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Mirrors > Home > MPE Home > Th. List > brdomg | Structured version Visualization version GIF version |
Description: Dominance relation. (Contributed by NM, 15-Jun-1998.) Extract brdom2g 8995 as an intermediate result. (Revised by BTernaryTau, 29-Nov-2024.) |
Ref | Expression |
---|---|
brdomg | ⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brdom2g 8995 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝐶) → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) | |
2 | 1 | ex 412 | . 2 ⊢ (𝐴 ∈ V → (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))) |
3 | reldom 8990 | . . . . 5 ⊢ Rel ≼ | |
4 | 3 | brrelex1i 5745 | . . . 4 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ∈ V) |
5 | f1f 6805 | . . . . . 6 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴⟶𝐵) | |
6 | fdm 6746 | . . . . . . 7 ⊢ (𝑓:𝐴⟶𝐵 → dom 𝑓 = 𝐴) | |
7 | vex 3482 | . . . . . . . 8 ⊢ 𝑓 ∈ V | |
8 | 7 | dmex 7932 | . . . . . . 7 ⊢ dom 𝑓 ∈ V |
9 | 6, 8 | eqeltrrdi 2848 | . . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → 𝐴 ∈ V) |
10 | 5, 9 | syl 17 | . . . . 5 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝐴 ∈ V) |
11 | 10 | exlimiv 1928 | . . . 4 ⊢ (∃𝑓 𝑓:𝐴–1-1→𝐵 → 𝐴 ∈ V) |
12 | 4, 11 | pm5.21ni 377 | . . 3 ⊢ (¬ 𝐴 ∈ V → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) |
13 | 12 | a1d 25 | . 2 ⊢ (¬ 𝐴 ∈ V → (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))) |
14 | 2, 13 | pm2.61i 182 | 1 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∃wex 1776 ∈ wcel 2106 Vcvv 3478 class class class wbr 5148 dom cdm 5689 ⟶wf 6559 –1-1→wf1 6560 ≼ cdom 8982 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-dm 5699 df-rn 5700 df-fn 6566 df-f 6567 df-f1 6568 df-dom 8986 |
This theorem is referenced by: brdomiOLD 8999 brdom 9000 f1dom3g 9007 f1domg 9011 dom3d 9033 domdifsn 9093 fidomtri 10031 hashdom 14415 hashge3el3dif 14523 sizusglecusg 29496 erdsze2lem1 35188 hashnexinj 42110 |
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