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Theorem cfsmo 9682
Description: The map in cff1 9669 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 5736 . . . . 5 (𝑥 = 𝑧 → dom 𝑥 = dom 𝑧)
21fveq2d 6649 . . . 4 (𝑥 = 𝑧 → (‘dom 𝑥) = (‘dom 𝑧))
3 fveq2 6645 . . . . . . 7 (𝑛 = 𝑚 → (𝑥𝑛) = (𝑥𝑚))
4 suceq 6224 . . . . . . 7 ((𝑥𝑛) = (𝑥𝑚) → suc (𝑥𝑛) = suc (𝑥𝑚))
53, 4syl 17 . . . . . 6 (𝑛 = 𝑚 → suc (𝑥𝑛) = suc (𝑥𝑚))
65cbviunv 4927 . . . . 5 𝑛 ∈ dom 𝑥 suc (𝑥𝑛) = 𝑚 ∈ dom 𝑥 suc (𝑥𝑚)
7 fveq1 6644 . . . . . . 7 (𝑥 = 𝑧 → (𝑥𝑚) = (𝑧𝑚))
8 suceq 6224 . . . . . . 7 ((𝑥𝑚) = (𝑧𝑚) → suc (𝑥𝑚) = suc (𝑧𝑚))
97, 8syl 17 . . . . . 6 (𝑥 = 𝑧 → suc (𝑥𝑚) = suc (𝑧𝑚))
101, 9iuneq12d 4909 . . . . 5 (𝑥 = 𝑧 𝑚 ∈ dom 𝑥 suc (𝑥𝑚) = 𝑚 ∈ dom 𝑧 suc (𝑧𝑚))
116, 10syl5eq 2845 . . . 4 (𝑥 = 𝑧 𝑛 ∈ dom 𝑥 suc (𝑥𝑛) = 𝑚 ∈ dom 𝑧 suc (𝑧𝑚))
122, 11uneq12d 4091 . . 3 (𝑥 = 𝑧 → ((‘dom 𝑥) ∪ 𝑛 ∈ dom 𝑥 suc (𝑥𝑛)) = ((‘dom 𝑧) ∪ 𝑚 ∈ dom 𝑧 suc (𝑧𝑚)))
1312cbvmptv 5133 . 2 (𝑥 ∈ V ↦ ((‘dom 𝑥) ∪ 𝑛 ∈ dom 𝑥 suc (𝑥𝑛))) = (𝑧 ∈ V ↦ ((‘dom 𝑧) ∪ 𝑚 ∈ dom 𝑧 suc (𝑧𝑚)))
14 eqid 2798 . 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 9681 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 1538  wex 1781  wcel 2111  wral 3106  wrex 3107  Vcvv 3441  cun 3879  wss 3881   ciun 4881  cmpt 5110  dom cdm 5519  cres 5521  Oncon0 6159  suc csuc 6161  wf 6320  cfv 6324  Smo wsmo 7965  recscrecs 7990  cfccf 9350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-smo 7966  df-recs 7991  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-card 9352  df-cf 9354  df-acn 9355
This theorem is referenced by:  cfidm  9686  pwcfsdom  9994
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