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Mirrors > Home > MPE Home > Th. List > cfsmo | Structured version Visualization version GIF version |
Description: The map in cff1 10297 can be assumed to be a strictly monotone ordinal function without loss of generality. (Contributed by Mario Carneiro, 28-Feb-2013.) |
Ref | Expression |
---|---|
cfsmo | ⊢ (𝐴 ∈ On → ∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓‘𝑤))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmeq 5909 | . . . . 5 ⊢ (𝑥 = 𝑧 → dom 𝑥 = dom 𝑧) | |
2 | 1 | fveq2d 6904 | . . . 4 ⊢ (𝑥 = 𝑧 → (ℎ‘dom 𝑥) = (ℎ‘dom 𝑧)) |
3 | fveq2 6900 | . . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝑥‘𝑛) = (𝑥‘𝑚)) | |
4 | suceq 6441 | . . . . . . 7 ⊢ ((𝑥‘𝑛) = (𝑥‘𝑚) → suc (𝑥‘𝑛) = suc (𝑥‘𝑚)) | |
5 | 3, 4 | syl 17 | . . . . . 6 ⊢ (𝑛 = 𝑚 → suc (𝑥‘𝑛) = suc (𝑥‘𝑚)) |
6 | 5 | cbviunv 5047 | . . . . 5 ⊢ ∪ 𝑛 ∈ dom 𝑥 suc (𝑥‘𝑛) = ∪ 𝑚 ∈ dom 𝑥 suc (𝑥‘𝑚) |
7 | fveq1 6899 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥‘𝑚) = (𝑧‘𝑚)) | |
8 | suceq 6441 | . . . . . . 7 ⊢ ((𝑥‘𝑚) = (𝑧‘𝑚) → suc (𝑥‘𝑚) = suc (𝑧‘𝑚)) | |
9 | 7, 8 | syl 17 | . . . . . 6 ⊢ (𝑥 = 𝑧 → suc (𝑥‘𝑚) = suc (𝑧‘𝑚)) |
10 | 1, 9 | iuneq12d 5028 | . . . . 5 ⊢ (𝑥 = 𝑧 → ∪ 𝑚 ∈ dom 𝑥 suc (𝑥‘𝑚) = ∪ 𝑚 ∈ dom 𝑧 suc (𝑧‘𝑚)) |
11 | 6, 10 | eqtrid 2777 | . . . 4 ⊢ (𝑥 = 𝑧 → ∪ 𝑛 ∈ dom 𝑥 suc (𝑥‘𝑛) = ∪ 𝑚 ∈ dom 𝑧 suc (𝑧‘𝑚)) |
12 | 2, 11 | uneq12d 4163 | . . 3 ⊢ (𝑥 = 𝑧 → ((ℎ‘dom 𝑥) ∪ ∪ 𝑛 ∈ dom 𝑥 suc (𝑥‘𝑛)) = ((ℎ‘dom 𝑧) ∪ ∪ 𝑚 ∈ dom 𝑧 suc (𝑧‘𝑚))) |
13 | 12 | cbvmptv 5265 | . 2 ⊢ (𝑥 ∈ V ↦ ((ℎ‘dom 𝑥) ∪ ∪ 𝑛 ∈ dom 𝑥 suc (𝑥‘𝑛))) = (𝑧 ∈ V ↦ ((ℎ‘dom 𝑧) ∪ ∪ 𝑚 ∈ dom 𝑧 suc (𝑧‘𝑚))) |
14 | eqid 2725 | . 2 ⊢ (recs((𝑥 ∈ V ↦ ((ℎ‘dom 𝑥) ∪ ∪ 𝑛 ∈ dom 𝑥 suc (𝑥‘𝑛)))) ↾ (cf‘𝐴)) = (recs((𝑥 ∈ V ↦ ((ℎ‘dom 𝑥) ∪ ∪ 𝑛 ∈ dom 𝑥 suc (𝑥‘𝑛)))) ↾ (cf‘𝐴)) | |
15 | 13, 14 | cfsmolem 10309 | 1 ⊢ (𝐴 ∈ On → ∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓‘𝑤))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ∀wral 3050 ∃wrex 3059 Vcvv 3461 ∪ cun 3944 ⊆ wss 3946 ∪ ciun 5000 ↦ cmpt 5235 dom cdm 5681 ↾ cres 5683 Oncon0 6375 suc csuc 6377 ⟶wf 6549 ‘cfv 6553 Smo wsmo 8374 recscrecs 8399 cfccf 9976 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-se 5637 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-1st 8002 df-2nd 8003 df-frecs 8295 df-wrecs 8326 df-smo 8375 df-recs 8400 df-er 8733 df-map 8856 df-en 8974 df-dom 8975 df-sdom 8976 df-card 9978 df-cf 9980 df-acn 9981 |
This theorem is referenced by: cfidm 10314 pwcfsdom 10622 |
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