Theorem axdc 10475

Description: This theorem derives ax-dc 10400 using ax-ac 10413 and ax-inf 9590. 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 5103	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → (𝑢𝑥𝑤 ↔ 𝑢𝑥𝑧))
2	1	cbvabv 2831	. . . . . . . . 9 ⊢ {𝑤 ∣ 𝑢𝑥𝑤} = {𝑧 ∣ 𝑢𝑥𝑧}
3		breq1 5102	. . . . . . . . . 10 ⊢ (𝑢 = 𝑣 → (𝑢𝑥𝑧 ↔ 𝑣𝑥𝑧))
4	3	abbidv 2827	. . . . . . . . 9 ⊢ (𝑢 = 𝑣 → {𝑧 ∣ 𝑢𝑥𝑧} = {𝑧 ∣ 𝑣𝑥𝑧})
5	2, 4	eqtrid 2808	. . . . . . . 8 ⊢ (𝑢 = 𝑣 → {𝑤 ∣ 𝑢𝑥𝑤} = {𝑧 ∣ 𝑣𝑥𝑧})
6	5	fveq2d 6867	. . . . . . 7 ⊢ (𝑢 = 𝑣 → (𝑔‘{𝑤 ∣ 𝑢𝑥𝑤}) = (𝑔‘{𝑧 ∣ 𝑣𝑥𝑧}))
7	6	cbvmptv 5203	. . . . . 6 ⊢ (𝑢 ∈ V ↦ (𝑔‘{𝑤 ∣ 𝑢𝑥𝑤})) = (𝑣 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑣𝑥𝑧}))
8		rdgeq1 8377	. . . . . 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 5959	. . . 4 ⊢ (rec((𝑢 ∈ V ↦ (𝑔‘{𝑤 ∣ 𝑢𝑥𝑤})), 𝑦) ↾ ω) = (rec((𝑣 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑣𝑥𝑧})), 𝑦) ↾ ω)
11	10	axdclem2 10474	. . 3 ⊢ (∃𝑧 𝑦𝑥𝑧 → (ran 𝑥 ⊆ dom 𝑥 → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)))
12	11	exlimiv 1949	. 2 ⊢ (∃𝑦∃𝑧 𝑦𝑥𝑧 → (ran 𝑥 ⊆ dom 𝑥 → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)))
13	12	imp 410	1 ⊢ ((∃𝑦∃𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559  ∃wex 1798  {cab 2739  ∀wral 3075  Vcvv 3453   ⊆ wss 3904   class class class wbr 5099   ↦ cmpt 5180  dom cdm 5645  ran crn 5646   ↾ cres 5647  suc csuc 6344  ‘cfv 6517  ωcom 7842  reccrdg 8375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-inf2 9593  ax-ac2 10417

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-om 7843  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-ac 10069

This theorem is referenced by: (None)