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Theorem freccllem 6403
Description: Lemma for freccl 6404. 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 6392 . . . 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 4904 . . . 4 (𝐺 ↾ ω) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})) ↾ ω)
41, 3eqtr4i 2201 . . 3 frec(𝐹, 𝐴) = (𝐺 ↾ ω)
54fveq1i 5517 . 2 (frec(𝐹, 𝐴)‘𝐵) = ((𝐺 ↾ ω)‘𝐵)
6 freccl.b . . . 4 (𝜑𝐵 ∈ ω)
7 fvres 5540 . . . 4 (𝐵 ∈ ω → ((𝐺 ↾ ω)‘𝐵) = (𝐺𝐵))
86, 7syl 14 . . 3 (𝜑 → ((𝐺 ↾ ω)‘𝐵) = (𝐺𝐵))
9 funmpt 5255 . . . . 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 4607 . . . . 5 Ord ω
1211a1i 9 . . . 4 (𝜑 → Ord ω)
13 vex 2741 . . . . . 6 𝑓 ∈ V
14 simp2 998 . . . . . . 7 ((𝜑𝑦 ∈ ω ∧ 𝑓:𝑦𝑆) → 𝑦 ∈ ω)
15 simp3 999 . . . . . . 7 ((𝜑𝑦 ∈ ω ∧ 𝑓:𝑦𝑆) → 𝑓:𝑦𝑆)
16 freccl.cl . . . . . . . . 9 ((𝜑𝑧𝑆) → (𝐹𝑧) ∈ 𝑆)
1716ralrimiva 2550 . . . . . . . 8 (𝜑 → ∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆)
18173ad2ant1 1018 . . . . . . 7 ((𝜑𝑦 ∈ ω ∧ 𝑓:𝑦𝑆) → ∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆)
19 freccl.a . . . . . . . 8 (𝜑𝐴𝑆)
20193ad2ant1 1018 . . . . . . 7 ((𝜑𝑦 ∈ ω ∧ 𝑓:𝑦𝑆) → 𝐴𝑆)
2114, 15, 18, 20frecabcl 6400 . . . . . 6 ((𝜑𝑦 ∈ ω ∧ 𝑓:𝑦𝑆) → {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚𝑥 ∈ (𝐹‘(𝑓𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑥𝐴))} ∈ 𝑆)
22 dmeq 4828 . . . . . . . . . . . 12 (𝑔 = 𝑓 → dom 𝑔 = dom 𝑓)
2322eqeq1d 2186 . . . . . . . . . . 11 (𝑔 = 𝑓 → (dom 𝑔 = suc 𝑚 ↔ dom 𝑓 = suc 𝑚))
24 fveq1 5515 . . . . . . . . . . . . 13 (𝑔 = 𝑓 → (𝑔𝑚) = (𝑓𝑚))
2524fveq2d 5520 . . . . . . . . . . . 12 (𝑔 = 𝑓 → (𝐹‘(𝑔𝑚)) = (𝐹‘(𝑓𝑚)))
2625eleq2d 2247 . . . . . . . . . . 11 (𝑔 = 𝑓 → (𝑥 ∈ (𝐹‘(𝑔𝑚)) ↔ 𝑥 ∈ (𝐹‘(𝑓𝑚))))
2723, 26anbi12d 473 . . . . . . . . . 10 (𝑔 = 𝑓 → ((dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ↔ (dom 𝑓 = suc 𝑚𝑥 ∈ (𝐹‘(𝑓𝑚)))))
2827rexbidv 2478 . . . . . . . . 9 (𝑔 = 𝑓 → (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ↔ ∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚𝑥 ∈ (𝐹‘(𝑓𝑚)))))
2922eqeq1d 2186 . . . . . . . . . 10 (𝑔 = 𝑓 → (dom 𝑔 = ∅ ↔ dom 𝑓 = ∅))
3029anbi1d 465 . . . . . . . . 9 (𝑔 = 𝑓 → ((dom 𝑔 = ∅ ∧ 𝑥𝐴) ↔ (dom 𝑓 = ∅ ∧ 𝑥𝐴)))
3128, 30orbi12d 793 . . . . . . . 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 2295 . . . . . . 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 2177 . . . . . . 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 5593 . . . . . 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 414 . . . . 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 2254 . . . 4 ((𝜑𝑦 ∈ ω ∧ 𝑓:𝑦𝑆) → ((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})‘𝑓) ∈ 𝑆)
37 limom 4614 . . . . . . 7 Lim ω
38 limuni 4397 . . . . . . 7 (Lim ω → ω = ω)
3937, 38ax-mp 5 . . . . . 6 ω = ω
4039eleq2i 2244 . . . . 5 (𝑦 ∈ ω ↔ 𝑦 ω)
41 peano2 4595 . . . . . 6 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
4241adantl 277 . . . . 5 ((𝜑𝑦 ∈ ω) → suc 𝑦 ∈ ω)
4340, 42sylan2br 288 . . . 4 ((𝜑𝑦 ω) → suc 𝑦 ∈ ω)
446, 39eleqtrdi 2270 . . . 4 (𝜑𝐵 ω)
452, 10, 12, 36, 43, 44tfrcl 6365 . . 3 (𝜑 → (𝐺𝐵) ∈ 𝑆)
468, 45eqeltrd 2254 . 2 (𝜑 → ((𝐺 ↾ ω)‘𝐵) ∈ 𝑆)
475, 46eqeltrid 2264 1 (𝜑 → (frec(𝐹, 𝐴)‘𝐵) ∈ 𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wo 708  w3a 978   = wceq 1353  wcel 2148  {cab 2163  wral 2455  wrex 2456  Vcvv 2738  c0 3423   cuni 3810  cmpt 4065  Ord word 4363  Lim wlim 4365  suc csuc 4366  ωcom 4590  dom cdm 4627  cres 4629  Fun wfun 5211  wf 5213  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-in1 614  ax-in2 615  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-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-ilim 4370  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-recs 6306  df-frec 6392
This theorem is referenced by:  freccl  6404
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