Theorem frecsuc 6386

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 4811	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → dom 𝑓 = dom 𝑔)
2	1	eqeq1d 2179	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → (dom 𝑓 = suc 𝑛 ↔ dom 𝑔 = suc 𝑛))
3		fveq1 5495	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑛) = (𝑔‘𝑛))
4	3	fveq2d 5500	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → (𝐹‘(𝑓‘𝑛)) = (𝐹‘(𝑔‘𝑛)))
5	4	eleq2d 2240	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → (𝑦 ∈ (𝐹‘(𝑓‘𝑛)) ↔ 𝑦 ∈ (𝐹‘(𝑔‘𝑛))))
6	2, 5	anbi12d 470	. . . . . . 7 ⊢ (𝑓 = 𝑔 → ((dom 𝑓 = suc 𝑛 ∧ 𝑦 ∈ (𝐹‘(𝑓‘𝑛))) ↔ (dom 𝑔 = suc 𝑛 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑛)))))
7	6	rexbidv 2471	. . . . . 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 2179	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (dom 𝑓 = ∅ ↔ dom 𝑔 = ∅))
9	8	anbi1d 462	. . . . . 6 ⊢ (𝑓 = 𝑔 → ((dom 𝑓 = ∅ ∧ 𝑦 ∈ 𝐴) ↔ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴)))
10	7, 9	orbi12d 788	. . . . 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 2288	. . . 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 4085	. . 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 2233	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ (𝐹‘(𝑔‘𝑛)) ↔ 𝑥 ∈ (𝐹‘(𝑔‘𝑛))))
14	13	anbi2d 461	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((dom 𝑔 = suc 𝑛 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑛))) ↔ (dom 𝑔 = suc 𝑛 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑛)))))
15	14	rexbidv 2471	. . . . . 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 2233	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
17	16	anbi2d 461	. . . . . 6 ⊢ (𝑦 = 𝑥 → ((dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴) ↔ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴)))
18	15, 17	orbi12d 788	. . . . 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 2295	. . . 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 4076	. . 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 4387	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → suc 𝑛 = suc 𝑚)
22	21	eqeq2d 2182	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (dom 𝑔 = suc 𝑛 ↔ dom 𝑔 = suc 𝑚))
23		fveq2 5496	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝑔‘𝑛) = (𝑔‘𝑚))
24	23	fveq2d 5500	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝐹‘(𝑔‘𝑛)) = (𝐹‘(𝑔‘𝑚)))
25	24	eleq2d 2240	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝑥 ∈ (𝐹‘(𝑔‘𝑛)) ↔ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))))
26	22, 25	anbi12d 470	. . . . . . 7 ⊢ (𝑛 = 𝑚 → ((dom 𝑔 = suc 𝑛 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑛))) ↔ (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚)))))
27	26	cbvrexv 2697	. . . . . 6 ⊢ (∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑛))) ↔ ∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))))
28	27	orbi1i 758	. . . . 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 2286	. . . 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 4076	. . 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 2195	. 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 6385	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 103   ∨ wo 703   ∧ w3a 973   = wceq 1348   ∈ wcel 2141  {cab 2156  ∀wral 2448  ∃wrex 2449  Vcvv 2730  ∅c0 3414   ↦ cmpt 4050  suc csuc 4350  ωcom 4574  dom cdm 4611  ‘cfv 5198  freccfrec 6369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-recs 6284  df-frec 6370

This theorem is referenced by:  frecrdg  6387  frec2uzsucd  10357  frec2uzrdg  10365  frecuzrdgsuc  10370  frecuzrdgg  10372  frecuzrdgsuctlem  10379  seq3val  10414  seqvalcd  10415