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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.) (Proof shortened by BTernaryTau, 29-Nov-2024.) |
Ref | Expression |
---|---|
brdomg | ⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brdom2g 8745 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝐶) → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) | |
2 | 1 | ex 413 | . 2 ⊢ (𝐴 ∈ V → (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))) |
3 | reldom 8739 | . . . . 5 ⊢ Rel ≼ | |
4 | 3 | brrelex1i 5643 | . . . 4 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ∈ V) |
5 | f1f 6670 | . . . . . 6 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴⟶𝐵) | |
6 | fdm 6609 | . . . . . . 7 ⊢ (𝑓:𝐴⟶𝐵 → dom 𝑓 = 𝐴) | |
7 | vex 3436 | . . . . . . . 8 ⊢ 𝑓 ∈ V | |
8 | 7 | dmex 7758 | . . . . . . 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 1933 | . . . 4 ⊢ (∃𝑓 𝑓:𝐴–1-1→𝐵 → 𝐴 ∈ V) |
12 | 4, 11 | pm5.21ni 379 | . . 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 205 ∃wex 1782 ∈ wcel 2106 Vcvv 3432 class class class wbr 5074 dom cdm 5589 ⟶wf 6429 –1-1→wf1 6430 ≼ cdom 8731 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-xp 5595 df-rel 5596 df-cnv 5597 df-dm 5599 df-rn 5600 df-fn 6436 df-f 6437 df-f1 6438 df-dom 8735 |
This theorem is referenced by: brdomiOLD 8749 brdom 8750 f1dom3g 8755 f1dom2gOLD 8758 f1domg 8760 dom3d 8782 domdifsn 8841 fidomtri 9751 hashdom 14094 hashge3el3dif 14200 sizusglecusg 27830 erdsze2lem1 33165 |
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