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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 8953 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 8953 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝐶) → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) | |
2 | 1 | ex 411 | . 2 ⊢ (𝐴 ∈ V → (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))) |
3 | reldom 8947 | . . . . 5 ⊢ Rel ≼ | |
4 | 3 | brrelex1i 5731 | . . . 4 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ∈ V) |
5 | f1f 6786 | . . . . . 6 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴⟶𝐵) | |
6 | fdm 6725 | . . . . . . 7 ⊢ (𝑓:𝐴⟶𝐵 → dom 𝑓 = 𝐴) | |
7 | vex 3476 | . . . . . . . 8 ⊢ 𝑓 ∈ V | |
8 | 7 | dmex 7904 | . . . . . . 7 ⊢ dom 𝑓 ∈ V |
9 | 6, 8 | eqeltrrdi 2840 | . . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → 𝐴 ∈ V) |
10 | 5, 9 | syl 17 | . . . . 5 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝐴 ∈ V) |
11 | 10 | exlimiv 1931 | . . . 4 ⊢ (∃𝑓 𝑓:𝐴–1-1→𝐵 → 𝐴 ∈ V) |
12 | 4, 11 | pm5.21ni 376 | . . 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 1779 ∈ wcel 2104 Vcvv 3472 class class class wbr 5147 dom cdm 5675 ⟶wf 6538 –1-1→wf1 6539 ≼ cdom 8939 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-xp 5681 df-rel 5682 df-cnv 5683 df-dm 5685 df-rn 5686 df-fn 6545 df-f 6546 df-f1 6547 df-dom 8943 |
This theorem is referenced by: brdomiOLD 8957 brdom 8958 f1dom3g 8965 f1dom2gOLD 8968 f1domg 8970 dom3d 8992 domdifsn 9056 fidomtri 9990 hashdom 14343 hashge3el3dif 14451 sizusglecusg 28987 erdsze2lem1 34492 |
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