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Mirrors > Home > ILE Home > Th. List > frec2uzled | GIF version |
Description: The mapping 𝐺 (see frec2uz0d 10324) preserves order. (Contributed by Jim Kingdon, 24-Feb-2022.) |
Ref | Expression |
---|---|
frec2uzled.1 | ⊢ (𝜑 → 𝐶 ∈ ℤ) |
frec2uzled.2 | ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) |
frec2uzled.a | ⊢ (𝜑 → 𝐴 ∈ ω) |
frec2uzled.b | ⊢ (𝜑 → 𝐵 ∈ ω) |
Ref | Expression |
---|---|
frec2uzled | ⊢ (𝜑 → (𝐴 ⊆ 𝐵 ↔ (𝐺‘𝐴) ≤ (𝐺‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frec2uzled.1 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℤ) | |
2 | frec2uzled.2 | . . . 4 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) | |
3 | frec2uzled.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ω) | |
4 | frec2uzled.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ω) | |
5 | 1, 2, 3, 4 | frec2uzlt2d 10329 | . . 3 ⊢ (𝜑 → (𝐴 ∈ 𝐵 ↔ (𝐺‘𝐴) < (𝐺‘𝐵))) |
6 | 1, 2 | frec2uzf1od 10331 | . . . . . 6 ⊢ (𝜑 → 𝐺:ω–1-1-onto→(ℤ≥‘𝐶)) |
7 | f1of1 5425 | . . . . . 6 ⊢ (𝐺:ω–1-1-onto→(ℤ≥‘𝐶) → 𝐺:ω–1-1→(ℤ≥‘𝐶)) | |
8 | 6, 7 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝐺:ω–1-1→(ℤ≥‘𝐶)) |
9 | f1fveq 5734 | . . . . 5 ⊢ ((𝐺:ω–1-1→(ℤ≥‘𝐶) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → ((𝐺‘𝐴) = (𝐺‘𝐵) ↔ 𝐴 = 𝐵)) | |
10 | 8, 3, 4, 9 | syl12anc 1225 | . . . 4 ⊢ (𝜑 → ((𝐺‘𝐴) = (𝐺‘𝐵) ↔ 𝐴 = 𝐵)) |
11 | 10 | bicomd 140 | . . 3 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ (𝐺‘𝐴) = (𝐺‘𝐵))) |
12 | 5, 11 | orbi12d 783 | . 2 ⊢ (𝜑 → ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵) ↔ ((𝐺‘𝐴) < (𝐺‘𝐵) ∨ (𝐺‘𝐴) = (𝐺‘𝐵)))) |
13 | nnsseleq 6460 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) | |
14 | 3, 4, 13 | syl2anc 409 | . 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) |
15 | 1, 2, 3 | frec2uzzd 10325 | . . 3 ⊢ (𝜑 → (𝐺‘𝐴) ∈ ℤ) |
16 | 1, 2, 4 | frec2uzzd 10325 | . . 3 ⊢ (𝜑 → (𝐺‘𝐵) ∈ ℤ) |
17 | zleloe 9229 | . . 3 ⊢ (((𝐺‘𝐴) ∈ ℤ ∧ (𝐺‘𝐵) ∈ ℤ) → ((𝐺‘𝐴) ≤ (𝐺‘𝐵) ↔ ((𝐺‘𝐴) < (𝐺‘𝐵) ∨ (𝐺‘𝐴) = (𝐺‘𝐵)))) | |
18 | 15, 16, 17 | syl2anc 409 | . 2 ⊢ (𝜑 → ((𝐺‘𝐴) ≤ (𝐺‘𝐵) ↔ ((𝐺‘𝐴) < (𝐺‘𝐵) ∨ (𝐺‘𝐴) = (𝐺‘𝐵)))) |
19 | 12, 14, 18 | 3bitr4d 219 | 1 ⊢ (𝜑 → (𝐴 ⊆ 𝐵 ↔ (𝐺‘𝐴) ≤ (𝐺‘𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∨ wo 698 = wceq 1342 ∈ wcel 2135 ⊆ wss 3111 class class class wbr 3976 ↦ cmpt 4037 ωcom 4561 –1-1→wf1 5179 –1-1-onto→wf1o 5181 ‘cfv 5182 (class class class)co 5836 freccfrec 6349 1c1 7745 + caddc 7747 < clt 7924 ≤ cle 7925 ℤcz 9182 ℤ≥cuz 9457 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-recs 6264 df-frec 6350 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-inn 8849 df-n0 9106 df-z 9183 df-uz 9458 |
This theorem is referenced by: fihashdom 10705 ennnfonelemkh 12282 ctinfomlemom 12297 |
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