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Theorem axdc 9735
Description: This theorem derives ax-dc 9660 using ax-ac 9673 and ax-inf 8889. 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 4927 . . . . . . . . 9 (𝑤 = 𝑧 → (𝑢𝑥𝑤𝑢𝑥𝑧))
21cbvabv 2904 . . . . . . . 8 {𝑤𝑢𝑥𝑤} = {𝑧𝑢𝑥𝑧}
3 breq1 4926 . . . . . . . . 9 (𝑢 = 𝑣 → (𝑢𝑥𝑧𝑣𝑥𝑧))
43abbidv 2837 . . . . . . . 8 (𝑢 = 𝑣 → {𝑧𝑢𝑥𝑧} = {𝑧𝑣𝑥𝑧})
52, 4syl5eq 2820 . . . . . . 7 (𝑢 = 𝑣 → {𝑤𝑢𝑥𝑤} = {𝑧𝑣𝑥𝑧})
65fveq2d 6497 . . . . . 6 (𝑢 = 𝑣 → (𝑔‘{𝑤𝑢𝑥𝑤}) = (𝑔‘{𝑧𝑣𝑥𝑧}))
76cbvmptv 5022 . . . . 5 (𝑢 ∈ V ↦ (𝑔‘{𝑤𝑢𝑥𝑤})) = (𝑣 ∈ V ↦ (𝑔‘{𝑧𝑣𝑥𝑧}))
8 rdgeq1 7845 . . . . 5 ((𝑢 ∈ V ↦ (𝑔‘{𝑤𝑢𝑥𝑤})) = (𝑣 ∈ V ↦ (𝑔‘{𝑧𝑣𝑥𝑧})) → rec((𝑢 ∈ V ↦ (𝑔‘{𝑤𝑢𝑥𝑤})), 𝑦) = rec((𝑣 ∈ V ↦ (𝑔‘{𝑧𝑣𝑥𝑧})), 𝑦))
9 reseq1 5683 . . . . 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 9734 . . 3 (∃𝑧 𝑦𝑥𝑧 → (ran 𝑥 ⊆ dom 𝑥 → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)))
1211exlimiv 1889 . 2 (∃𝑦𝑧 𝑦𝑥𝑧 → (ran 𝑥 ⊆ dom 𝑥 → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)))
1312imp 398 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 387   = wceq 1507  wex 1742  {cab 2752  wral 3082  Vcvv 3409  wss 3823   class class class wbr 4923  cmpt 5002  dom cdm 5401  ran crn 5402  cres 5403  suc csuc 6025  cfv 6182  ωcom 7390  reccrdg 7843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-inf2 8892  ax-ac2 9677
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-pss 3839  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5306  df-eprel 5311  df-po 5320  df-so 5321  df-fr 5360  df-we 5362  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-om 7391  df-wrecs 7744  df-recs 7806  df-rdg 7844  df-ac 9330
This theorem is referenced by: (None)
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