Theorem frecsuc 6616

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 4937	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → dom 𝑓 = dom 𝑔)
2	1	eqeq1d 2240	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → (dom 𝑓 = suc 𝑛 ↔ dom 𝑔 = suc 𝑛))
3		fveq1 5647	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑛) = (𝑔‘𝑛))
4	3	fveq2d 5652	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → (𝐹‘(𝑓‘𝑛)) = (𝐹‘(𝑔‘𝑛)))
5	4	eleq2d 2301	. . . . . . . 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 2534	. . . . . 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 2240	. . . . . . 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 2350	. . . 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 4190	. . 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 2294	. . . . . . . 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 2534	. . . . . 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 2294	. . . . . . 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 2357	. . . 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 4181	. . 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 4505	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → suc 𝑛 = suc 𝑚)
22	21	eqeq2d 2243	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (dom 𝑔 = suc 𝑛 ↔ dom 𝑔 = suc 𝑚))
23		fveq2 5648	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝑔‘𝑛) = (𝑔‘𝑚))
24	23	fveq2d 5652	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝐹‘(𝑔‘𝑛)) = (𝐹‘(𝑔‘𝑚)))
25	24	eleq2d 2301	. . . . . . . 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 2769	. . . . . 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 2347	. . . 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 4181	. . 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 2256	. 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 6615	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 2202  {cab 2217  ∀wral 2511  ∃wrex 2512  Vcvv 2803  ∅c0 3496   ↦ cmpt 4155  suc csuc 4468  ωcom 4694  dom cdm 4731  ‘cfv 5333  freccfrec 6599

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-recs 6514  df-frec 6600

This theorem is referenced by:  frecrdg  6617  frec2uzsucd  10709  frec2uzrdg  10717  frecuzrdgsuc  10722  frecuzrdgg  10724  frecuzrdgsuctlem  10731  seq3val  10768  seqvalcd  10769