Theorem axdc 10513

Description: This theorem derives ax-dc 10438 using ax-ac 10451 and ax-inf 9630. Thus, AC implies DC, but not vice-versa (so that ZFC is strictly stronger than ZF+DC). (New usage is discouraged.) (Contributed by Mario Carneiro, 25-Jan-2013.)

Assertion

Ref	Expression
axdc	⊢ ((∃𝑦∃𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛))

Distinct variable group:   𝑓,𝑛,𝑥,𝑦,𝑧

Proof of Theorem axdc

Dummy variables 𝑣 𝑔 𝑢 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 5143	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (𝑢𝑥𝑤 ↔ 𝑢𝑥𝑧))
2	1	cbvabv 2797	. . . . . . . 8 ⊢ {𝑤 ∣ 𝑢𝑥𝑤} = {𝑧 ∣ 𝑢𝑥𝑧}
3		breq1 5142	. . . . . . . . 9 ⊢ (𝑢 = 𝑣 → (𝑢𝑥𝑧 ↔ 𝑣𝑥𝑧))
4	3	abbidv 2793	. . . . . . . 8 ⊢ (𝑢 = 𝑣 → {𝑧 ∣ 𝑢𝑥𝑧} = {𝑧 ∣ 𝑣𝑥𝑧})
5	2, 4	eqtrid 2776	. . . . . . 7 ⊢ (𝑢 = 𝑣 → {𝑤 ∣ 𝑢𝑥𝑤} = {𝑧 ∣ 𝑣𝑥𝑧})
6	5	fveq2d 6886	. . . . . 6 ⊢ (𝑢 = 𝑣 → (𝑔‘{𝑤 ∣ 𝑢𝑥𝑤}) = (𝑔‘{𝑧 ∣ 𝑣𝑥𝑧}))
7	6	cbvmptv 5252	. . . . 5 ⊢ (𝑢 ∈ V ↦ (𝑔‘{𝑤 ∣ 𝑢𝑥𝑤})) = (𝑣 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑣𝑥𝑧}))
8		rdgeq1 8407	. . . . 5 ⊢ ((𝑢 ∈ V ↦ (𝑔‘{𝑤 ∣ 𝑢𝑥𝑤})) = (𝑣 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑣𝑥𝑧})) → rec((𝑢 ∈ V ↦ (𝑔‘{𝑤 ∣ 𝑢𝑥𝑤})), 𝑦) = rec((𝑣 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑣𝑥𝑧})), 𝑦))
9		reseq1 5966	. . . . 5 ⊢ (rec((𝑢 ∈ V ↦ (𝑔‘{𝑤 ∣ 𝑢𝑥𝑤})), 𝑦) = rec((𝑣 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑣𝑥𝑧})), 𝑦) → (rec((𝑢 ∈ V ↦ (𝑔‘{𝑤 ∣ 𝑢𝑥𝑤})), 𝑦) ↾ ω) = (rec((𝑣 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑣𝑥𝑧})), 𝑦) ↾ ω))
10	7, 8, 9	mp2b 10	. . . 4 ⊢ (rec((𝑢 ∈ V ↦ (𝑔‘{𝑤 ∣ 𝑢𝑥𝑤})), 𝑦) ↾ ω) = (rec((𝑣 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑣𝑥𝑧})), 𝑦) ↾ ω)
11	10	axdclem2 10512	. . 3 ⊢ (∃𝑧 𝑦𝑥𝑧 → (ran 𝑥 ⊆ dom 𝑥 → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)))
12	11	exlimiv 1925	. 2 ⊢ (∃𝑦∃𝑧 𝑦𝑥𝑧 → (ran 𝑥 ⊆ dom 𝑥 → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)))
13	12	imp 406	1 ⊢ ((∃𝑦∃𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533  ∃wex 1773  {cab 2701  ∀wral 3053  Vcvv 3466   ⊆ wss 3941   class class class wbr 5139   ↦ cmpt 5222  dom cdm 5667  ran crn 5668   ↾ cres 5669  suc csuc 6357  ‘cfv 6534  ωcom 7849  reccrdg 8405

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-inf2 9633  ax-ac2 10455

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-om 7850  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-ac 10108

This theorem is referenced by: (None)