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Theorem axdc 10458
Description: This theorem derives ax-dc 10383 using ax-ac 10396 and ax-inf 9575. 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 5110 . . . . . . . . 9 (𝑤 = 𝑧 → (𝑢𝑥𝑤𝑢𝑥𝑧))
21cbvabv 2810 . . . . . . . 8 {𝑤𝑢𝑥𝑤} = {𝑧𝑢𝑥𝑧}
3 breq1 5109 . . . . . . . . 9 (𝑢 = 𝑣 → (𝑢𝑥𝑧𝑣𝑥𝑧))
43abbidv 2806 . . . . . . . 8 (𝑢 = 𝑣 → {𝑧𝑢𝑥𝑧} = {𝑧𝑣𝑥𝑧})
52, 4eqtrid 2789 . . . . . . 7 (𝑢 = 𝑣 → {𝑤𝑢𝑥𝑤} = {𝑧𝑣𝑥𝑧})
65fveq2d 6847 . . . . . 6 (𝑢 = 𝑣 → (𝑔‘{𝑤𝑢𝑥𝑤}) = (𝑔‘{𝑧𝑣𝑥𝑧}))
76cbvmptv 5219 . . . . 5 (𝑢 ∈ V ↦ (𝑔‘{𝑤𝑢𝑥𝑤})) = (𝑣 ∈ V ↦ (𝑔‘{𝑧𝑣𝑥𝑧}))
8 rdgeq1 8358 . . . . 5 ((𝑢 ∈ V ↦ (𝑔‘{𝑤𝑢𝑥𝑤})) = (𝑣 ∈ V ↦ (𝑔‘{𝑧𝑣𝑥𝑧})) → rec((𝑢 ∈ V ↦ (𝑔‘{𝑤𝑢𝑥𝑤})), 𝑦) = rec((𝑣 ∈ V ↦ (𝑔‘{𝑧𝑣𝑥𝑧})), 𝑦))
9 reseq1 5932 . . . . 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 10457 . . 3 (∃𝑧 𝑦𝑥𝑧 → (ran 𝑥 ⊆ dom 𝑥 → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)))
1211exlimiv 1934 . 2 (∃𝑦𝑧 𝑦𝑥𝑧 → (ran 𝑥 ⊆ dom 𝑥 → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)))
1312imp 408 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 397   = wceq 1542  wex 1782  {cab 2714  wral 3065  Vcvv 3446  wss 3911   class class class wbr 5106  cmpt 5189  dom cdm 5634  ran crn 5635  cres 5636  suc csuc 6320  cfv 6497  ωcom 7803  reccrdg 8356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-inf2 9578  ax-ac2 10400
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-om 7804  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-ac 10053
This theorem is referenced by: (None)
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