Theorem axdc3lem3 10208

Description: Simple substitution lemma for axdc3 10210. (Contributed by Mario Carneiro, 27-Jan-2013.)

Hypotheses

Ref	Expression
axdc3lem3.1	⊢ 𝐴 ∈ V
axdc3lem3.2	⊢ 𝑆 = {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛⟶𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘)))}
axdc3lem3.3	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
axdc3lem3	⊢ (𝐵 ∈ 𝑆 ↔ ∃𝑚 ∈ ω (𝐵:suc 𝑚⟶𝐴 ∧ (𝐵‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑚 (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵‘𝑘))))

Distinct variable groups:   𝐴,𝑚,𝑛   𝐴,𝑠,𝑛   𝐵,𝑘,𝑚,𝑛   𝐵,𝑠,𝑘   𝐶,𝑚,𝑛   𝐶,𝑠   𝑚,𝐹,𝑛   𝐹,𝑠

Allowed substitution hints:   𝐴(𝑘)   𝐶(𝑘)   𝑆(𝑘,𝑚,𝑛,𝑠)   𝐹(𝑘)

Proof of Theorem axdc3lem3

Step	Hyp	Ref	Expression
1		axdc3lem3.2	. . 3 ⊢ 𝑆 = {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛⟶𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘)))}
2	1	eleq2i 2830	. 2 ⊢ (𝐵 ∈ 𝑆 ↔ 𝐵 ∈ {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛⟶𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘)))})
3		axdc3lem3.3	. . 3 ⊢ 𝐵 ∈ V
4		feq1 6581	. . . . 5 ⊢ (𝑠 = 𝐵 → (𝑠:suc 𝑛⟶𝐴 ↔ 𝐵:suc 𝑛⟶𝐴))
5		fveq1 6773	. . . . . 6 ⊢ (𝑠 = 𝐵 → (𝑠‘∅) = (𝐵‘∅))
6	5	eqeq1d 2740	. . . . 5 ⊢ (𝑠 = 𝐵 → ((𝑠‘∅) = 𝐶 ↔ (𝐵‘∅) = 𝐶))
7		fveq1 6773	. . . . . . 7 ⊢ (𝑠 = 𝐵 → (𝑠‘suc 𝑘) = (𝐵‘suc 𝑘))
8		fveq1 6773	. . . . . . . 8 ⊢ (𝑠 = 𝐵 → (𝑠‘𝑘) = (𝐵‘𝑘))
9	8	fveq2d 6778	. . . . . . 7 ⊢ (𝑠 = 𝐵 → (𝐹‘(𝑠‘𝑘)) = (𝐹‘(𝐵‘𝑘)))
10	7, 9	eleq12d 2833	. . . . . 6 ⊢ (𝑠 = 𝐵 → ((𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘)) ↔ (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵‘𝑘))))
11	10	ralbidv 3112	. . . . 5 ⊢ (𝑠 = 𝐵 → (∀𝑘 ∈ 𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘)) ↔ ∀𝑘 ∈ 𝑛 (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵‘𝑘))))
12	4, 6, 11	3anbi123d 1435	. . . 4 ⊢ (𝑠 = 𝐵 → ((𝑠:suc 𝑛⟶𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘))) ↔ (𝐵:suc 𝑛⟶𝐴 ∧ (𝐵‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑛 (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵‘𝑘)))))
13	12	rexbidv 3226	. . 3 ⊢ (𝑠 = 𝐵 → (∃𝑛 ∈ ω (𝑠:suc 𝑛⟶𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘))) ↔ ∃𝑛 ∈ ω (𝐵:suc 𝑛⟶𝐴 ∧ (𝐵‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑛 (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵‘𝑘)))))
14	3, 13	elab 3609	. 2 ⊢ (𝐵 ∈ {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛⟶𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘)))} ↔ ∃𝑛 ∈ ω (𝐵:suc 𝑛⟶𝐴 ∧ (𝐵‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑛 (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵‘𝑘))))
15		suceq 6331	. . . . 5 ⊢ (𝑛 = 𝑚 → suc 𝑛 = suc 𝑚)
16	15	feq2d 6586	. . . 4 ⊢ (𝑛 = 𝑚 → (𝐵:suc 𝑛⟶𝐴 ↔ 𝐵:suc 𝑚⟶𝐴))
17		raleq 3342	. . . 4 ⊢ (𝑛 = 𝑚 → (∀𝑘 ∈ 𝑛 (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵‘𝑘)) ↔ ∀𝑘 ∈ 𝑚 (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵‘𝑘))))
18	16, 17	3anbi13d 1437	. . 3 ⊢ (𝑛 = 𝑚 → ((𝐵:suc 𝑛⟶𝐴 ∧ (𝐵‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑛 (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵‘𝑘))) ↔ (𝐵:suc 𝑚⟶𝐴 ∧ (𝐵‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑚 (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵‘𝑘)))))
19	18	cbvrexvw 3384	. 2 ⊢ (∃𝑛 ∈ ω (𝐵:suc 𝑛⟶𝐴 ∧ (𝐵‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑛 (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵‘𝑘))) ↔ ∃𝑚 ∈ ω (𝐵:suc 𝑚⟶𝐴 ∧ (𝐵‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑚 (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵‘𝑘))))
20	2, 14, 19	3bitri 297	1 ⊢ (𝐵 ∈ 𝑆 ↔ ∃𝑚 ∈ ω (𝐵:suc 𝑚⟶𝐴 ∧ (𝐵‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑚 (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵‘𝑘))))

Syntax hints:   ↔ wb 205   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  {cab 2715  ∀wral 3064  ∃wrex 3065  Vcvv 3432  ∅c0 4256  suc csuc 6268  ⟶wf 6429  ‘cfv 6433  ωcom 7712

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-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441

This theorem is referenced by:  axdc3lem4  10209