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Mathbox for Glauco Siliprandi |
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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 2797 | . 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 41006 | 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 2083 ∀wral 3107 ∃wrex 3108 𝒫 cpw 4459 {csn 4478 class class class wbr 4968 ↦ cmpt 5047 We wwe 5408 ◡ccnv 5449 ran crn 5451 ↾ cres 5452 “ cima 5453 Fn wfn 6227 –1-1-onto→wf1o 6231 ℩crio 6983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pr 5228 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-br 4969 df-opab 5031 df-mpt 5048 df-id 5355 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 |
This theorem is referenced by: ssnnf1octb 41017 sge0resrn 42250 nnfoctbdj 42302 |
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