Theorem freceq1 6393

Description: Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)

Assertion

Ref	Expression
freceq1	⊢ (𝐹 = 𝐺 → frec(𝐹, 𝐴) = frec(𝐺, 𝐴))

Proof of Theorem freceq1

Dummy variables 𝑥 𝑔 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 109	. . . . . . . . . . 11 ⊢ ((𝐹 = 𝐺 ∧ 𝑔 ∈ V) → 𝐹 = 𝐺)
2	1	fveq1d 5518	. . . . . . . . . 10 ⊢ ((𝐹 = 𝐺 ∧ 𝑔 ∈ V) → (𝐹‘(𝑔‘𝑚)) = (𝐺‘(𝑔‘𝑚)))
3	2	eleq2d 2247	. . . . . . . . 9 ⊢ ((𝐹 = 𝐺 ∧ 𝑔 ∈ V) → (𝑥 ∈ (𝐹‘(𝑔‘𝑚)) ↔ 𝑥 ∈ (𝐺‘(𝑔‘𝑚))))
4	3	anbi2d 464	. . . . . . . 8 ⊢ ((𝐹 = 𝐺 ∧ 𝑔 ∈ V) → ((dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ↔ (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐺‘(𝑔‘𝑚)))))
5	4	rexbidv 2478	. . . . . . 7 ⊢ ((𝐹 = 𝐺 ∧ 𝑔 ∈ V) → (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ↔ ∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐺‘(𝑔‘𝑚)))))
6	5	orbi1d 791	. . . . . 6 ⊢ ((𝐹 = 𝐺 ∧ 𝑔 ∈ V) → ((∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴)) ↔ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐺‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))))
7	6	abbidv 2295	. . . . 5 ⊢ ((𝐹 = 𝐺 ∧ 𝑔 ∈ V) → {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))} = {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐺‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})
8	7	mpteq2dva 4094	. . . 4 ⊢ (𝐹 = 𝐺 → (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))}) = (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐺‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))}))
9		recseq 6307	. . . 4 ⊢ ((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))}) = (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐺‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))}) → recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})) = recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐺‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})))
10	8, 9	syl 14	. . 3 ⊢ (𝐹 = 𝐺 → recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})) = recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐺‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})))
11	10	reseq1d 4907	. 2 ⊢ (𝐹 = 𝐺 → (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})) ↾ ω) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐺‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})) ↾ ω))
12		df-frec 6392	. 2 ⊢ frec(𝐹, 𝐴) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})) ↾ ω)
13		df-frec 6392	. 2 ⊢ frec(𝐺, 𝐴) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐺‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})) ↾ ω)
14	11, 12, 13	3eqtr4g 2235	1 ⊢ (𝐹 = 𝐺 → frec(𝐹, 𝐴) = frec(𝐺, 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 708   = wceq 1353   ∈ wcel 2148  {cab 2163  ∃wrex 2456  Vcvv 2738  ∅c0 3423   ↦ cmpt 4065  suc csuc 4366  ωcom 4590  dom cdm 4627   ↾ cres 4629  ‘cfv 5217  recscrecs 6305  freccfrec 6391

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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-in 3136  df-uni 3811  df-br 4005  df-opab 4066  df-mpt 4067  df-res 4639  df-iota 5179  df-fv 5225  df-recs 6306  df-frec 6392

This theorem is referenced by:  frecuzrdgdom  10418  frecuzrdgfun  10420  frecuzrdgsuct  10424  seqeq1  10448  seqeq2  10449  seqeq3  10450  iseqvalcbv  10457  hashfz1  10763  ennnfonelemr  12424  ctinfom  12429  isomninn  14782  iswomninn  14801  ismkvnn  14804