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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 2821 | . 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 41443 | 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 2110 ∀wral 3138 ∃wrex 3139 𝒫 cpw 4538 {csn 4566 class class class wbr 5065 ↦ cmpt 5145 We wwe 5512 ◡ccnv 5553 ran crn 5555 ↾ cres 5556 “ cima 5557 Fn wfn 6349 –1-1-onto→wf1o 6353 ℩crio 7112 |
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 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 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-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 |
This theorem is referenced by: ssnnf1octb 41454 sge0resrn 42685 nnfoctbdj 42737 |
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