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Mirrors > Home > MPE Home > Th. List > Mathboxes > wessf1orn | Structured version Visualization version GIF version |
Description: Given a function 𝐹 on a well ordered domain 𝐴 there exists a subset of 𝐴 such that 𝐹 restricted to such subset is injective and onto the range of 𝐹 (without using the axiom of choice). (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
wessf1orn.f | ⊢ (𝜑 → 𝐹 Fn 𝐴) |
wessf1orn.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
wessf1orn.r | ⊢ (𝜑 → 𝑅 We 𝐴) |
Ref | Expression |
---|---|
wessf1orn | ⊢ (𝜑 → ∃𝑥 ∈ 𝒫 𝐴(𝐹 ↾ 𝑥):𝑥–1-1-onto→ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wessf1orn.f | . 2 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
2 | wessf1orn.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | wessf1orn.r | . 2 ⊢ (𝜑 → 𝑅 We 𝐴) | |
4 | eqid 2610 | . 2 ⊢ (𝑦 ∈ ran 𝐹 ↦ (℩𝑥 ∈ (◡𝐹 “ {𝑦})∀𝑧 ∈ (◡𝐹 “ {𝑦}) ¬ 𝑧𝑅𝑥)) = (𝑦 ∈ ran 𝐹 ↦ (℩𝑥 ∈ (◡𝐹 “ {𝑦})∀𝑧 ∈ (◡𝐹 “ {𝑦}) ¬ 𝑧𝑅𝑥)) | |
5 | 1, 2, 3, 4 | wessf1ornlem 38366 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝒫 𝐴(𝐹 ↾ 𝑥):𝑥–1-1-onto→ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 𝒫 cpw 4108 {csn 4125 class class class wbr 4583 ↦ cmpt 4643 We wwe 4996 ◡ccnv 5037 ran crn 5039 ↾ cres 5040 “ cima 5041 Fn wfn 5799 –1-1-onto→wf1o 5803 ℩crio 6510 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 |
This theorem is referenced by: ssnnf1octb 38377 sge0resrn 39297 nnfoctbdj 39349 |
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