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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)
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