 ∈ V) |
10 | 9 | a1i 11 | . 2 ⊢ (𝑋 ≠ ∅ → (∃𝑧 𝑧:𝑌–onto→𝑋 → 𝑌 ∈ V)) |
11 | brwdom 9559 | . . . 4 ⊢ (𝑌 ∈ V → (𝑋 ≼* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋))) | |
12 | df-ne 2933 | . . . . . 6 ⊢ (𝑋 ≠ ∅ ↔ ¬ 𝑋 = ∅) | |
13 | biorf 933 | . . . . . 6 ⊢ (¬ 𝑋 = ∅ → (∃𝑧 𝑧:𝑌–onto→𝑋 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋))) | |
14 | 12, 13 | sylbi 216 | . . . . 5 ⊢ (𝑋 ≠ ∅ → (∃𝑧 𝑧:𝑌–onto→𝑋 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋))) |
15 | 14 | bicomd 222 | . . . 4 ⊢ (𝑋 ≠ ∅ → ((𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋) ↔ ∃𝑧 𝑧:𝑌–onto→𝑋)) |
16 | 11, 15 | sylan9bbr 510 | . . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝑌 ∈ V) → (𝑋 ≼* 𝑌 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋)) |
17 | 16 | ex 412 | . 2 ⊢ (𝑋 ≠ ∅ → (𝑌 ∈ V → (𝑋 ≼* 𝑌 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋))) |
18 | 3, 10, 17 | pm5.21ndd 379 | 1 ⊢ (𝑋 ≠ ∅ → (𝑋 ≼* 𝑌 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∨ wo 844 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ≠ wne 2932 Vcvv 3466 ∅c0 4315 class class class wbr 5139 dom cdm 5667 –onto→wfo 6532 ≼* cwdom 9556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-xp 5673 df-rel 5674 df-cnv 5675 df-dm 5677 df-rn 5678 df-fn 6537 df-f 6538 df-fo 6540 df-wdom 9557 |
This theorem is referenced by: brwdom2 9565 wdomtr 9567 wdompwdom 9570 canthwdom 9571 wdomfil 10053 fin1a2lem7 10398 |
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