Theorem frecsuc 6516

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 4897	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → dom 𝑓 = dom 𝑔)
2	1	eqeq1d 2216	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → (dom 𝑓 = suc 𝑛 ↔ dom 𝑔 = suc 𝑛))
3		fveq1 5598	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑛) = (𝑔‘𝑛))
4	3	fveq2d 5603	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → (𝐹‘(𝑓‘𝑛)) = (𝐹‘(𝑔‘𝑛)))
5	4	eleq2d 2277	. . . . . . . 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 2509	. . . . . 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 2216	. . . . . . 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 795	. . . . 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 2325	. . . 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 4156	. . 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 2270	. . . . . . . 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 2509	. . . . . 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 2270	. . . . . . 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 795	. . . . 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 2332	. . . 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 4147	. . 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 4467	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → suc 𝑛 = suc 𝑚)
22	21	eqeq2d 2219	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (dom 𝑔 = suc 𝑛 ↔ dom 𝑔 = suc 𝑚))
23		fveq2 5599	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝑔‘𝑛) = (𝑔‘𝑚))
24	23	fveq2d 5603	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝐹‘(𝑔‘𝑛)) = (𝐹‘(𝑔‘𝑚)))
25	24	eleq2d 2277	. . . . . . . 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 2743	. . . . . 6 ⊢ (∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑛))) ↔ ∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))))
28	27	orbi1i 765	. . . . 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 2323	. . . 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 4147	. . 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 2232	. 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 6515	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 710   ∧ w3a 981   = wceq 1373   ∈ wcel 2178  {cab 2193  ∀wral 2486  ∃wrex 2487  Vcvv 2776  ∅c0 3468   ↦ cmpt 4121  suc csuc 4430  ωcom 4656  dom cdm 4693  ‘cfv 5290  freccfrec 6499

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-recs 6414  df-frec 6500

This theorem is referenced by:  frecrdg  6517  frec2uzsucd  10583  frec2uzrdg  10591  frecuzrdgsuc  10596  frecuzrdgg  10598  frecuzrdgsuctlem  10605  seq3val  10642  seqvalcd  10643