Theorem axdclem 10491

Description: Lemma for axdc 10493. (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 3022	. . . . . . 7 ⊢ (𝑦 = {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} → (𝑦 ≠ ∅ ↔ {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} ≠ ∅))
2		abn0 4341	. . . . . . 7 ⊢ ({𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} ≠ ∅ ↔ ∃𝑧(𝐹‘𝐾)𝑥𝑧)
3	1, 2	bitrdi 290	. . . . . 6 ⊢ (𝑦 = {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} → (𝑦 ≠ ∅ ↔ ∃𝑧(𝐹‘𝐾)𝑥𝑧))
4		eleq2 2854	. . . . . . . . 9 ⊢ (𝑦 = {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} → ((𝑔‘𝑦) ∈ 𝑦 ↔ (𝑔‘𝑦) ∈ {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧}))
5		breq2 5109	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → ((𝐹‘𝐾)𝑥𝑤 ↔ (𝐹‘𝐾)𝑥𝑧))
6	5	cbvabv 2835	. . . . . . . . . 10 ⊢ {𝑤 ∣ (𝐹‘𝐾)𝑥𝑤} = {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧}
7	6	eleq2i 2857	. . . . . . . . 9 ⊢ ((𝑔‘𝑦) ∈ {𝑤 ∣ (𝐹‘𝐾)𝑥𝑤} ↔ (𝑔‘𝑦) ∈ {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧})
8	4, 7	bitr4di 292	. . . . . . . 8 ⊢ (𝑦 = {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} → ((𝑔‘𝑦) ∈ 𝑦 ↔ (𝑔‘𝑦) ∈ {𝑤 ∣ (𝐹‘𝐾)𝑥𝑤}))
9		fvex 6884	. . . . . . . . 9 ⊢ (𝑔‘𝑦) ∈ V
10		breq2 5109	. . . . . . . . 9 ⊢ (𝑤 = (𝑔‘𝑦) → ((𝐹‘𝐾)𝑥𝑤 ↔ (𝐹‘𝐾)𝑥(𝑔‘𝑦)))
11	9, 10	elab 3641	. . . . . . . 8 ⊢ ((𝑔‘𝑦) ∈ {𝑤 ∣ (𝐹‘𝐾)𝑥𝑤} ↔ (𝐹‘𝐾)𝑥(𝑔‘𝑦))
12	8, 11	bitrdi 290	. . . . . . 7 ⊢ (𝑦 = {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} → ((𝑔‘𝑦) ∈ 𝑦 ↔ (𝐹‘𝐾)𝑥(𝑔‘𝑦)))
13		fveq2 6871	. . . . . . . 8 ⊢ (𝑦 = {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} → (𝑔‘𝑦) = (𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧}))
14	13	breq2d 5117	. . . . . . 7 ⊢ (𝑦 = {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} → ((𝐹‘𝐾)𝑥(𝑔‘𝑦) ↔ (𝐹‘𝐾)𝑥(𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧})))
15	12, 14	bitrd 282	. . . . . 6 ⊢ (𝑦 = {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} → ((𝑔‘𝑦) ∈ 𝑦 ↔ (𝐹‘𝐾)𝑥(𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧})))
16	3, 15	imbi12d 347	. . . . 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 3580	. . . 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 6884	. . . . . . . 8 ⊢ (𝐹‘𝐾) ∈ V
19		vex 3461	. . . . . . . 8 ⊢ 𝑧 ∈ V
20	18, 19	brelrn 5923	. . . . . . 7 ⊢ ((𝐹‘𝐾)𝑥𝑧 → 𝑧 ∈ ran 𝑥)
21	20	abssi 4024	. . . . . 6 ⊢ {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} ⊆ ran 𝑥
22		sstr 3947	. . . . . 6 ⊢ (({𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} ⊆ ran 𝑥 ∧ ran 𝑥 ⊆ dom 𝑥) → {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} ⊆ dom 𝑥)
23	21, 22	mpan 702	. . . . 5 ⊢ (ran 𝑥 ⊆ dom 𝑥 → {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} ⊆ dom 𝑥)
24		vex 3461	. . . . . . 7 ⊢ 𝑥 ∈ V
25	24	dmex 7894	. . . . . 6 ⊢ dom 𝑥 ∈ V
26	25	elpw2 5295	. . . . 5 ⊢ ({𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} ∈ 𝒫 dom 𝑥 ↔ {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} ⊆ dom 𝑥)
27	23, 26	sylibr 237	. . . 4 ⊢ (ran 𝑥 ⊆ dom 𝑥 → {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧} ∈ 𝒫 dom 𝑥)
28	17, 27	syl11 34	. . 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 1126	. 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 6884	. . . 4 ⊢ (𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧}) ∈ V
31		nfcv 2927	. . . . 5 ⊢ Ⅎ𝑦𝑠
32		nfcv 2927	. . . . 5 ⊢ Ⅎ𝑦𝐾
33		nfcv 2927	. . . . 5 ⊢ Ⅎ𝑦(𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧})
34		axdclem.1	. . . . 5 ⊢ 𝐹 = (rec((𝑦 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑦𝑥𝑧})), 𝑠) ↾ ω)
35		breq1 5108	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝐾) → (𝑦𝑥𝑧 ↔ (𝐹‘𝐾)𝑥𝑧))
36	35	abbidv 2831	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝐾) → {𝑧 ∣ 𝑦𝑥𝑧} = {𝑧 ∣ (𝐹‘𝐾)𝑥𝑧})
37	36	fveq2d 6875	. . . . 5 ⊢ (𝑦 = (𝐹‘𝐾) → (𝑔‘{𝑧 ∣ 𝑦𝑥𝑧}) = (𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧}))
38	31, 32, 33, 34, 37	frsucmpt 8413	. . . 4 ⊢ ((𝐾 ∈ ω ∧ (𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧}) ∈ V) → (𝐹‘suc 𝐾) = (𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧}))
39	30, 38	mpan2 703	. . 3 ⊢ (𝐾 ∈ ω → (𝐹‘suc 𝐾) = (𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧}))
40	39	breq2d 5117	. 2 ⊢ (𝐾 ∈ ω → ((𝐹‘𝐾)𝑥(𝐹‘suc 𝐾) ↔ (𝐹‘𝐾)𝑥(𝑔‘{𝑧 ∣ (𝐹‘𝐾)𝑥𝑧})))
41	29, 40	syl5ibrcom 250	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 1101   = wceq 1563  ∃wex 1802   ∈ wcel 2145  {cab 2743   ≠ wne 2960  ∀wral 3079  Vcvv 3457   ⊆ wss 3907  ∅c0 4288  𝒫 cpw 4558   class class class wbr 5105   ↦ cmpt 5186  dom cdm 5652  ran crn 5653   ↾ cres 5654  suc csuc 6352  ‘cfv 6525  ωcom 7850  reccrdg 8384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-om 7851  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385

This theorem is referenced by:  axdclem2  10492