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Theorem freccllem 6299
Description: Lemma for freccl 6300. Just giving a name to a common expression to simplify the proof. (Contributed by Jim Kingdon, 27-Mar-2022.)
Hypotheses
Ref Expression
freccl.a (𝜑𝐴𝑆)
freccl.cl ((𝜑𝑧𝑆) → (𝐹𝑧) ∈ 𝑆)
freccl.b (𝜑𝐵 ∈ ω)
freccllem.g 𝐺 = recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))}))
Assertion
Ref Expression
freccllem (𝜑 → (frec(𝐹, 𝐴)‘𝐵) ∈ 𝑆)
Distinct variable groups:   𝐴,𝑔,𝑚,𝑥   𝑧,𝐴,𝑚,𝑥   𝑥,𝐵   𝑔,𝐹,𝑚,𝑥   𝑧,𝐹   𝑆,𝑚,𝑥,𝑧   𝜑,𝑚,𝑥,𝑧
Allowed substitution hints:   𝜑(𝑔)   𝐵(𝑧,𝑔,𝑚)   𝑆(𝑔)   𝐺(𝑥,𝑧,𝑔,𝑚)

Proof of Theorem freccllem
Dummy variables 𝑓 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-frec 6288 . . . 4 frec(𝐹, 𝐴) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})) ↾ ω)
2 freccllem.g . . . . 5 𝐺 = recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))}))
32reseq1i 4815 . . . 4 (𝐺 ↾ ω) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})) ↾ ω)
41, 3eqtr4i 2163 . . 3 frec(𝐹, 𝐴) = (𝐺 ↾ ω)
54fveq1i 5422 . 2 (frec(𝐹, 𝐴)‘𝐵) = ((𝐺 ↾ ω)‘𝐵)
6 freccl.b . . . 4 (𝜑𝐵 ∈ ω)
7 fvres 5445 . . . 4 (𝐵 ∈ ω → ((𝐺 ↾ ω)‘𝐵) = (𝐺𝐵))
86, 7syl 14 . . 3 (𝜑 → ((𝐺 ↾ ω)‘𝐵) = (𝐺𝐵))
9 funmpt 5161 . . . . 5 Fun (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})
109a1i 9 . . . 4 (𝜑 → Fun (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))}))
11 ordom 4520 . . . . 5 Ord ω
1211a1i 9 . . . 4 (𝜑 → Ord ω)
13 vex 2689 . . . . . 6 𝑓 ∈ V
14 simp2 982 . . . . . . 7 ((𝜑𝑦 ∈ ω ∧ 𝑓:𝑦𝑆) → 𝑦 ∈ ω)
15 simp3 983 . . . . . . 7 ((𝜑𝑦 ∈ ω ∧ 𝑓:𝑦𝑆) → 𝑓:𝑦𝑆)
16 freccl.cl . . . . . . . . 9 ((𝜑𝑧𝑆) → (𝐹𝑧) ∈ 𝑆)
1716ralrimiva 2505 . . . . . . . 8 (𝜑 → ∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆)
18173ad2ant1 1002 . . . . . . 7 ((𝜑𝑦 ∈ ω ∧ 𝑓:𝑦𝑆) → ∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆)
19 freccl.a . . . . . . . 8 (𝜑𝐴𝑆)
20193ad2ant1 1002 . . . . . . 7 ((𝜑𝑦 ∈ ω ∧ 𝑓:𝑦𝑆) → 𝐴𝑆)
2114, 15, 18, 20frecabcl 6296 . . . . . 6 ((𝜑𝑦 ∈ ω ∧ 𝑓:𝑦𝑆) → {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚𝑥 ∈ (𝐹‘(𝑓𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑥𝐴))} ∈ 𝑆)
22 dmeq 4739 . . . . . . . . . . . 12 (𝑔 = 𝑓 → dom 𝑔 = dom 𝑓)
2322eqeq1d 2148 . . . . . . . . . . 11 (𝑔 = 𝑓 → (dom 𝑔 = suc 𝑚 ↔ dom 𝑓 = suc 𝑚))
24 fveq1 5420 . . . . . . . . . . . . 13 (𝑔 = 𝑓 → (𝑔𝑚) = (𝑓𝑚))
2524fveq2d 5425 . . . . . . . . . . . 12 (𝑔 = 𝑓 → (𝐹‘(𝑔𝑚)) = (𝐹‘(𝑓𝑚)))
2625eleq2d 2209 . . . . . . . . . . 11 (𝑔 = 𝑓 → (𝑥 ∈ (𝐹‘(𝑔𝑚)) ↔ 𝑥 ∈ (𝐹‘(𝑓𝑚))))
2723, 26anbi12d 464 . . . . . . . . . 10 (𝑔 = 𝑓 → ((dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ↔ (dom 𝑓 = suc 𝑚𝑥 ∈ (𝐹‘(𝑓𝑚)))))
2827rexbidv 2438 . . . . . . . . 9 (𝑔 = 𝑓 → (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ↔ ∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚𝑥 ∈ (𝐹‘(𝑓𝑚)))))
2922eqeq1d 2148 . . . . . . . . . 10 (𝑔 = 𝑓 → (dom 𝑔 = ∅ ↔ dom 𝑓 = ∅))
3029anbi1d 460 . . . . . . . . 9 (𝑔 = 𝑓 → ((dom 𝑔 = ∅ ∧ 𝑥𝐴) ↔ (dom 𝑓 = ∅ ∧ 𝑥𝐴)))
3128, 30orbi12d 782 . . . . . . . 8 (𝑔 = 𝑓 → ((∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴)) ↔ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚𝑥 ∈ (𝐹‘(𝑓𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑥𝐴))))
3231abbidv 2257 . . . . . . 7 (𝑔 = 𝑓 → {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))} = {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚𝑥 ∈ (𝐹‘(𝑓𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑥𝐴))})
33 eqid 2139 . . . . . . 7 (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))}) = (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})
3432, 33fvmptg 5497 . . . . . 6 ((𝑓 ∈ V ∧ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚𝑥 ∈ (𝐹‘(𝑓𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑥𝐴))} ∈ 𝑆) → ((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})‘𝑓) = {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚𝑥 ∈ (𝐹‘(𝑓𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑥𝐴))})
3513, 21, 34sylancr 410 . . . . 5 ((𝜑𝑦 ∈ ω ∧ 𝑓:𝑦𝑆) → ((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})‘𝑓) = {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚𝑥 ∈ (𝐹‘(𝑓𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑥𝐴))})
3635, 21eqeltrd 2216 . . . 4 ((𝜑𝑦 ∈ ω ∧ 𝑓:𝑦𝑆) → ((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})‘𝑓) ∈ 𝑆)
37 limom 4527 . . . . . . 7 Lim ω
38 limuni 4318 . . . . . . 7 (Lim ω → ω = ω)
3937, 38ax-mp 5 . . . . . 6 ω = ω
4039eleq2i 2206 . . . . 5 (𝑦 ∈ ω ↔ 𝑦 ω)
41 peano2 4509 . . . . . 6 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
4241adantl 275 . . . . 5 ((𝜑𝑦 ∈ ω) → suc 𝑦 ∈ ω)
4340, 42sylan2br 286 . . . 4 ((𝜑𝑦 ω) → suc 𝑦 ∈ ω)
446, 39eleqtrdi 2232 . . . 4 (𝜑𝐵 ω)
452, 10, 12, 36, 43, 44tfrcl 6261 . . 3 (𝜑 → (𝐺𝐵) ∈ 𝑆)
468, 45eqeltrd 2216 . 2 (𝜑 → ((𝐺 ↾ ω)‘𝐵) ∈ 𝑆)
475, 46eqeltrid 2226 1 (𝜑 → (frec(𝐹, 𝐴)‘𝐵) ∈ 𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wo 697  w3a 962   = wceq 1331  wcel 1480  {cab 2125  wral 2416  wrex 2417  Vcvv 2686  c0 3363   cuni 3736  cmpt 3989  Ord word 4284  Lim wlim 4286  suc csuc 4287  ωcom 4504  dom cdm 4539  cres 4541  Fun wfun 5117  wf 5119  cfv 5123  recscrecs 6201  freccfrec 6287
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-recs 6202  df-frec 6288
This theorem is referenced by:  freccl  6300
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