Theorem axdclem 10472

Description: Lemma for axdc 10474. (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 2987	. . . . . . 7 ⊢ (𝑦 = {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} → (𝑦 ≠ ∅ ↔ {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} ≠ ∅))
2		abn0 4348	. . . . . . 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 2817	. . . . . . . . 9 ⊢ (𝑦 = {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} → ((𝑔‘𝑦) ∈ 𝑦 ↔ (𝑔‘𝑦) ∈ {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧}))
5		breq2 5111	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → ((𝐹‘𝐾)𝑥𝑤 ↔ (𝐹‘𝐾)𝑥𝑧))
6	5	cbvabv 2799	. . . . . . . . . 10 ⊢ {𝑤 ∣ (𝐹‘𝐾)𝑥𝑤} = {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧}
7	6	eleq2i 2820	. . . . . . . . 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 6871	. . . . . . . . 9 ⊢ (𝑔‘𝑦) ∈ V
10		breq2 5111	. . . . . . . . 9 ⊢ (𝑤 = (𝑔‘𝑦) → ((𝐹‘𝐾)𝑥𝑤 ↔ (𝐹‘𝐾)𝑥(𝑔‘𝑦)))
11	9, 10	elab 3646	. . . . . . . 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 6858	. . . . . . . 8 ⊢ (𝑦 = {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} → (𝑔‘𝑦) = (𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧}))
14	13	breq2d 5119	. . . . . . 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 3584	. . . 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 6871	. . . . . . . 8 ⊢ (𝐹‘𝐾) ∈ V
19		vex 3451	. . . . . . . 8 ⊢ 𝑧 ∈ V
20	18, 19	brelrn 5906	. . . . . . 7 ⊢ ((𝐹‘𝐾)𝑥𝑧 → 𝑧 ∈ ran 𝑥)
21	20	abssi 4033	. . . . . 6 ⊢ {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} ⊆ ran 𝑥
22		sstr 3955	. . . . . 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 3451	. . . . . . 7 ⊢ 𝑥 ∈ V
25	24	dmex 7885	. . . . . 6 ⊢ dom 𝑥 ∈ V
26	25	elpw2 5289	. . . . 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 6871	. . . 4 ⊢ (𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧}) ∈ V
31		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑦𝑠
32		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑦𝐾
33		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑦(𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧})
34		axdclem.1	. . . . 5 ⊢ 𝐹 = (rec((𝑦 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑦𝑥𝑧})), 𝑠) ↾ ω)
35		breq1 5110	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝐾) → (𝑦𝑥𝑧 ↔ (𝐹‘𝐾)𝑥𝑧))
36	35	abbidv 2795	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝐾) → {𝑧 ∣ 𝑦𝑥𝑧} = {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧})
37	36	fveq2d 6862	. . . . 5 ⊢ (𝑦 = (𝐹‘𝐾) → (𝑔‘{𝑧 ∣ 𝑦𝑥𝑧}) = (𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧}))
38	31, 32, 33, 34, 37	frsucmpt 8406	. . . 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 5119	. 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 1540  ∃wex 1779   ∈ wcel 2109  {cab 2707   ≠ wne 2925  ∀wral 3044  Vcvv 3447   ⊆ wss 3914  ∅c0 4296  𝒫 cpw 4563   class class class wbr 5107   ↦ cmpt 5188  dom cdm 5638  ran crn 5639   ↾ cres 5640  suc csuc 6334  ‘cfv 6511  ωcom 7842  reccrdg 8377

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-om 7843  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378

This theorem is referenced by:  axdclem2  10473