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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 9032 | . . . 4 ⊢ Rel ≼* | |
2 | 1 | brrelex2i 5611 | . . 3 ⊢ (𝑋 ≼* 𝑌 → 𝑌 ∈ V) |
3 | 2 | a1i 11 | . 2 ⊢ (𝑋 ≠ ∅ → (𝑋 ≼* 𝑌 → 𝑌 ∈ V)) |
4 | fof 6592 | . . . . . 6 ⊢ (𝑧:𝑌–onto→𝑋 → 𝑧:𝑌⟶𝑋) | |
5 | 4 | fdmd 6525 | . . . . 5 ⊢ (𝑧:𝑌–onto→𝑋 → dom 𝑧 = 𝑌) |
6 | vex 3499 | . . . . . 6 ⊢ 𝑧 ∈ V | |
7 | 6 | dmex 7618 | . . . . 5 ⊢ dom 𝑧 ∈ V |
8 | 5, 7 | eqeltrrdi 2924 | . . . 4 ⊢ (𝑧:𝑌–onto→𝑋 → 𝑌 ∈ V) |
9 | 8 | exlimiv 1931 | . . 3 ⊢ (∃𝑧 𝑧:𝑌–onto→𝑋 → 𝑌 ∈ V) |
10 | 9 | a1i 11 | . 2 ⊢ (𝑋 ≠ ∅ → (∃𝑧 𝑧:𝑌–onto→𝑋 → 𝑌 ∈ V)) |
11 | brwdom 9033 | . . . 4 ⊢ (𝑌 ∈ V → (𝑋 ≼* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋))) | |
12 | df-ne 3019 | . . . . . 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 1537 ∃wex 1780 ∈ wcel 2114 ≠ wne 3018 Vcvv 3496 ∅c0 4293 class class class wbr 5068 dom cdm 5557 –onto→wfo 6355 ≼* cwdom 9023 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-cnv 5565 df-dm 5567 df-rn 5568 df-fn 6360 df-f 6361 df-fo 6363 df-wdom 9025 |
This theorem is referenced by: brwdom2 9039 wdomtr 9041 wdompwdom 9044 canthwdom 9045 wdomfil 9489 fin1a2lem7 9830 |
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