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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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