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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 9606 | . . . 4 ⊢ Rel ≼* | |
| 2 | 1 | brrelex2i 5742 | . . 3 ⊢ (𝑋 ≼* 𝑌 → 𝑌 ∈ V) |
| 3 | 2 | a1i 11 | . 2 ⊢ (𝑋 ≠ ∅ → (𝑋 ≼* 𝑌 → 𝑌 ∈ V)) |
| 4 | fof 6820 | . . . . . 6 ⊢ (𝑧:𝑌–onto→𝑋 → 𝑧:𝑌⟶𝑋) | |
| 5 | 4 | fdmd 6746 | . . . . 5 ⊢ (𝑧:𝑌–onto→𝑋 → dom 𝑧 = 𝑌) |
| 6 | vex 3484 | . . . . . 6 ⊢ 𝑧 ∈ V | |
| 7 | 6 | dmex 7931 | . . . . 5 ⊢ dom 𝑧 ∈ V |
| 8 | 5, 7 | eqeltrrdi 2850 | . . . 4 ⊢ (𝑧:𝑌–onto→𝑋 → 𝑌 ∈ V) |
| 9 | 8 | exlimiv 1930 | . . 3 ⊢ (∃𝑧 𝑧:𝑌–onto→𝑋 → 𝑌 ∈ V) |
| 10 | 9 | a1i 11 | . 2 ⊢ (𝑋 ≠ ∅ → (∃𝑧 𝑧:𝑌–onto→𝑋 → 𝑌 ∈ V)) |
| 11 | brwdom 9607 | . . . 4 ⊢ (𝑌 ∈ V → (𝑋 ≼* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋))) | |
| 12 | df-ne 2941 | . . . . . 6 ⊢ (𝑋 ≠ ∅ ↔ ¬ 𝑋 = ∅) | |
| 13 | biorf 937 | . . . . . 6 ⊢ (¬ 𝑋 = ∅ → (∃𝑧 𝑧:𝑌–onto→𝑋 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋))) | |
| 14 | 12, 13 | sylbi 217 | . . . . 5 ⊢ (𝑋 ≠ ∅ → (∃𝑧 𝑧:𝑌–onto→𝑋 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋))) |
| 15 | 14 | bicomd 223 | . . . 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 206 ∨ wo 848 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ∅c0 4333 class class class wbr 5143 dom cdm 5685 –onto→wfo 6559 ≼* cwdom 9604 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-fn 6564 df-f 6565 df-fo 6567 df-wdom 9605 |
| This theorem is referenced by: brwdom2 9613 wdomtr 9615 wdompwdom 9618 canthwdom 9619 wdomfil 10101 fin1a2lem7 10446 |
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