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Mirrors > Home > MPE Home > Th. List > axdc4uz | Structured version Visualization version GIF version |
Description: A version of axdc4 9724 that works on an upper set of integers instead of ω. (Contributed by Mario Carneiro, 8-Jan-2014.) |
Ref | Expression |
---|---|
axdc4uz.1 | ⊢ 𝑀 ∈ ℤ |
axdc4uz.2 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
Ref | Expression |
---|---|
axdc4uz | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝐴 ∧ 𝐹:(𝑍 × 𝐴)⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:𝑍⟶𝐴 ∧ (𝑔‘𝑀) = 𝐶 ∧ ∀𝑘 ∈ 𝑍 (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq2 2871 | . . . . 5 ⊢ (𝑓 = 𝐴 → (𝐶 ∈ 𝑓 ↔ 𝐶 ∈ 𝐴)) | |
2 | xpeq2 5464 | . . . . . 6 ⊢ (𝑓 = 𝐴 → (𝑍 × 𝑓) = (𝑍 × 𝐴)) | |
3 | pweq 4456 | . . . . . . 7 ⊢ (𝑓 = 𝐴 → 𝒫 𝑓 = 𝒫 𝐴) | |
4 | 3 | difeq1d 4019 | . . . . . 6 ⊢ (𝑓 = 𝐴 → (𝒫 𝑓 ∖ {∅}) = (𝒫 𝐴 ∖ {∅})) |
5 | 2, 4 | feq23d 6377 | . . . . 5 ⊢ (𝑓 = 𝐴 → (𝐹:(𝑍 × 𝑓)⟶(𝒫 𝑓 ∖ {∅}) ↔ 𝐹:(𝑍 × 𝐴)⟶(𝒫 𝐴 ∖ {∅}))) |
6 | 1, 5 | anbi12d 630 | . . . 4 ⊢ (𝑓 = 𝐴 → ((𝐶 ∈ 𝑓 ∧ 𝐹:(𝑍 × 𝑓)⟶(𝒫 𝑓 ∖ {∅})) ↔ (𝐶 ∈ 𝐴 ∧ 𝐹:(𝑍 × 𝐴)⟶(𝒫 𝐴 ∖ {∅})))) |
7 | feq3 6365 | . . . . . 6 ⊢ (𝑓 = 𝐴 → (𝑔:𝑍⟶𝑓 ↔ 𝑔:𝑍⟶𝐴)) | |
8 | 7 | 3anbi1d 1432 | . . . . 5 ⊢ (𝑓 = 𝐴 → ((𝑔:𝑍⟶𝑓 ∧ (𝑔‘𝑀) = 𝐶 ∧ ∀𝑘 ∈ 𝑍 (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘))) ↔ (𝑔:𝑍⟶𝐴 ∧ (𝑔‘𝑀) = 𝐶 ∧ ∀𝑘 ∈ 𝑍 (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘))))) |
9 | 8 | exbidv 1899 | . . . 4 ⊢ (𝑓 = 𝐴 → (∃𝑔(𝑔:𝑍⟶𝑓 ∧ (𝑔‘𝑀) = 𝐶 ∧ ∀𝑘 ∈ 𝑍 (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘))) ↔ ∃𝑔(𝑔:𝑍⟶𝐴 ∧ (𝑔‘𝑀) = 𝐶 ∧ ∀𝑘 ∈ 𝑍 (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘))))) |
10 | 6, 9 | imbi12d 346 | . . 3 ⊢ (𝑓 = 𝐴 → (((𝐶 ∈ 𝑓 ∧ 𝐹:(𝑍 × 𝑓)⟶(𝒫 𝑓 ∖ {∅})) → ∃𝑔(𝑔:𝑍⟶𝑓 ∧ (𝑔‘𝑀) = 𝐶 ∧ ∀𝑘 ∈ 𝑍 (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)))) ↔ ((𝐶 ∈ 𝐴 ∧ 𝐹:(𝑍 × 𝐴)⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:𝑍⟶𝐴 ∧ (𝑔‘𝑀) = 𝐶 ∧ ∀𝑘 ∈ 𝑍 (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)))))) |
11 | axdc4uz.1 | . . . 4 ⊢ 𝑀 ∈ ℤ | |
12 | axdc4uz.2 | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
13 | vex 3440 | . . . 4 ⊢ 𝑓 ∈ V | |
14 | eqid 2795 | . . . 4 ⊢ (rec((𝑦 ∈ V ↦ (𝑦 + 1)), 𝑀) ↾ ω) = (rec((𝑦 ∈ V ↦ (𝑦 + 1)), 𝑀) ↾ ω) | |
15 | eqid 2795 | . . . 4 ⊢ (𝑛 ∈ ω, 𝑥 ∈ 𝑓 ↦ (((rec((𝑦 ∈ V ↦ (𝑦 + 1)), 𝑀) ↾ ω)‘𝑛)𝐹𝑥)) = (𝑛 ∈ ω, 𝑥 ∈ 𝑓 ↦ (((rec((𝑦 ∈ V ↦ (𝑦 + 1)), 𝑀) ↾ ω)‘𝑛)𝐹𝑥)) | |
16 | 11, 12, 13, 14, 15 | axdc4uzlem 13201 | . . 3 ⊢ ((𝐶 ∈ 𝑓 ∧ 𝐹:(𝑍 × 𝑓)⟶(𝒫 𝑓 ∖ {∅})) → ∃𝑔(𝑔:𝑍⟶𝑓 ∧ (𝑔‘𝑀) = 𝐶 ∧ ∀𝑘 ∈ 𝑍 (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)))) |
17 | 10, 16 | vtoclg 3510 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐶 ∈ 𝐴 ∧ 𝐹:(𝑍 × 𝐴)⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:𝑍⟶𝐴 ∧ (𝑔‘𝑀) = 𝐶 ∧ ∀𝑘 ∈ 𝑍 (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘))))) |
18 | 17 | 3impib 1109 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝐴 ∧ 𝐹:(𝑍 × 𝐴)⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:𝑍⟶𝐴 ∧ (𝑔‘𝑀) = 𝐶 ∧ ∀𝑘 ∈ 𝑍 (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1080 = wceq 1522 ∃wex 1761 ∈ wcel 2081 ∀wral 3105 Vcvv 3437 ∖ cdif 3856 ∅c0 4211 𝒫 cpw 4453 {csn 4472 ↦ cmpt 5041 × cxp 5441 ↾ cres 5445 ⟶wf 6221 ‘cfv 6225 (class class class)co 7016 ∈ cmpo 7018 ωcom 7436 reccrdg 7897 1c1 10384 + caddc 10386 ℤcz 11829 ℤ≥cuz 12093 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-inf2 8950 ax-dc 9714 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-nn 11487 df-n0 11746 df-z 11830 df-uz 12094 |
This theorem is referenced by: bcthlem5 23614 sdclem1 34550 |
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