Theorem axdclem 10275

Description: Lemma for axdc 10277. (Contributed by Mario Carneiro, 25-Jan-2013.)

Hypothesis

Ref	Expression
axdclem.1	⊢ 𝐹 = (rec((𝑦 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑦𝑥𝑧})), 𝑠) ↾ ω)

Assertion

Ref	Expression
axdclem	⊢ ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥 ∧ ∃𝑧(𝐹‘𝐾)𝑥𝑧) → (𝐾 ∈ ω → (𝐹‘𝐾)𝑥(𝐹‘suc 𝐾)))

Distinct variable groups:   𝑦,𝐹,𝑧   𝑦,𝐾,𝑧   𝑦,𝑔   𝑦,𝑠   𝑥,𝑦,𝑧

Allowed substitution hints:   𝐹(𝑥,𝑔,𝑠)   𝐾(𝑥,𝑔,𝑠)

Proof of Theorem axdclem

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		neeq1 3006	. . . . . . 7 ⊢ (𝑦 = {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} → (𝑦 ≠ ∅ ↔ {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} ≠ ∅))
2		abn0 4314	. . . . . . 7 ⊢ ({𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} ≠ ∅ ↔ ∃𝑧(𝐹‘𝐾)𝑥𝑧)
3	1, 2	bitrdi 287	. . . . . 6 ⊢ (𝑦 = {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} → (𝑦 ≠ ∅ ↔ ∃𝑧(𝐹‘𝐾)𝑥𝑧))
4		eleq2 2827	. . . . . . . . 9 ⊢ (𝑦 = {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} → ((𝑔‘𝑦) ∈ 𝑦 ↔ (𝑔‘𝑦) ∈ {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧}))
5		breq2 5078	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → ((𝐹‘𝐾)𝑥𝑤 ↔ (𝐹‘𝐾)𝑥𝑧))
6	5	cbvabv 2811	. . . . . . . . . 10 ⊢ {𝑤 ∣ (𝐹‘𝐾)𝑥𝑤} = {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧}
7	6	eleq2i 2830	. . . . . . . . 9 ⊢ ((𝑔‘𝑦) ∈ {𝑤 ∣ (𝐹‘𝐾)𝑥𝑤} ↔ (𝑔‘𝑦) ∈ {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧})
8	4, 7	bitr4di 289	. . . . . . . 8 ⊢ (𝑦 = {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} → ((𝑔‘𝑦) ∈ 𝑦 ↔ (𝑔‘𝑦) ∈ {𝑤 ∣ (𝐹‘𝐾)𝑥𝑤}))
9		fvex 6787	. . . . . . . . 9 ⊢ (𝑔‘𝑦) ∈ V
10		breq2 5078	. . . . . . . . 9 ⊢ (𝑤 = (𝑔‘𝑦) → ((𝐹‘𝐾)𝑥𝑤 ↔ (𝐹‘𝐾)𝑥(𝑔‘𝑦)))
11	9, 10	elab 3609	. . . . . . . 8 ⊢ ((𝑔‘𝑦) ∈ {𝑤 ∣ (𝐹‘𝐾)𝑥𝑤} ↔ (𝐹‘𝐾)𝑥(𝑔‘𝑦))
12	8, 11	bitrdi 287	. . . . . . 7 ⊢ (𝑦 = {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} → ((𝑔‘𝑦) ∈ 𝑦 ↔ (𝐹‘𝐾)𝑥(𝑔‘𝑦)))
13		fveq2 6774	. . . . . . . 8 ⊢ (𝑦 = {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} → (𝑔‘𝑦) = (𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧}))
14	13	breq2d 5086	. . . . . . 7 ⊢ (𝑦 = {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} → ((𝐹‘𝐾)𝑥(𝑔‘𝑦) ↔ (𝐹‘𝐾)𝑥(𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧})))
15	12, 14	bitrd 278	. . . . . 6 ⊢ (𝑦 = {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} → ((𝑔‘𝑦) ∈ 𝑦 ↔ (𝐹‘𝐾)𝑥(𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧})))
16	3, 15	imbi12d 345	. . . . 5 ⊢ (𝑦 = {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} → ((𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ↔ (∃𝑧(𝐹‘𝐾)𝑥𝑧 → (𝐹‘𝐾)𝑥(𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧}))))
17	16	rspcv 3557	. . . 4 ⊢ ({𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} ∈ 𝒫 dom 𝑥 → (∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) → (∃𝑧(𝐹‘𝐾)𝑥𝑧 → (𝐹‘𝐾)𝑥(𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧}))))
18		fvex 6787	. . . . . . . 8 ⊢ (𝐹‘𝐾) ∈ V
19		vex 3436	. . . . . . . 8 ⊢ 𝑧 ∈ V
20	18, 19	brelrn 5851	. . . . . . 7 ⊢ ((𝐹‘𝐾)𝑥𝑧 → 𝑧 ∈ ran 𝑥)
21	20	abssi 4003	. . . . . 6 ⊢ {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} ⊆ ran 𝑥
22		sstr 3929	. . . . . 6 ⊢ (({𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} ⊆ ran 𝑥 ∧ ran 𝑥 ⊆ dom 𝑥) → {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} ⊆ dom 𝑥)
23	21, 22	mpan 687	. . . . 5 ⊢ (ran 𝑥 ⊆ dom 𝑥 → {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} ⊆ dom 𝑥)
24		vex 3436	. . . . . . 7 ⊢ 𝑥 ∈ V
25	24	dmex 7758	. . . . . 6 ⊢ dom 𝑥 ∈ V
26	25	elpw2 5269	. . . . 5 ⊢ ({𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} ∈ 𝒫 dom 𝑥 ↔ {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} ⊆ dom 𝑥)
27	23, 26	sylibr 233	. . . 4 ⊢ (ran 𝑥 ⊆ dom 𝑥 → {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} ∈ 𝒫 dom 𝑥)
28	17, 27	syl11 33	. . 3 ⊢ (∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) → (ran 𝑥 ⊆ dom 𝑥 → (∃𝑧(𝐹‘𝐾)𝑥𝑧 → (𝐹‘𝐾)𝑥(𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧}))))
29	28	3imp 1110	. 2 ⊢ ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥 ∧ ∃𝑧(𝐹‘𝐾)𝑥𝑧) → (𝐹‘𝐾)𝑥(𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧}))
30		fvex 6787	. . . 4 ⊢ (𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧}) ∈ V
31		nfcv 2907	. . . . 5 ⊢ Ⅎ𝑦𝑠
32		nfcv 2907	. . . . 5 ⊢ Ⅎ𝑦𝐾
33		nfcv 2907	. . . . 5 ⊢ Ⅎ𝑦(𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧})
34		axdclem.1	. . . . 5 ⊢ 𝐹 = (rec((𝑦 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑦𝑥𝑧})), 𝑠) ↾ ω)
35		breq1 5077	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝐾) → (𝑦𝑥𝑧 ↔ (𝐹‘𝐾)𝑥𝑧))
36	35	abbidv 2807	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝐾) → {𝑧 ∣ 𝑦𝑥𝑧} = {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧})
37	36	fveq2d 6778	. . . . 5 ⊢ (𝑦 = (𝐹‘𝐾) → (𝑔‘{𝑧 ∣ 𝑦𝑥𝑧}) = (𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧}))
38	31, 32, 33, 34, 37	frsucmpt 8269	. . . 4 ⊢ ((𝐾 ∈ ω ∧ (𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧}) ∈ V) → (𝐹‘suc 𝐾) = (𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧}))
39	30, 38	mpan2 688	. . 3 ⊢ (𝐾 ∈ ω → (𝐹‘suc 𝐾) = (𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧}))
40	39	breq2d 5086	. 2 ⊢ (𝐾 ∈ ω → ((𝐹‘𝐾)𝑥(𝐹‘suc 𝐾) ↔ (𝐹‘𝐾)𝑥(𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧})))
41	29, 40	syl5ibrcom 246	1 ⊢ ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥 ∧ ∃𝑧(𝐹‘𝐾)𝑥𝑧) → (𝐾 ∈ ω → (𝐹‘𝐾)𝑥(𝐹‘suc 𝐾)))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1539  ∃wex 1782   ∈ wcel 2106  {cab 2715   ≠ wne 2943  ∀wral 3064  Vcvv 3432   ⊆ wss 3887  ∅c0 4256  𝒫 cpw 4533   class class class wbr 5074   ↦ cmpt 5157  dom cdm 5589  ran crn 5590   ↾ cres 5591  suc csuc 6268  ‘cfv 6433  ωcom 7712  reccrdg 8240

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241

This theorem is referenced by:  axdclem2  10276