Theorem frecsuc 6460

Description: The successor value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 31-Mar-2022.)

Assertion

Ref	Expression
frecsuc	⊢ ((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(frec(𝐹, 𝐴)‘𝐵)))

Distinct variable groups:   𝑧,𝐹   𝑧,𝑆

Allowed substitution hints:   𝐴(𝑧)   𝐵(𝑧)

Proof of Theorem frecsuc

Dummy variables 𝑓 𝑔 𝑚 𝑥 𝑦 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmeq 4862	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → dom 𝑓 = dom 𝑔)
2	1	eqeq1d 2202	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → (dom 𝑓 = suc 𝑛 ↔ dom 𝑔 = suc 𝑛))
3		fveq1 5553	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑛) = (𝑔‘𝑛))
4	3	fveq2d 5558	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → (𝐹‘(𝑓‘𝑛)) = (𝐹‘(𝑔‘𝑛)))
5	4	eleq2d 2263	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → (𝑦 ∈ (𝐹‘(𝑓‘𝑛)) ↔ 𝑦 ∈ (𝐹‘(𝑔‘𝑛))))
6	2, 5	anbi12d 473	. . . . . . 7 ⊢ (𝑓 = 𝑔 → ((dom 𝑓 = suc 𝑛 ∧ 𝑦 ∈ (𝐹‘(𝑓‘𝑛))) ↔ (dom 𝑔 = suc 𝑛 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑛)))))
7	6	rexbidv 2495	. . . . . 6 ⊢ (𝑓 = 𝑔 → (∃𝑛 ∈ ω (dom 𝑓 = suc 𝑛 ∧ 𝑦 ∈ (𝐹‘(𝑓‘𝑛))) ↔ ∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑛)))))
8	1	eqeq1d 2202	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (dom 𝑓 = ∅ ↔ dom 𝑔 = ∅))
9	8	anbi1d 465	. . . . . 6 ⊢ (𝑓 = 𝑔 → ((dom 𝑓 = ∅ ∧ 𝑦 ∈ 𝐴) ↔ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴)))
10	7, 9	orbi12d 794	. . . . 5 ⊢ (𝑓 = 𝑔 → ((∃𝑛 ∈ ω (dom 𝑓 = suc 𝑛 ∧ 𝑦 ∈ (𝐹‘(𝑓‘𝑛))) ∨ (dom 𝑓 = ∅ ∧ 𝑦 ∈ 𝐴)) ↔ (∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑛))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))))
11	10	abbidv 2311	. . . 4 ⊢ (𝑓 = 𝑔 → {𝑦 ∣ (∃𝑛 ∈ ω (dom 𝑓 = suc 𝑛 ∧ 𝑦 ∈ (𝐹‘(𝑓‘𝑛))) ∨ (dom 𝑓 = ∅ ∧ 𝑦 ∈ 𝐴))} = {𝑦 ∣ (∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑛))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))})
12	11	cbvmptv 4125	. . 3 ⊢ (𝑓 ∈ V ↦ {𝑦 ∣ (∃𝑛 ∈ ω (dom 𝑓 = suc 𝑛 ∧ 𝑦 ∈ (𝐹‘(𝑓‘𝑛))) ∨ (dom 𝑓 = ∅ ∧ 𝑦 ∈ 𝐴))}) = (𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑛))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))})
13		eleq1 2256	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ (𝐹‘(𝑔‘𝑛)) ↔ 𝑥 ∈ (𝐹‘(𝑔‘𝑛))))
14	13	anbi2d 464	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((dom 𝑔 = suc 𝑛 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑛))) ↔ (dom 𝑔 = suc 𝑛 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑛)))))
15	14	rexbidv 2495	. . . . . 6 ⊢ (𝑦 = 𝑥 → (∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑛))) ↔ ∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑛)))))
16		eleq1 2256	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
17	16	anbi2d 464	. . . . . 6 ⊢ (𝑦 = 𝑥 → ((dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴) ↔ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴)))
18	15, 17	orbi12d 794	. . . . 5 ⊢ (𝑦 = 𝑥 → ((∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑛))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴)) ↔ (∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑛))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))))
19	18	cbvabv 2318	. . . 4 ⊢ {𝑦 ∣ (∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑛))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))} = {𝑥 ∣ (∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑛))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))}
20	19	mpteq2i 4116	. . 3 ⊢ (𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑛))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))}) = (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑛))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})
21		suceq 4433	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → suc 𝑛 = suc 𝑚)
22	21	eqeq2d 2205	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (dom 𝑔 = suc 𝑛 ↔ dom 𝑔 = suc 𝑚))
23		fveq2 5554	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝑔‘𝑛) = (𝑔‘𝑚))
24	23	fveq2d 5558	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝐹‘(𝑔‘𝑛)) = (𝐹‘(𝑔‘𝑚)))
25	24	eleq2d 2263	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝑥 ∈ (𝐹‘(𝑔‘𝑛)) ↔ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))))
26	22, 25	anbi12d 473	. . . . . . 7 ⊢ (𝑛 = 𝑚 → ((dom 𝑔 = suc 𝑛 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑛))) ↔ (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚)))))
27	26	cbvrexv 2727	. . . . . 6 ⊢ (∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑛))) ↔ ∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))))
28	27	orbi1i 764	. . . . 5 ⊢ ((∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑛))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴)) ↔ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴)))
29	28	abbii 2309	. . . 4 ⊢ {𝑥 ∣ (∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑛))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))} = {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))}
30	29	mpteq2i 4116	. . 3 ⊢ (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑛))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))}) = (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})
31	12, 20, 30	3eqtri 2218	. 2 ⊢ (𝑓 ∈ V ↦ {𝑦 ∣ (∃𝑛 ∈ ω (dom 𝑓 = suc 𝑛 ∧ 𝑦 ∈ (𝐹‘(𝑓‘𝑛))) ∨ (dom 𝑓 = ∅ ∧ 𝑦 ∈ 𝐴))}) = (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})
32	31	frecsuclem 6459	1 ⊢ ((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(frec(𝐹, 𝐴)‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 709   ∧ w3a 980   = wceq 1364   ∈ wcel 2164  {cab 2179  ∀wral 2472  ∃wrex 2473  Vcvv 2760  ∅c0 3446   ↦ cmpt 4090  suc csuc 4396  ωcom 4622  dom cdm 4659  ‘cfv 5254  freccfrec 6443

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-recs 6358  df-frec 6444

This theorem is referenced by:  frecrdg  6461  frec2uzsucd  10472  frec2uzrdg  10480  frecuzrdgsuc  10485  frecuzrdgg  10487  frecuzrdgsuctlem  10494  seq3val  10531  seqvalcd  10532