Theorem cfsmo 9693

Description: The map in cff1 9680 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 5772	. . . . 5 ⊢ (𝑥 = 𝑧 → dom 𝑥 = dom 𝑧)
2	1	fveq2d 6674	. . . 4 ⊢ (𝑥 = 𝑧 → (ℎ‘dom 𝑥) = (ℎ‘dom 𝑧))
3		fveq2 6670	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝑥‘𝑛) = (𝑥‘𝑚))
4		suceq 6256	. . . . . . 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 4965	. . . . 5 ⊢ ∪ 𝑛 ∈ dom 𝑥 suc (𝑥‘𝑛) = ∪ 𝑚 ∈ dom 𝑥 suc (𝑥‘𝑚)
7		fveq1 6669	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥‘𝑚) = (𝑧‘𝑚))
8		suceq 6256	. . . . . . 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 4947	. . . . 5 ⊢ (𝑥 = 𝑧 → ∪ 𝑚 ∈ dom 𝑥 suc (𝑥‘𝑚) = ∪ 𝑚 ∈ dom 𝑧 suc (𝑧‘𝑚))
11	6, 10	syl5eq 2868	. . . 4 ⊢ (𝑥 = 𝑧 → ∪ 𝑛 ∈ dom 𝑥 suc (𝑥‘𝑛) = ∪ 𝑚 ∈ dom 𝑧 suc (𝑧‘𝑚))
12	2, 11	uneq12d 4140	. . 3 ⊢ (𝑥 = 𝑧 → ((ℎ‘dom 𝑥) ∪ ∪ 𝑛 ∈ dom 𝑥 suc (𝑥‘𝑛)) = ((ℎ‘dom 𝑧) ∪ ∪ 𝑚 ∈ dom 𝑧 suc (𝑧‘𝑚)))
13	12	cbvmptv 5169	. 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 2821	. 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 9692	1 ⊢ (𝐴 ∈ On → ∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓‘𝑤)))

Syntax hints:   → wi 4   ∧ w3a 1083   = wceq 1537  ∃wex 1780   ∈ wcel 2114  ∀wral 3138  ∃wrex 3139  Vcvv 3494   ∪ cun 3934   ⊆ wss 3936  ∪ ciun 4919   ↦ cmpt 5146  dom cdm 5555   ↾ cres 5557  Oncon0 6191  suc csuc 6193  ⟶wf 6351  ‘cfv 6355  Smo wsmo 7982  recscrecs 8007  cfccf 9366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-se 5515  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-isom 6364  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-smo 7983  df-recs 8008  df-er 8289  df-map 8408  df-en 8510  df-dom 8511  df-sdom 8512  df-card 9368  df-cf 9370  df-acn 9371

This theorem is referenced by:  cfidm  9697  pwcfsdom  10005