Theorem axdclem 10556

Description: Lemma for axdc 10558. (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 3000	. . . . . . 7 ⊢ (𝑦 = {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} → (𝑦 ≠ ∅ ↔ {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} ≠ ∅))
2		abn0 4390	. . . . . . 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 5151	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → ((𝐹‘𝐾)𝑥𝑤 ↔ (𝐹‘𝐾)𝑥𝑧))
6	5	cbvabv 2809	. . . . . . . . . 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 6919	. . . . . . . . 9 ⊢ (𝑔‘𝑦) ∈ V
10		breq2 5151	. . . . . . . . 9 ⊢ (𝑤 = (𝑔‘𝑦) → ((𝐹‘𝐾)𝑥𝑤 ↔ (𝐹‘𝐾)𝑥(𝑔‘𝑦)))
11	9, 10	elab 3680	. . . . . . . 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 6906	. . . . . . . 8 ⊢ (𝑦 = {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} → (𝑔‘𝑦) = (𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧}))
14	13	breq2d 5159	. . . . . . 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 3617	. . . 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 6919	. . . . . . . 8 ⊢ (𝐹‘𝐾) ∈ V
19		vex 3481	. . . . . . . 8 ⊢ 𝑧 ∈ V
20	18, 19	brelrn 5955	. . . . . . 7 ⊢ ((𝐹‘𝐾)𝑥𝑧 → 𝑧 ∈ ran 𝑥)
21	20	abssi 4079	. . . . . 6 ⊢ {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} ⊆ ran 𝑥
22		sstr 4003	. . . . . 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 3481	. . . . . . 7 ⊢ 𝑥 ∈ V
25	24	dmex 7931	. . . . . 6 ⊢ dom 𝑥 ∈ V
26	25	elpw2 5339	. . . . 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 6919	. . . 4 ⊢ (𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧}) ∈ V
31		nfcv 2902	. . . . 5 ⊢ Ⅎ𝑦𝑠
32		nfcv 2902	. . . . 5 ⊢ Ⅎ𝑦𝐾
33		nfcv 2902	. . . . 5 ⊢ Ⅎ𝑦(𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧})
34		axdclem.1	. . . . 5 ⊢ 𝐹 = (rec((𝑦 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑦𝑥𝑧})), 𝑠) ↾ ω)
35		breq1 5150	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝐾) → (𝑦𝑥𝑧 ↔ (𝐹‘𝐾)𝑥𝑧))
36	35	abbidv 2805	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝐾) → {𝑧 ∣ 𝑦𝑥𝑧} = {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧})
37	36	fveq2d 6910	. . . . 5 ⊢ (𝑦 = (𝐹‘𝐾) → (𝑔‘{𝑧 ∣ 𝑦𝑥𝑧}) = (𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧}))
38	31, 32, 33, 34, 37	frsucmpt 8476	. . . 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 5159	. 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 1536  ∃wex 1775   ∈ wcel 2105  {cab 2711   ≠ wne 2937  ∀wral 3058  Vcvv 3477   ⊆ wss 3962  ∅c0 4338  𝒫 cpw 4604   class class class wbr 5147   ↦ cmpt 5230  dom cdm 5688  ran crn 5689   ↾ cres 5690  suc csuc 6387  ‘cfv 6562  ωcom 7886  reccrdg 8447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448

This theorem is referenced by:  axdclem2  10557