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Mirrors > Home > MPE Home > Th. List > brdomi | Structured version Visualization version GIF version |
Description: Dominance relation. (Contributed by Mario Carneiro, 26-Apr-2015.) Avoid ax-un 7721. (Revised by BTernaryTau, 29-Nov-2024.) |
Ref | Expression |
---|---|
brdomi | ⊢ (𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reldom 8941 | . . . 4 ⊢ Rel ≼ | |
2 | 1 | brrelex12i 5729 | . . 3 ⊢ (𝐴 ≼ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
3 | brdom2g 8947 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝐴 ≼ 𝐵 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) |
5 | 4 | ibi 266 | 1 ⊢ (𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∃wex 1781 ∈ wcel 2106 Vcvv 3474 class class class wbr 5147 –1-1→wf1 6537 ≼ cdom 8933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 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-br 5148 df-opab 5210 df-xp 5681 df-rel 5682 df-fn 6543 df-f 6544 df-f1 6545 df-dom 8937 |
This theorem is referenced by: domssl 8990 domssr 8991 2dom 9026 undom 9055 xpdom2 9063 domunsncan 9068 sucdom2OLD 9078 dom0 9098 fodomr 9124 domssex 9134 domtrfil 9191 sucdom2 9202 sdom1 9238 1sdom2dom 9243 infn0 9303 hartogslem1 9533 infdifsn 9648 acndom 10042 acndom2 10045 fictb 10236 fin23lem41 10343 iundom2g 10531 pwfseq 10655 omssubadd 33287 |
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