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Mirrors > Home > ILE Home > Th. List > frec2uzled | GIF version |
Description: The mapping 𝐺 (see frec2uz0d 10140) 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 10145 | . . 3 ⊢ (𝜑 → (𝐴 ∈ 𝐵 ↔ (𝐺‘𝐴) < (𝐺‘𝐵))) |
6 | 1, 2 | frec2uzf1od 10147 | . . . . . 6 ⊢ (𝜑 → 𝐺:ω–1-1-onto→(ℤ≥‘𝐶)) |
7 | f1of1 5334 | . . . . . 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 5641 | . . . . 5 ⊢ ((𝐺:ω–1-1→(ℤ≥‘𝐶) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → ((𝐺‘𝐴) = (𝐺‘𝐵) ↔ 𝐴 = 𝐵)) | |
10 | 8, 3, 4, 9 | syl12anc 1199 | . . . 4 ⊢ (𝜑 → ((𝐺‘𝐴) = (𝐺‘𝐵) ↔ 𝐴 = 𝐵)) |
11 | 10 | bicomd 140 | . . 3 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ (𝐺‘𝐴) = (𝐺‘𝐵))) |
12 | 5, 11 | orbi12d 767 | . 2 ⊢ (𝜑 → ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵) ↔ ((𝐺‘𝐴) < (𝐺‘𝐵) ∨ (𝐺‘𝐴) = (𝐺‘𝐵)))) |
13 | nnsseleq 6365 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) | |
14 | 3, 4, 13 | syl2anc 408 | . 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) |
15 | 1, 2, 3 | frec2uzzd 10141 | . . 3 ⊢ (𝜑 → (𝐺‘𝐴) ∈ ℤ) |
16 | 1, 2, 4 | frec2uzzd 10141 | . . 3 ⊢ (𝜑 → (𝐺‘𝐵) ∈ ℤ) |
17 | zleloe 9069 | . . 3 ⊢ (((𝐺‘𝐴) ∈ ℤ ∧ (𝐺‘𝐵) ∈ ℤ) → ((𝐺‘𝐴) ≤ (𝐺‘𝐵) ↔ ((𝐺‘𝐴) < (𝐺‘𝐵) ∨ (𝐺‘𝐴) = (𝐺‘𝐵)))) | |
18 | 15, 16, 17 | syl2anc 408 | . 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 682 = wceq 1316 ∈ wcel 1465 ⊆ wss 3041 class class class wbr 3899 ↦ cmpt 3959 ωcom 4474 –1-1→wf1 5090 –1-1-onto→wf1o 5092 ‘cfv 5093 (class class class)co 5742 freccfrec 6255 1c1 7589 + caddc 7591 < clt 7768 ≤ cle 7769 ℤcz 9022 ℤ≥cuz 9294 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-ilim 4261 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-recs 6170 df-frec 6256 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-inn 8689 df-n0 8946 df-z 9023 df-uz 9295 |
This theorem is referenced by: fihashdom 10517 ennnfonelemkh 11852 ctinfomlemom 11867 |
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