Theorem frecsuc 6014

Description: The successor value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 15-Aug-2019.)

Assertion

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

Distinct variable groups:   𝑧,𝐴   𝑧,𝐵   𝑧,𝐹

Allowed substitution hint:   𝑉(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		suceq 4157	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → suc 𝑛 = suc 𝑚)
2	1	eqeq2d 2092	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (dom 𝑓 = suc 𝑛 ↔ dom 𝑓 = suc 𝑚))
3		fveq2 5198	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → (𝑓‘𝑛) = (𝑓‘𝑚))
4	3	fveq2d 5202	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝐹‘(𝑓‘𝑛)) = (𝐹‘(𝑓‘𝑚)))
5	4	eleq2d 2148	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑥 ∈ (𝐹‘(𝑓‘𝑛)) ↔ 𝑥 ∈ (𝐹‘(𝑓‘𝑚))))
6	2, 5	anbi12d 456	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → ((dom 𝑓 = suc 𝑛 ∧ 𝑥 ∈ (𝐹‘(𝑓‘𝑛))) ↔ (dom 𝑓 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑓‘𝑚)))))
7	6	cbvrexv 2578	. . . . . . 7 ⊢ (∃𝑛 ∈ ω (dom 𝑓 = suc 𝑛 ∧ 𝑥 ∈ (𝐹‘(𝑓‘𝑛))) ↔ ∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑓‘𝑚))))
8	7	orbi1i 712	. . . . . 6 ⊢ ((∃𝑛 ∈ ω (dom 𝑓 = suc 𝑛 ∧ 𝑥 ∈ (𝐹‘(𝑓‘𝑛))) ∨ (dom 𝑓 = ∅ ∧ 𝑥 ∈ 𝐴)) ↔ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑓‘𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑥 ∈ 𝐴)))
9	8	abbii 2194	. . . . 5 ⊢ {𝑥 ∣ (∃𝑛 ∈ ω (dom 𝑓 = suc 𝑛 ∧ 𝑥 ∈ (𝐹‘(𝑓‘𝑛))) ∨ (dom 𝑓 = ∅ ∧ 𝑥 ∈ 𝐴))} = {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑓‘𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑥 ∈ 𝐴))}
10		eleq1 2141	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ (𝐹‘(𝑓‘𝑚)) ↔ 𝑦 ∈ (𝐹‘(𝑓‘𝑚))))
11	10	anbi2d 451	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((dom 𝑓 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑓‘𝑚))) ↔ (dom 𝑓 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑓‘𝑚)))))
12	11	rexbidv 2369	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑓‘𝑚))) ↔ ∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑓‘𝑚)))))
13		eleq1 2141	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
14	13	anbi2d 451	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((dom 𝑓 = ∅ ∧ 𝑥 ∈ 𝐴) ↔ (dom 𝑓 = ∅ ∧ 𝑦 ∈ 𝐴)))
15	12, 14	orbi12d 739	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑓‘𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑥 ∈ 𝐴)) ↔ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑓‘𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑦 ∈ 𝐴))))
16	15	cbvabv 2202	. . . . 5 ⊢ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑓‘𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑥 ∈ 𝐴))} = {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑓‘𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑦 ∈ 𝐴))}
17	9, 16	eqtri 2101	. . . 4 ⊢ {𝑥 ∣ (∃𝑛 ∈ ω (dom 𝑓 = suc 𝑛 ∧ 𝑥 ∈ (𝐹‘(𝑓‘𝑛))) ∨ (dom 𝑓 = ∅ ∧ 𝑥 ∈ 𝐴))} = {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑓‘𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑦 ∈ 𝐴))}
18	17	mpteq2i 3865	. . 3 ⊢ (𝑓 ∈ V ↦ {𝑥 ∣ (∃𝑛 ∈ ω (dom 𝑓 = suc 𝑛 ∧ 𝑥 ∈ (𝐹‘(𝑓‘𝑛))) ∨ (dom 𝑓 = ∅ ∧ 𝑥 ∈ 𝐴))}) = (𝑓 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑓‘𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑦 ∈ 𝐴))})
19		dmeq 4553	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → dom 𝑓 = dom 𝑔)
20	19	eqeq1d 2089	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → (dom 𝑓 = suc 𝑚 ↔ dom 𝑔 = suc 𝑚))
21		fveq1 5197	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑚) = (𝑔‘𝑚))
22	21	fveq2d 5202	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → (𝐹‘(𝑓‘𝑚)) = (𝐹‘(𝑔‘𝑚)))
23	22	eleq2d 2148	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → (𝑦 ∈ (𝐹‘(𝑓‘𝑚)) ↔ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))))
24	20, 23	anbi12d 456	. . . . . . 7 ⊢ (𝑓 = 𝑔 → ((dom 𝑓 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑓‘𝑚))) ↔ (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚)))))
25	24	rexbidv 2369	. . . . . 6 ⊢ (𝑓 = 𝑔 → (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑓‘𝑚))) ↔ ∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚)))))
26	19	eqeq1d 2089	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (dom 𝑓 = ∅ ↔ dom 𝑔 = ∅))
27	26	anbi1d 452	. . . . . 6 ⊢ (𝑓 = 𝑔 → ((dom 𝑓 = ∅ ∧ 𝑦 ∈ 𝐴) ↔ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴)))
28	25, 27	orbi12d 739	. . . . 5 ⊢ (𝑓 = 𝑔 → ((∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑓‘𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑦 ∈ 𝐴)) ↔ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))))
29	28	abbidv 2196	. . . 4 ⊢ (𝑓 = 𝑔 → {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑓‘𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑦 ∈ 𝐴))} = {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))})
30	29	cbvmptv 3873	. . 3 ⊢ (𝑓 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑓‘𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑦 ∈ 𝐴))}) = (𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))})
31	18, 30	eqtri 2101	. 2 ⊢ (𝑓 ∈ V ↦ {𝑥 ∣ (∃𝑛 ∈ ω (dom 𝑓 = suc 𝑛 ∧ 𝑥 ∈ (𝐹‘(𝑓‘𝑛))) ∨ (dom 𝑓 = ∅ ∧ 𝑥 ∈ 𝐴))}) = (𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))})
32	31	frecsuclem3 6013	1 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(frec(𝐹, 𝐴)‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 102   ∨ wo 661   ∧ w3a 919  ∀wal 1282   = wceq 1284   ∈ wcel 1433  {cab 2067  ∃wrex 2349  Vcvv 2601  ∅c0 3251   ↦ cmpt 3839  suc csuc 4120  ωcom 4331  dom cdm 4363  ‘cfv 4922  freccfrec 6000

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-recs 5943  df-frec 6001

This theorem is referenced by:  frecrdg  6015  freccl  6016  frec2uzzd  9402  frec2uzsucd  9403  frec2uzrdg  9411  frecuzrdgsuc  9417