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Mirrors > Home > MPE Home > Th. List > cfsmo | Structured version Visualization version GIF version |
Description: The map in cff1 9837 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 5757 | . . . . 5 ⊢ (𝑥 = 𝑧 → dom 𝑥 = dom 𝑧) | |
2 | 1 | fveq2d 6699 | . . . 4 ⊢ (𝑥 = 𝑧 → (ℎ‘dom 𝑥) = (ℎ‘dom 𝑧)) |
3 | fveq2 6695 | . . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝑥‘𝑛) = (𝑥‘𝑚)) | |
4 | suceq 6256 | . . . . . . 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 4935 | . . . . 5 ⊢ ∪ 𝑛 ∈ dom 𝑥 suc (𝑥‘𝑛) = ∪ 𝑚 ∈ dom 𝑥 suc (𝑥‘𝑚) |
7 | fveq1 6694 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥‘𝑚) = (𝑧‘𝑚)) | |
8 | suceq 6256 | . . . . . . 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 4918 | . . . . 5 ⊢ (𝑥 = 𝑧 → ∪ 𝑚 ∈ dom 𝑥 suc (𝑥‘𝑚) = ∪ 𝑚 ∈ dom 𝑧 suc (𝑧‘𝑚)) |
11 | 6, 10 | syl5eq 2783 | . . . 4 ⊢ (𝑥 = 𝑧 → ∪ 𝑛 ∈ dom 𝑥 suc (𝑥‘𝑛) = ∪ 𝑚 ∈ dom 𝑧 suc (𝑧‘𝑚)) |
12 | 2, 11 | uneq12d 4064 | . . 3 ⊢ (𝑥 = 𝑧 → ((ℎ‘dom 𝑥) ∪ ∪ 𝑛 ∈ dom 𝑥 suc (𝑥‘𝑛)) = ((ℎ‘dom 𝑧) ∪ ∪ 𝑚 ∈ dom 𝑧 suc (𝑧‘𝑚))) |
13 | 12 | cbvmptv 5143 | . 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 2736 | . 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 9849 | 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 1089 = wceq 1543 ∃wex 1787 ∈ wcel 2112 ∀wral 3051 ∃wrex 3052 Vcvv 3398 ∪ cun 3851 ⊆ wss 3853 ∪ ciun 4890 ↦ cmpt 5120 dom cdm 5536 ↾ cres 5538 Oncon0 6191 suc csuc 6193 ⟶wf 6354 ‘cfv 6358 Smo wsmo 8060 recscrecs 8085 cfccf 9518 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-smo 8061 df-recs 8086 df-er 8369 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-card 9520 df-cf 9522 df-acn 9523 |
This theorem is referenced by: cfidm 9854 pwcfsdom 10162 |
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