MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  axdc Structured version   Visualization version   GIF version

Theorem axdc 9936
Description: This theorem derives ax-dc 9861 using ax-ac 9874 and ax-inf 9089. 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 ((∃𝑦𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)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.
StepHypRef Expression
1 breq2 5037 . . . . . . . . 9 (𝑤 = 𝑧 → (𝑢𝑥𝑤𝑢𝑥𝑧))
21cbvabv 2869 . . . . . . . 8 {𝑤𝑢𝑥𝑤} = {𝑧𝑢𝑥𝑧}
3 breq1 5036 . . . . . . . . 9 (𝑢 = 𝑣 → (𝑢𝑥𝑧𝑣𝑥𝑧))
43abbidv 2865 . . . . . . . 8 (𝑢 = 𝑣 → {𝑧𝑢𝑥𝑧} = {𝑧𝑣𝑥𝑧})
52, 4syl5eq 2848 . . . . . . 7 (𝑢 = 𝑣 → {𝑤𝑢𝑥𝑤} = {𝑧𝑣𝑥𝑧})
65fveq2d 6653 . . . . . 6 (𝑢 = 𝑣 → (𝑔‘{𝑤𝑢𝑥𝑤}) = (𝑔‘{𝑧𝑣𝑥𝑧}))
76cbvmptv 5136 . . . . 5 (𝑢 ∈ V ↦ (𝑔‘{𝑤𝑢𝑥𝑤})) = (𝑣 ∈ V ↦ (𝑔‘{𝑧𝑣𝑥𝑧}))
8 rdgeq1 8034 . . . . 5 ((𝑢 ∈ V ↦ (𝑔‘{𝑤𝑢𝑥𝑤})) = (𝑣 ∈ V ↦ (𝑔‘{𝑧𝑣𝑥𝑧})) → rec((𝑢 ∈ V ↦ (𝑔‘{𝑤𝑢𝑥𝑤})), 𝑦) = rec((𝑣 ∈ V ↦ (𝑔‘{𝑧𝑣𝑥𝑧})), 𝑦))
9 reseq1 5816 . . . . 5 (rec((𝑢 ∈ V ↦ (𝑔‘{𝑤𝑢𝑥𝑤})), 𝑦) = rec((𝑣 ∈ V ↦ (𝑔‘{𝑧𝑣𝑥𝑧})), 𝑦) → (rec((𝑢 ∈ V ↦ (𝑔‘{𝑤𝑢𝑥𝑤})), 𝑦) ↾ ω) = (rec((𝑣 ∈ V ↦ (𝑔‘{𝑧𝑣𝑥𝑧})), 𝑦) ↾ ω))
107, 8, 9mp2b 10 . . . 4 (rec((𝑢 ∈ V ↦ (𝑔‘{𝑤𝑢𝑥𝑤})), 𝑦) ↾ ω) = (rec((𝑣 ∈ V ↦ (𝑔‘{𝑧𝑣𝑥𝑧})), 𝑦) ↾ ω)
1110axdclem2 9935 . . 3 (∃𝑧 𝑦𝑥𝑧 → (ran 𝑥 ⊆ dom 𝑥 → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)))
1211exlimiv 1931 . 2 (∃𝑦𝑧 𝑦𝑥𝑧 → (ran 𝑥 ⊆ dom 𝑥 → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)))
1312imp 410 1 ((∃𝑦𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wex 1781  {cab 2779  wral 3109  Vcvv 3444  wss 3884   class class class wbr 5033  cmpt 5113  dom cdm 5523  ran crn 5524  cres 5525  suc csuc 6165  cfv 6328  ωcom 7564  reccrdg 8032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092  ax-ac2 9878
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-om 7565  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-ac 9531
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator