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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.) |
Ref | Expression |
---|---|
brdomi | ⊢ (𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reldom 8509 | . . . 4 ⊢ Rel ≼ | |
2 | 1 | brrelex2i 5604 | . . 3 ⊢ (𝐴 ≼ 𝐵 → 𝐵 ∈ V) |
3 | brdomg 8513 | . . 3 ⊢ (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 269 | 1 ⊢ (𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∃wex 1776 ∈ wcel 2110 Vcvv 3495 class class class wbr 5059 –1-1→wf1 6347 ≼ cdom 8501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-xp 5556 df-rel 5557 df-cnv 5558 df-dm 5560 df-rn 5561 df-fn 6353 df-f 6354 df-f1 6355 df-dom 8505 |
This theorem is referenced by: 2dom 8576 xpdom2 8606 domunsncan 8611 fodomr 8662 domssex 8672 sucdom2 8708 hartogslem1 9000 infdifsn 9114 acndom 9471 acndom2 9474 fictb 9661 fin23lem41 9768 iundom2g 9956 pwfseq 10080 omssubadd 31553 |
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