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Theorem cfsmo 10310
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.)
Assertion
Ref Expression
cfsmo (𝐴 ∈ On → ∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤)))
Distinct variable group:   𝐴,𝑓,𝑤,𝑧

Proof of Theorem cfsmo
Dummy variables 𝑚 𝑥 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmeq 5909 . . . . 5 (𝑥 = 𝑧 → dom 𝑥 = dom 𝑧)
21fveq2d 6904 . . . 4 (𝑥 = 𝑧 → (‘dom 𝑥) = (‘dom 𝑧))
3 fveq2 6900 . . . . . . 7 (𝑛 = 𝑚 → (𝑥𝑛) = (𝑥𝑚))
4 suceq 6441 . . . . . . 7 ((𝑥𝑛) = (𝑥𝑚) → suc (𝑥𝑛) = suc (𝑥𝑚))
53, 4syl 17 . . . . . 6 (𝑛 = 𝑚 → suc (𝑥𝑛) = suc (𝑥𝑚))
65cbviunv 5047 . . . . 5 𝑛 ∈ dom 𝑥 suc (𝑥𝑛) = 𝑚 ∈ dom 𝑥 suc (𝑥𝑚)
7 fveq1 6899 . . . . . . 7 (𝑥 = 𝑧 → (𝑥𝑚) = (𝑧𝑚))
8 suceq 6441 . . . . . . 7 ((𝑥𝑚) = (𝑧𝑚) → suc (𝑥𝑚) = suc (𝑧𝑚))
97, 8syl 17 . . . . . 6 (𝑥 = 𝑧 → suc (𝑥𝑚) = suc (𝑧𝑚))
101, 9iuneq12d 5028 . . . . 5 (𝑥 = 𝑧 𝑚 ∈ dom 𝑥 suc (𝑥𝑚) = 𝑚 ∈ dom 𝑧 suc (𝑧𝑚))
116, 10eqtrid 2777 . . . 4 (𝑥 = 𝑧 𝑛 ∈ dom 𝑥 suc (𝑥𝑛) = 𝑚 ∈ dom 𝑧 suc (𝑧𝑚))
122, 11uneq12d 4163 . . 3 (𝑥 = 𝑧 → ((‘dom 𝑥) ∪ 𝑛 ∈ dom 𝑥 suc (𝑥𝑛)) = ((‘dom 𝑧) ∪ 𝑚 ∈ dom 𝑧 suc (𝑧𝑚)))
1312cbvmptv 5265 . 2 (𝑥 ∈ V ↦ ((‘dom 𝑥) ∪ 𝑛 ∈ dom 𝑥 suc (𝑥𝑛))) = (𝑧 ∈ V ↦ ((‘dom 𝑧) ∪ 𝑚 ∈ dom 𝑧 suc (𝑧𝑚)))
14 eqid 2725 . 2 (recs((𝑥 ∈ V ↦ ((‘dom 𝑥) ∪ 𝑛 ∈ dom 𝑥 suc (𝑥𝑛)))) ↾ (cf‘𝐴)) = (recs((𝑥 ∈ V ↦ ((‘dom 𝑥) ∪ 𝑛 ∈ dom 𝑥 suc (𝑥𝑛)))) ↾ (cf‘𝐴))
1513, 14cfsmolem 10309 1 (𝐴 ∈ On → ∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤)))
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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