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Theorem axdc 10559
Description: This theorem derives ax-dc 10484 using ax-ac 10497 and ax-inf 9676. 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 5152 . . . . . . . . 9 (𝑤 = 𝑧 → (𝑢𝑥𝑤𝑢𝑥𝑧))
21cbvabv 2810 . . . . . . . 8 {𝑤𝑢𝑥𝑤} = {𝑧𝑢𝑥𝑧}
3 breq1 5151 . . . . . . . . 9 (𝑢 = 𝑣 → (𝑢𝑥𝑧𝑣𝑥𝑧))
43abbidv 2806 . . . . . . . 8 (𝑢 = 𝑣 → {𝑧𝑢𝑥𝑧} = {𝑧𝑣𝑥𝑧})
52, 4eqtrid 2787 . . . . . . 7 (𝑢 = 𝑣 → {𝑤𝑢𝑥𝑤} = {𝑧𝑣𝑥𝑧})
65fveq2d 6911 . . . . . 6 (𝑢 = 𝑣 → (𝑔‘{𝑤𝑢𝑥𝑤}) = (𝑔‘{𝑧𝑣𝑥𝑧}))
76cbvmptv 5261 . . . . 5 (𝑢 ∈ V ↦ (𝑔‘{𝑤𝑢𝑥𝑤})) = (𝑣 ∈ V ↦ (𝑔‘{𝑧𝑣𝑥𝑧}))
8 rdgeq1 8450 . . . . 5 ((𝑢 ∈ V ↦ (𝑔‘{𝑤𝑢𝑥𝑤})) = (𝑣 ∈ V ↦ (𝑔‘{𝑧𝑣𝑥𝑧})) → rec((𝑢 ∈ V ↦ (𝑔‘{𝑤𝑢𝑥𝑤})), 𝑦) = rec((𝑣 ∈ V ↦ (𝑔‘{𝑧𝑣𝑥𝑧})), 𝑦))
9 reseq1 5994 . . . . 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 10558 . . 3 (∃𝑧 𝑦𝑥𝑧 → (ran 𝑥 ⊆ dom 𝑥 → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)))
1211exlimiv 1928 . 2 (∃𝑦𝑧 𝑦𝑥𝑧 → (ran 𝑥 ⊆ dom 𝑥 → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)))
1312imp 406 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 395   = wceq 1537  wex 1776  {cab 2712  wral 3059  Vcvv 3478  wss 3963   class class class wbr 5148  cmpt 5231  dom cdm 5689  ran crn 5690  cres 5691  suc csuc 6388  cfv 6563  ωcom 7887  reccrdg 8448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-ac2 10501
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-ac 10154
This theorem is referenced by: (None)
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