Theorem cfsmo 9489

Description: The map in cff1 9476 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 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓‘𝑤)))

Distinct variable group:   𝐴,𝑓,𝑤,𝑧

Proof of Theorem cfsmo

Dummy variables 𝑚 ℎ 𝑥 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmeq 5618	. . . . 5 ⊢ (𝑥 = 𝑧 → dom 𝑥 = dom 𝑧)
2	1	fveq2d 6500	. . . 4 ⊢ (𝑥 = 𝑧 → (ℎ‘dom 𝑥) = (ℎ‘dom 𝑧))
3		fveq2 6496	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝑥‘𝑛) = (𝑥‘𝑚))
4		suceq 6091	. . . . . . 7 ⊢ ((𝑥‘𝑛) = (𝑥‘𝑚) → suc (𝑥‘𝑛) = suc (𝑥‘𝑚))
5	3, 4	syl 17	. . . . . 6 ⊢ (𝑛 = 𝑚 → suc (𝑥‘𝑛) = suc (𝑥‘𝑚))
6	5	cbviunv 4829	. . . . 5 ⊢ ∪ 𝑛 ∈ dom 𝑥 suc (𝑥‘𝑛) = ∪ 𝑚 ∈ dom 𝑥 suc (𝑥‘𝑚)
7		fveq1 6495	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥‘𝑚) = (𝑧‘𝑚))
8		suceq 6091	. . . . . . 7 ⊢ ((𝑥‘𝑚) = (𝑧‘𝑚) → suc (𝑥‘𝑚) = suc (𝑧‘𝑚))
9	7, 8	syl 17	. . . . . 6 ⊢ (𝑥 = 𝑧 → suc (𝑥‘𝑚) = suc (𝑧‘𝑚))
10	1, 9	iuneq12d 4815	. . . . 5 ⊢ (𝑥 = 𝑧 → ∪ 𝑚 ∈ dom 𝑥 suc (𝑥‘𝑚) = ∪ 𝑚 ∈ dom 𝑧 suc (𝑧‘𝑚))
11	6, 10	syl5eq 2820	. . . 4 ⊢ (𝑥 = 𝑧 → ∪ 𝑛 ∈ dom 𝑥 suc (𝑥‘𝑛) = ∪ 𝑚 ∈ dom 𝑧 suc (𝑧‘𝑚))
12	2, 11	uneq12d 4023	. . 3 ⊢ (𝑥 = 𝑧 → ((ℎ‘dom 𝑥) ∪ ∪ 𝑛 ∈ dom 𝑥 suc (𝑥‘𝑛)) = ((ℎ‘dom 𝑧) ∪ ∪ 𝑚 ∈ dom 𝑧 suc (𝑧‘𝑚)))
13	12	cbvmptv 5024	. 2 ⊢ (𝑥 ∈ V ↦ ((ℎ‘dom 𝑥) ∪ ∪ 𝑛 ∈ dom 𝑥 suc (𝑥‘𝑛))) = (𝑧 ∈ V ↦ ((ℎ‘dom 𝑧) ∪ ∪ 𝑚 ∈ dom 𝑧 suc (𝑧‘𝑚)))
14		eqid 2772	. 2 ⊢ (recs((𝑥 ∈ V ↦ ((ℎ‘dom 𝑥) ∪ ∪ 𝑛 ∈ dom 𝑥 suc (𝑥‘𝑛)))) ↾ (cf‘𝐴)) = (recs((𝑥 ∈ V ↦ ((ℎ‘dom 𝑥) ∪ ∪ 𝑛 ∈ dom 𝑥 suc (𝑥‘𝑛)))) ↾ (cf‘𝐴))
15	13, 14	cfsmolem 9488	1 ⊢ (𝐴 ∈ On → ∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓‘𝑤)))

Syntax hints:   → wi 4   ∧ w3a 1068   = wceq 1507  ∃wex 1742   ∈ wcel 2050  ∀wral 3082  ∃wrex 3083  Vcvv 3409   ∪ cun 3821   ⊆ wss 3823  ∪ ciun 4788   ↦ cmpt 5004  dom cdm 5403   ↾ cres 5405  Oncon0 6026  suc csuc 6028  ⟶wf 6181  ‘cfv 6185  Smo wsmo 7784  recscrecs 7809  cfccf 9158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-pss 3839  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-int 4746  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-se 5363  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-pred 5983  df-ord 6029  df-on 6030  df-suc 6032  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-isom 6194  df-riota 6935  df-ov 6977  df-oprab 6978  df-mpo 6979  df-1st 7499  df-2nd 7500  df-wrecs 7748  df-smo 7785  df-recs 7810  df-er 8087  df-map 8206  df-en 8305  df-dom 8306  df-sdom 8307  df-card 9160  df-cf 9162  df-acn 9163

This theorem is referenced by:  cfidm  9493  pwcfsdom  9801