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Mirrors > Home > MPE Home > Th. List > cfsmo | Structured version Visualization version GIF version |
Description: The map in cff1 9476 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 5618 | . . . . 5 ⊢ (𝑥 = 𝑧 → dom 𝑥 = dom 𝑧) | |
2 | 1 | fveq2d 6500 | . . . 4 ⊢ (𝑥 = 𝑧 → (ℎ‘dom 𝑥) = (ℎ‘dom 𝑧)) |
3 | fveq2 6496 | . . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝑥‘𝑛) = (𝑥‘𝑚)) | |
4 | suceq 6091 | . . . . . . 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 4829 | . . . . 5 ⊢ ∪ 𝑛 ∈ dom 𝑥 suc (𝑥‘𝑛) = ∪ 𝑚 ∈ dom 𝑥 suc (𝑥‘𝑚) |
7 | fveq1 6495 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥‘𝑚) = (𝑧‘𝑚)) | |
8 | suceq 6091 | . . . . . . 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 4815 | . . . . 5 ⊢ (𝑥 = 𝑧 → ∪ 𝑚 ∈ dom 𝑥 suc (𝑥‘𝑚) = ∪ 𝑚 ∈ dom 𝑧 suc (𝑧‘𝑚)) |
11 | 6, 10 | syl5eq 2820 | . . . 4 ⊢ (𝑥 = 𝑧 → ∪ 𝑛 ∈ dom 𝑥 suc (𝑥‘𝑛) = ∪ 𝑚 ∈ dom 𝑧 suc (𝑧‘𝑚)) |
12 | 2, 11 | uneq12d 4023 | . . 3 ⊢ (𝑥 = 𝑧 → ((ℎ‘dom 𝑥) ∪ ∪ 𝑛 ∈ dom 𝑥 suc (𝑥‘𝑛)) = ((ℎ‘dom 𝑧) ∪ ∪ 𝑚 ∈ dom 𝑧 suc (𝑧‘𝑚))) |
13 | 12 | cbvmptv 5024 | . 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 2772 | . 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 9488 | 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 1068 = wceq 1507 ∃wex 1742 ∈ wcel 2050 ∀wral 3082 ∃wrex 3083 Vcvv 3409 ∪ cun 3821 ⊆ wss 3823 ∪ ciun 4788 ↦ cmpt 5004 dom cdm 5403 ↾ cres 5405 Oncon0 6026 suc csuc 6028 ⟶wf 6181 ‘cfv 6185 Smo wsmo 7784 recscrecs 7809 cfccf 9158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-se 5363 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-isom 6194 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-1st 7499 df-2nd 7500 df-wrecs 7748 df-smo 7785 df-recs 7810 df-er 8087 df-map 8206 df-en 8305 df-dom 8306 df-sdom 8307 df-card 9160 df-cf 9162 df-acn 9163 |
This theorem is referenced by: cfidm 9493 pwcfsdom 9801 |
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