Theorem frecsuc 6638

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 4956	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → dom 𝑓 = dom 𝑔)
2	1	eqeq1d 2241	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → (dom 𝑓 = suc 𝑛 ↔ dom 𝑔 = suc 𝑛))
3		fveq1 5669	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑛) = (𝑔‘𝑛))
4	3	fveq2d 5674	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → (𝐹‘(𝑓‘𝑛)) = (𝐹‘(𝑔‘𝑛)))
5	4	eleq2d 2302	. . . . . . . 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 2543	. . . . . 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 2241	. . . . . . 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 801	. . . . 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 2352	. . . 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 4206	. . 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 2295	. . . . . . . 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 2543	. . . . . 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 2295	. . . . . . 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 801	. . . . 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 2359	. . . 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 4197	. . 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 4523	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → suc 𝑛 = suc 𝑚)
22	21	eqeq2d 2244	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (dom 𝑔 = suc 𝑛 ↔ dom 𝑔 = suc 𝑚))
23		fveq2 5670	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝑔‘𝑛) = (𝑔‘𝑚))
24	23	fveq2d 5674	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝐹‘(𝑔‘𝑛)) = (𝐹‘(𝑔‘𝑚)))
25	24	eleq2d 2302	. . . . . . . 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 2779	. . . . . 6 ⊢ (∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑛))) ↔ ∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))))
28	27	orbi1i 771	. . . . 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 2348	. . . 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 4197	. . 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 2257	. 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 6637	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 716   ∧ w3a 1005   = wceq 1398   ∈ wcel 2203  {cab 2218  ∀wral 2520  ∃wrex 2521  Vcvv 2813  ∅c0 3508   ↦ cmpt 4171  suc csuc 4486  ωcom 4712  dom cdm 4749  ‘cfv 5352  freccfrec 6621

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-recs 6536  df-frec 6622

This theorem is referenced by:  frecrdg  6639  frec2uzsucd  10763  frec2uzrdg  10771  frecuzrdgsuc  10776  frecuzrdgg  10778  frecuzrdgsuctlem  10785  seq3val  10822  seqvalcd  10823