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Mirrors > Home > MPE Home > Th. List > brdomiOLD | Structured version Visualization version GIF version |
Description: Obsolete version of brdomi 8832 as of 29-Nov-2024. (Contributed by Mario Carneiro, 26-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
brdomiOLD | ⊢ (𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reldom 8823 | . . . 4 ⊢ Rel ≼ | |
2 | 1 | brrelex2i 5686 | . . 3 ⊢ (𝐴 ≼ 𝐵 → 𝐵 ∈ V) |
3 | brdomg 8830 | . . 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 267 | 1 ⊢ (𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∃wex 1782 ∈ wcel 2107 Vcvv 3444 class class class wbr 5104 –1-1→wf1 6489 ≼ cdom 8815 |
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 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 ax-un 7663 |
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 2716 df-cleq 2730 df-clel 2816 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-xp 5637 df-rel 5638 df-cnv 5639 df-dm 5641 df-rn 5642 df-fn 6495 df-f 6496 df-f1 6497 df-dom 8819 |
This theorem is referenced by: (None) |
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