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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 7755. (Revised by BTernaryTau, 29-Nov-2024.) | 
| Ref | Expression | 
|---|---|
| brdomi | ⊢ (𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | reldom 8991 | . . . 4 ⊢ Rel ≼ | |
| 2 | 1 | brrelex12i 5740 | . . 3 ⊢ (𝐴 ≼ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | 
| 3 | brdom2g 8996 | . . 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 267 | 1 ⊢ (𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 –1-1→wf1 6558 ≼ cdom 8983 | 
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-fn 6564 df-f 6565 df-f1 6566 df-dom 8987 | 
| This theorem is referenced by: domssl 9038 domssr 9039 2dom 9070 undom 9099 xpdom2 9107 domunsncan 9112 sucdom2OLD 9122 dom0 9142 fodomr 9168 domssex 9178 domtrfil 9232 sucdom2 9243 sdom1 9278 1sdom2dom 9283 infn0 9340 fodomfir 9368 hartogslem1 9582 infdifsn 9697 acndom 10091 acndom2 10094 fictb 10284 fin23lem41 10392 iundom2g 10580 pwfseq 10704 omssubadd 34302 | 
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