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Theorem cfsmo 9078
Description: The map in cff1 9065 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 5313 . . . . 5 (𝑥 = 𝑧 → dom 𝑥 = dom 𝑧)
21fveq2d 6182 . . . 4 (𝑥 = 𝑧 → (‘dom 𝑥) = (‘dom 𝑧))
3 fveq2 6178 . . . . . . 7 (𝑛 = 𝑚 → (𝑥𝑛) = (𝑥𝑚))
4 suceq 5778 . . . . . . 7 ((𝑥𝑛) = (𝑥𝑚) → suc (𝑥𝑛) = suc (𝑥𝑚))
53, 4syl 17 . . . . . 6 (𝑛 = 𝑚 → suc (𝑥𝑛) = suc (𝑥𝑚))
65cbviunv 4550 . . . . 5 𝑛 ∈ dom 𝑥 suc (𝑥𝑛) = 𝑚 ∈ dom 𝑥 suc (𝑥𝑚)
7 fveq1 6177 . . . . . . 7 (𝑥 = 𝑧 → (𝑥𝑚) = (𝑧𝑚))
8 suceq 5778 . . . . . . 7 ((𝑥𝑚) = (𝑧𝑚) → suc (𝑥𝑚) = suc (𝑧𝑚))
97, 8syl 17 . . . . . 6 (𝑥 = 𝑧 → suc (𝑥𝑚) = suc (𝑧𝑚))
101, 9iuneq12d 4537 . . . . 5 (𝑥 = 𝑧 𝑚 ∈ dom 𝑥 suc (𝑥𝑚) = 𝑚 ∈ dom 𝑧 suc (𝑧𝑚))
116, 10syl5eq 2666 . . . 4 (𝑥 = 𝑧 𝑛 ∈ dom 𝑥 suc (𝑥𝑛) = 𝑚 ∈ dom 𝑧 suc (𝑧𝑚))
122, 11uneq12d 3760 . . 3 (𝑥 = 𝑧 → ((‘dom 𝑥) ∪ 𝑛 ∈ dom 𝑥 suc (𝑥𝑛)) = ((‘dom 𝑧) ∪ 𝑚 ∈ dom 𝑧 suc (𝑧𝑚)))
1312cbvmptv 4741 . 2 (𝑥 ∈ V ↦ ((‘dom 𝑥) ∪ 𝑛 ∈ dom 𝑥 suc (𝑥𝑛))) = (𝑧 ∈ V ↦ ((‘dom 𝑧) ∪ 𝑚 ∈ dom 𝑧 suc (𝑧𝑚)))
14 eqid 2620 . 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 9077 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 1036   = wceq 1481  wex 1702  wcel 1988  wral 2909  wrex 2910  Vcvv 3195  cun 3565  wss 3567   ciun 4511  cmpt 4720  dom cdm 5104  cres 5106  Oncon0 5711  suc csuc 5713  wf 5872  cfv 5876  Smo wsmo 7427  recscrecs 7452  cfccf 8748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-smo 7428  df-recs 7453  df-er 7727  df-map 7844  df-en 7941  df-dom 7942  df-sdom 7943  df-card 8750  df-cf 8752  df-acn 8753
This theorem is referenced by:  cfidm  9082  pwcfsdom  9390
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