Theorem axdclem 10407

Description: Lemma for axdc 10409. (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 2990	. . . . . . 7 ⊢ (𝑦 = {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} → (𝑦 ≠ ∅ ↔ {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} ≠ ∅))
2		abn0 4335	. . . . . . 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 2820	. . . . . . . . 9 ⊢ (𝑦 = {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} → ((𝑔‘𝑦) ∈ 𝑦 ↔ (𝑔‘𝑦) ∈ {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧}))
5		breq2 5095	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → ((𝐹‘𝐾)𝑥𝑤 ↔ (𝐹‘𝐾)𝑥𝑧))
6	5	cbvabv 2801	. . . . . . . . . 10 ⊢ {𝑤 ∣ (𝐹‘𝐾)𝑥𝑤} = {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧}
7	6	eleq2i 2823	. . . . . . . . 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 6835	. . . . . . . . 9 ⊢ (𝑔‘𝑦) ∈ V
10		breq2 5095	. . . . . . . . 9 ⊢ (𝑤 = (𝑔‘𝑦) → ((𝐹‘𝐾)𝑥𝑤 ↔ (𝐹‘𝐾)𝑥(𝑔‘𝑦)))
11	9, 10	elab 3635	. . . . . . . 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 6822	. . . . . . . 8 ⊢ (𝑦 = {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} → (𝑔‘𝑦) = (𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧}))
14	13	breq2d 5103	. . . . . . 7 ⊢ (𝑦 = {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} → ((𝐹‘𝐾)𝑥(𝑔‘𝑦) ↔ (𝐹‘𝐾)𝑥(𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧})))
15	12, 14	bitrd 279	. . . . . 6 ⊢ (𝑦 = {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} → ((𝑔‘𝑦) ∈ 𝑦 ↔ (𝐹‘𝐾)𝑥(𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧})))
16	3, 15	imbi12d 344	. . . . 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 3573	. . . 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 6835	. . . . . . . 8 ⊢ (𝐹‘𝐾) ∈ V
19		vex 3440	. . . . . . . 8 ⊢ 𝑧 ∈ V
20	18, 19	brelrn 5882	. . . . . . 7 ⊢ ((𝐹‘𝐾)𝑥𝑧 → 𝑧 ∈ ran 𝑥)
21	20	abssi 4020	. . . . . 6 ⊢ {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} ⊆ ran 𝑥
22		sstr 3943	. . . . . 6 ⊢ (({𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} ⊆ ran 𝑥 ∧ ran 𝑥 ⊆ dom 𝑥) → {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} ⊆ dom 𝑥)
23	21, 22	mpan 690	. . . . 5 ⊢ (ran 𝑥 ⊆ dom 𝑥 → {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} ⊆ dom 𝑥)
24		vex 3440	. . . . . . 7 ⊢ 𝑥 ∈ V
25	24	dmex 7839	. . . . . 6 ⊢ dom 𝑥 ∈ V
26	25	elpw2 5272	. . . . 5 ⊢ ({𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} ∈ 𝒫 dom 𝑥 ↔ {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} ⊆ dom 𝑥)
27	23, 26	sylibr 234	. . . 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 6835	. . . 4 ⊢ (𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧}) ∈ V
31		nfcv 2894	. . . . 5 ⊢ Ⅎ𝑦𝑠
32		nfcv 2894	. . . . 5 ⊢ Ⅎ𝑦𝐾
33		nfcv 2894	. . . . 5 ⊢ Ⅎ𝑦(𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧})
34		axdclem.1	. . . . 5 ⊢ 𝐹 = (rec((𝑦 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑦𝑥𝑧})), 𝑠) ↾ ω)
35		breq1 5094	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝐾) → (𝑦𝑥𝑧 ↔ (𝐹‘𝐾)𝑥𝑧))
36	35	abbidv 2797	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝐾) → {𝑧 ∣ 𝑦𝑥𝑧} = {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧})
37	36	fveq2d 6826	. . . . 5 ⊢ (𝑦 = (𝐹‘𝐾) → (𝑔‘{𝑧 ∣ 𝑦𝑥𝑧}) = (𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧}))
38	31, 32, 33, 34, 37	frsucmpt 8357	. . . 4 ⊢ ((𝐾 ∈ ω ∧ (𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧}) ∈ V) → (𝐹‘suc 𝐾) = (𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧}))
39	30, 38	mpan2 691	. . 3 ⊢ (𝐾 ∈ ω → (𝐹‘suc 𝐾) = (𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧}))
40	39	breq2d 5103	. 2 ⊢ (𝐾 ∈ ω → ((𝐹‘𝐾)𝑥(𝐹‘suc 𝐾) ↔ (𝐹‘𝐾)𝑥(𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧})))
41	29, 40	syl5ibrcom 247	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 1541  ∃wex 1780   ∈ wcel 2111  {cab 2709   ≠ wne 2928  ∀wral 3047  Vcvv 3436   ⊆ wss 3902  ∅c0 4283  𝒫 cpw 4550   class class class wbr 5091   ↦ cmpt 5172  dom cdm 5616  ran crn 5617   ↾ cres 5618  suc csuc 6308  ‘cfv 6481  ωcom 7796  reccrdg 8328

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-om 7797  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329

This theorem is referenced by:  axdclem2  10408