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Theorem freccllem 6267
Description: Lemma for freccl 6268. 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 6256 . . . 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 4785 . . . 4 (𝐺 ↾ ω) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})) ↾ ω)
41, 3eqtr4i 2141 . . 3 frec(𝐹, 𝐴) = (𝐺 ↾ ω)
54fveq1i 5390 . 2 (frec(𝐹, 𝐴)‘𝐵) = ((𝐺 ↾ ω)‘𝐵)
6 freccl.b . . . 4 (𝜑𝐵 ∈ ω)
7 fvres 5413 . . . 4 (𝐵 ∈ ω → ((𝐺 ↾ ω)‘𝐵) = (𝐺𝐵))
86, 7syl 14 . . 3 (𝜑 → ((𝐺 ↾ ω)‘𝐵) = (𝐺𝐵))
9 funmpt 5131 . . . . 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 4490 . . . . 5 Ord ω
1211a1i 9 . . . 4 (𝜑 → Ord ω)
13 vex 2663 . . . . . 6 𝑓 ∈ V
14 simp2 967 . . . . . . 7 ((𝜑𝑦 ∈ ω ∧ 𝑓:𝑦𝑆) → 𝑦 ∈ ω)
15 simp3 968 . . . . . . 7 ((𝜑𝑦 ∈ ω ∧ 𝑓:𝑦𝑆) → 𝑓:𝑦𝑆)
16 freccl.cl . . . . . . . . 9 ((𝜑𝑧𝑆) → (𝐹𝑧) ∈ 𝑆)
1716ralrimiva 2482 . . . . . . . 8 (𝜑 → ∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆)
18173ad2ant1 987 . . . . . . 7 ((𝜑𝑦 ∈ ω ∧ 𝑓:𝑦𝑆) → ∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆)
19 freccl.a . . . . . . . 8 (𝜑𝐴𝑆)
20193ad2ant1 987 . . . . . . 7 ((𝜑𝑦 ∈ ω ∧ 𝑓:𝑦𝑆) → 𝐴𝑆)
2114, 15, 18, 20frecabcl 6264 . . . . . 6 ((𝜑𝑦 ∈ ω ∧ 𝑓:𝑦𝑆) → {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚𝑥 ∈ (𝐹‘(𝑓𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑥𝐴))} ∈ 𝑆)
22 dmeq 4709 . . . . . . . . . . . 12 (𝑔 = 𝑓 → dom 𝑔 = dom 𝑓)
2322eqeq1d 2126 . . . . . . . . . . 11 (𝑔 = 𝑓 → (dom 𝑔 = suc 𝑚 ↔ dom 𝑓 = suc 𝑚))
24 fveq1 5388 . . . . . . . . . . . . 13 (𝑔 = 𝑓 → (𝑔𝑚) = (𝑓𝑚))
2524fveq2d 5393 . . . . . . . . . . . 12 (𝑔 = 𝑓 → (𝐹‘(𝑔𝑚)) = (𝐹‘(𝑓𝑚)))
2625eleq2d 2187 . . . . . . . . . . 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 2415 . . . . . . . . 9 (𝑔 = 𝑓 → (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ↔ ∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚𝑥 ∈ (𝐹‘(𝑓𝑚)))))
2922eqeq1d 2126 . . . . . . . . . 10 (𝑔 = 𝑓 → (dom 𝑔 = ∅ ↔ dom 𝑓 = ∅))
3029anbi1d 460 . . . . . . . . 9 (𝑔 = 𝑓 → ((dom 𝑔 = ∅ ∧ 𝑥𝐴) ↔ (dom 𝑓 = ∅ ∧ 𝑥𝐴)))
3128, 30orbi12d 767 . . . . . . . 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 2235 . . . . . . 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 2117 . . . . . . 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 5465 . . . . . 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 2194 . . . 4 ((𝜑𝑦 ∈ ω ∧ 𝑓:𝑦𝑆) → ((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})‘𝑓) ∈ 𝑆)
37 limom 4497 . . . . . . 7 Lim ω
38 limuni 4288 . . . . . . 7 (Lim ω → ω = ω)
3937, 38ax-mp 5 . . . . . 6 ω = ω
4039eleq2i 2184 . . . . 5 (𝑦 ∈ ω ↔ 𝑦 ω)
41 peano2 4479 . . . . . 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 2210 . . . 4 (𝜑𝐵 ω)
452, 10, 12, 36, 43, 44tfrcl 6229 . . 3 (𝜑 → (𝐺𝐵) ∈ 𝑆)
468, 45eqeltrd 2194 . 2 (𝜑 → ((𝐺 ↾ ω)‘𝐵) ∈ 𝑆)
475, 46eqeltrid 2204 1 (𝜑 → (frec(𝐹, 𝐴)‘𝐵) ∈ 𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wo 682  w3a 947   = wceq 1316  wcel 1465  {cab 2103  wral 2393  wrex 2394  Vcvv 2660  c0 3333   cuni 3706  cmpt 3959  Ord word 4254  Lim wlim 4256  suc csuc 4257  ωcom 4474  dom cdm 4509  cres 4511  Fun wfun 5087  wf 5089  cfv 5093  recscrecs 6169  freccfrec 6255
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-recs 6170  df-frec 6256
This theorem is referenced by:  freccl  6268
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