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Theorem cfsmo 8976
Description: The map in cff1 8963 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 5246 . . . . 5 (𝑥 = 𝑧 → dom 𝑥 = dom 𝑧)
21fveq2d 6107 . . . 4 (𝑥 = 𝑧 → (‘dom 𝑥) = (‘dom 𝑧))
3 fveq2 6103 . . . . . . 7 (𝑛 = 𝑚 → (𝑥𝑛) = (𝑥𝑚))
4 suceq 5707 . . . . . . 7 ((𝑥𝑛) = (𝑥𝑚) → suc (𝑥𝑛) = suc (𝑥𝑚))
53, 4syl 17 . . . . . 6 (𝑛 = 𝑚 → suc (𝑥𝑛) = suc (𝑥𝑚))
65cbviunv 4495 . . . . 5 𝑛 ∈ dom 𝑥 suc (𝑥𝑛) = 𝑚 ∈ dom 𝑥 suc (𝑥𝑚)
7 fveq1 6102 . . . . . . 7 (𝑥 = 𝑧 → (𝑥𝑚) = (𝑧𝑚))
8 suceq 5707 . . . . . . 7 ((𝑥𝑚) = (𝑧𝑚) → suc (𝑥𝑚) = suc (𝑧𝑚))
97, 8syl 17 . . . . . 6 (𝑥 = 𝑧 → suc (𝑥𝑚) = suc (𝑧𝑚))
101, 9iuneq12d 4482 . . . . 5 (𝑥 = 𝑧 𝑚 ∈ dom 𝑥 suc (𝑥𝑚) = 𝑚 ∈ dom 𝑧 suc (𝑧𝑚))
116, 10syl5eq 2656 . . . 4 (𝑥 = 𝑧 𝑛 ∈ dom 𝑥 suc (𝑥𝑛) = 𝑚 ∈ dom 𝑧 suc (𝑧𝑚))
122, 11uneq12d 3730 . . 3 (𝑥 = 𝑧 → ((‘dom 𝑥) ∪ 𝑛 ∈ dom 𝑥 suc (𝑥𝑛)) = ((‘dom 𝑧) ∪ 𝑚 ∈ dom 𝑧 suc (𝑧𝑚)))
1312cbvmptv 4678 . 2 (𝑥 ∈ V ↦ ((‘dom 𝑥) ∪ 𝑛 ∈ dom 𝑥 suc (𝑥𝑛))) = (𝑧 ∈ V ↦ ((‘dom 𝑧) ∪ 𝑚 ∈ dom 𝑧 suc (𝑧𝑚)))
14 eqid 2610 . 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 8975 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 1031   = wceq 1475  wex 1695  wcel 1977  wral 2896  wrex 2897  Vcvv 3173  cun 3538  wss 3540   ciun 4455  cmpt 4643  dom cdm 5038  cres 5040  Oncon0 5640  suc csuc 5642  wf 5800  cfv 5804  Smo wsmo 7329  recscrecs 7354  cfccf 8646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-smo 7330  df-recs 7355  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-card 8648  df-cf 8650  df-acn 8651
This theorem is referenced by:  cfidm  8980  pwcfsdom  9284
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