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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 8951 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 8951 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝐶) → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) | |
2 | 1 | ex 414 | . 2 ⊢ (𝐴 ∈ V → (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))) |
3 | reldom 8945 | . . . . 5 ⊢ Rel ≼ | |
4 | 3 | brrelex1i 5733 | . . . 4 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ∈ V) |
5 | f1f 6788 | . . . . . 6 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴⟶𝐵) | |
6 | fdm 6727 | . . . . . . 7 ⊢ (𝑓:𝐴⟶𝐵 → dom 𝑓 = 𝐴) | |
7 | vex 3479 | . . . . . . . 8 ⊢ 𝑓 ∈ V | |
8 | 7 | dmex 7902 | . . . . . . 7 ⊢ dom 𝑓 ∈ V |
9 | 6, 8 | eqeltrrdi 2843 | . . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → 𝐴 ∈ V) |
10 | 5, 9 | syl 17 | . . . . 5 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝐴 ∈ V) |
11 | 10 | exlimiv 1934 | . . . 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 2107 Vcvv 3475 class class class wbr 5149 dom cdm 5677 ⟶wf 6540 –1-1→wf1 6541 ≼ cdom 8937 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-xp 5683 df-rel 5684 df-cnv 5685 df-dm 5687 df-rn 5688 df-fn 6547 df-f 6548 df-f1 6549 df-dom 8941 |
This theorem is referenced by: brdomiOLD 8955 brdom 8956 f1dom3g 8963 f1dom2gOLD 8966 f1domg 8968 dom3d 8990 domdifsn 9054 fidomtri 9988 hashdom 14339 hashge3el3dif 14447 sizusglecusg 28720 erdsze2lem1 34194 |
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