Theorem axdc 10602

Description: This theorem derives ax-dc 10527 using ax-ac 10540 and ax-inf 9642. 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	⊢ ((∃𝑦∃𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)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.

Step	Hyp	Ref	Expression
1		breq2 5106	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → (𝑢𝑥𝑤 ↔ 𝑢𝑥𝑧))
2	1	cbvabv 2830	. . . . . . . . 9 ⊢ {𝑤 ∣ 𝑢𝑥𝑤} = {𝑧 ∣ 𝑢𝑥𝑧}
3		breq1 5105	. . . . . . . . . 10 ⊢ (𝑢 = 𝑣 → (𝑢𝑥𝑧 ↔ 𝑣𝑥𝑧))
4	3	abbidv 2826	. . . . . . . . 9 ⊢ (𝑢 = 𝑣 → {𝑧 ∣ 𝑢𝑥𝑧} = {𝑧 ∣ 𝑣𝑥𝑧})
5	2, 4	eqtrid 2807	. . . . . . . 8 ⊢ (𝑢 = 𝑣 → {𝑤 ∣ 𝑢𝑥𝑤} = {𝑧 ∣ 𝑣𝑥𝑧})
6	5	fveq2d 6889	. . . . . . 7 ⊢ (𝑢 = 𝑣 → (𝑔‘{𝑤 ∣ 𝑢𝑥𝑤}) = (𝑔‘{𝑧 ∣ 𝑣𝑥𝑧}))
7	6	cbvmptv 5208	. . . . . 6 ⊢ (𝑢 ∈ V ↦ (𝑔‘{𝑤 ∣ 𝑢𝑥𝑤})) = (𝑣 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑣𝑥𝑧}))
8		rdgeq1 8422	. . . . . 6 ⊢ ((𝑢 ∈ V ↦ (𝑔‘{𝑤 ∣ 𝑢𝑥𝑤})) = (𝑣 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑣𝑥𝑧})) → rec((𝑢 ∈ V ↦ (𝑔‘{𝑤 ∣ 𝑢𝑥𝑤})), 𝑦) = rec((𝑣 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑣𝑥𝑧})), 𝑦))
9	7, 8	ax-mp 5	. . . . 5 ⊢ rec((𝑢 ∈ V ↦ (𝑔‘{𝑤 ∣ 𝑢𝑥𝑤})), 𝑦) = rec((𝑣 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑣𝑥𝑧})), 𝑦)
10	9	reseq1i 5965	. . . 4 ⊢ (rec((𝑢 ∈ V ↦ (𝑔‘{𝑤 ∣ 𝑢𝑥𝑤})), 𝑦) ↾ ω) = (rec((𝑣 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑣𝑥𝑧})), 𝑦) ↾ ω)
11	10	axdclem2 10601	. . 3 ⊢ (∃𝑧 𝑦𝑥𝑧 → (ran 𝑥 ⊆ dom 𝑥 → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)))
12	11	exlimiv 1962	. 2 ⊢ (∃𝑦∃𝑧 𝑦𝑥𝑧 → (ran 𝑥 ⊆ dom 𝑥 → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)))
13	12	imp 412	1 ⊢ ((∃𝑦∃𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569  ∃wex 1811  {cab 2738  ∀wral 3076  Vcvv 3450   ⊆ wss 3895   class class class wbr 5102   ↦ cmpt 5185  dom cdm 5650  ran crn 5651   ↾ cres 5652  suc csuc 6364  ‘cfv 6538  ωcom 7877  reccrdg 8420

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-inf2 9645  ax-ac2 10544

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-om 7878  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-ac 10198

This theorem is used by: (None)