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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 8883 | . . . 4 ⊢ Rel ≼* | |
2 | 1 | brrelex2i 5502 | . . 3 ⊢ (𝑋 ≼* 𝑌 → 𝑌 ∈ V) |
3 | 2 | a1i 11 | . 2 ⊢ (𝑋 ≠ ∅ → (𝑋 ≼* 𝑌 → 𝑌 ∈ V)) |
4 | fof 6465 | . . . . . 6 ⊢ (𝑧:𝑌–onto→𝑋 → 𝑧:𝑌⟶𝑋) | |
5 | 4 | fdmd 6398 | . . . . 5 ⊢ (𝑧:𝑌–onto→𝑋 → dom 𝑧 = 𝑌) |
6 | vex 3443 | . . . . . 6 ⊢ 𝑧 ∈ V | |
7 | 6 | dmex 7479 | . . . . 5 ⊢ dom 𝑧 ∈ V |
8 | 5, 7 | syl6eqelr 2894 | . . . 4 ⊢ (𝑧:𝑌–onto→𝑋 → 𝑌 ∈ V) |
9 | 8 | exlimiv 1912 | . . 3 ⊢ (∃𝑧 𝑧:𝑌–onto→𝑋 → 𝑌 ∈ V) |
10 | 9 | a1i 11 | . 2 ⊢ (𝑋 ≠ ∅ → (∃𝑧 𝑧:𝑌–onto→𝑋 → 𝑌 ∈ V)) |
11 | brwdom 8884 | . . . 4 ⊢ (𝑌 ∈ V → (𝑋 ≼* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋))) | |
12 | df-ne 2987 | . . . . . 6 ⊢ (𝑋 ≠ ∅ ↔ ¬ 𝑋 = ∅) | |
13 | biorf 931 | . . . . . 6 ⊢ (¬ 𝑋 = ∅ → (∃𝑧 𝑧:𝑌–onto→𝑋 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋))) | |
14 | 12, 13 | sylbi 218 | . . . . 5 ⊢ (𝑋 ≠ ∅ → (∃𝑧 𝑧:𝑌–onto→𝑋 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋))) |
15 | 14 | bicomd 224 | . . . 4 ⊢ (𝑋 ≠ ∅ → ((𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋) ↔ ∃𝑧 𝑧:𝑌–onto→𝑋)) |
16 | 11, 15 | sylan9bbr 511 | . . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝑌 ∈ V) → (𝑋 ≼* 𝑌 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋)) |
17 | 16 | ex 413 | . 2 ⊢ (𝑋 ≠ ∅ → (𝑌 ∈ V → (𝑋 ≼* 𝑌 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋))) |
18 | 3, 10, 17 | pm5.21ndd 381 | 1 ⊢ (𝑋 ≠ ∅ → (𝑋 ≼* 𝑌 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∨ wo 842 = wceq 1525 ∃wex 1765 ∈ wcel 2083 ≠ wne 2986 Vcvv 3440 ∅c0 4217 class class class wbr 4968 dom cdm 5450 –onto→wfo 6230 ≼* cwdom 8874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pr 5228 ax-un 7326 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-ral 3112 df-rex 3113 df-rab 3116 df-v 3442 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-br 4969 df-opab 5031 df-xp 5456 df-rel 5457 df-cnv 5458 df-dm 5460 df-rn 5461 df-fn 6235 df-f 6236 df-fo 6238 df-wdom 8876 |
This theorem is referenced by: brwdom2 8890 wdomtr 8892 wdompwdom 8895 canthwdom 8896 wdomfil 9340 fin1a2lem7 9681 |
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