Theorem axdc 9343

Description: This theorem derives ax-dc 9268 using ax-ac 9281 and ax-inf 8535. 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 4657	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (𝑢𝑥𝑤 ↔ 𝑢𝑥𝑧))
2	1	cbvabv 2747	. . . . . . . 8 ⊢ {𝑤 ∣ 𝑢𝑥𝑤} = {𝑧 ∣ 𝑢𝑥𝑧}
3		breq1 4656	. . . . . . . . 9 ⊢ (𝑢 = 𝑣 → (𝑢𝑥𝑧 ↔ 𝑣𝑥𝑧))
4	3	abbidv 2741	. . . . . . . 8 ⊢ (𝑢 = 𝑣 → {𝑧 ∣ 𝑢𝑥𝑧} = {𝑧 ∣ 𝑣𝑥𝑧})
5	2, 4	syl5eq 2668	. . . . . . 7 ⊢ (𝑢 = 𝑣 → {𝑤 ∣ 𝑢𝑥𝑤} = {𝑧 ∣ 𝑣𝑥𝑧})
6	5	fveq2d 6195	. . . . . 6 ⊢ (𝑢 = 𝑣 → (𝑔‘{𝑤 ∣ 𝑢𝑥𝑤}) = (𝑔‘{𝑧 ∣ 𝑣𝑥𝑧}))
7	6	cbvmptv 4750	. . . . 5 ⊢ (𝑢 ∈ V ↦ (𝑔‘{𝑤 ∣ 𝑢𝑥𝑤})) = (𝑣 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑣𝑥𝑧}))
8		rdgeq1 7507	. . . . 5 ⊢ ((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		reseq1 5390	. . . . 5 ⊢ (rec((𝑢 ∈ V ↦ (𝑔‘{𝑤 ∣ 𝑢𝑥𝑤})), 𝑦) = rec((𝑣 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑣𝑥𝑧})), 𝑦) → (rec((𝑢 ∈ V ↦ (𝑔‘{𝑤 ∣ 𝑢𝑥𝑤})), 𝑦) ↾ ω) = (rec((𝑣 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑣𝑥𝑧})), 𝑦) ↾ ω))
10	7, 8, 9	mp2b 10	. . . 4 ⊢ (rec((𝑢 ∈ V ↦ (𝑔‘{𝑤 ∣ 𝑢𝑥𝑤})), 𝑦) ↾ ω) = (rec((𝑣 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑣𝑥𝑧})), 𝑦) ↾ ω)
11	10	axdclem2 9342	. . 3 ⊢ (∃𝑧 𝑦𝑥𝑧 → (ran 𝑥 ⊆ dom 𝑥 → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)))
12	11	exlimiv 1858	. 2 ⊢ (∃𝑦∃𝑧 𝑦𝑥𝑧 → (ran 𝑥 ⊆ dom 𝑥 → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)))
13	12	imp 445	1 ⊢ ((∃𝑦∃𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483  ∃wex 1704  {cab 2608  ∀wral 2912  Vcvv 3200   ⊆ wss 3574   class class class wbr 4653   ↦ cmpt 4729  dom cdm 5114  ran crn 5115   ↾ cres 5116  suc csuc 5725  ‘cfv 5888  ωcom 7065  reccrdg 7505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-ac2 9285

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-ac 8939

This theorem is referenced by: (None)