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Theorem frecsuclem3 5929
Description: Lemma for frecsuc 5930. (Contributed by Jim Kingdon, 15-Aug-2019.)
Hypothesis
Ref Expression
frecsuclem1.h 𝐺 = (g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})
Assertion
Ref Expression
frecsuclem3 ((z(𝐹z) V A 𝑉 B 𝜔) → (frec(𝐹, A)‘suc B) = (𝐹‘(frec(𝐹, A)‘B)))
Distinct variable groups:   A,g,𝑚,x,z   B,g,𝑚,x,z   g,𝐹,𝑚,x,z   g,𝐺,𝑚,x,z   g,𝑉,𝑚,x
Allowed substitution hint:   𝑉(z)

Proof of Theorem frecsuclem3
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 eqid 2037 . . . . . . . . . . . . 13 recs(𝐺) = recs(𝐺)
2 frecsuclem1.h . . . . . . . . . . . . . 14 𝐺 = (g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})
32frectfr 5924 . . . . . . . . . . . . 13 ((z(𝐹z) V A 𝑉) → y(Fun 𝐺 (𝐺y) V))
41, 3tfri1d 5890 . . . . . . . . . . . 12 ((z(𝐹z) V A 𝑉) → recs(𝐺) Fn On)
5 fnfun 4939 . . . . . . . . . . . 12 (recs(𝐺) Fn On → Fun recs(𝐺))
64, 5syl 14 . . . . . . . . . . 11 ((z(𝐹z) V A 𝑉) → Fun recs(𝐺))
763adant3 923 . . . . . . . . . 10 ((z(𝐹z) V A 𝑉 B 𝜔) → Fun recs(𝐺))
8 peano2 4261 . . . . . . . . . . 11 (B 𝜔 → suc B 𝜔)
983ad2ant3 926 . . . . . . . . . 10 ((z(𝐹z) V A 𝑉 B 𝜔) → suc B 𝜔)
10 resfunexg 5325 . . . . . . . . . 10 ((Fun recs(𝐺) suc B 𝜔) → (recs(𝐺) ↾ suc B) V)
117, 9, 10syl2anc 391 . . . . . . . . 9 ((z(𝐹z) V A 𝑉 B 𝜔) → (recs(𝐺) ↾ suc B) V)
12 simp1 903 . . . . . . . . 9 ((z(𝐹z) V A 𝑉 B 𝜔) → z(𝐹z) V)
13 simp2 904 . . . . . . . . 9 ((z(𝐹z) V A 𝑉 B 𝜔) → A 𝑉)
1411, 12, 13frecabex 5923 . . . . . . . 8 ((z(𝐹z) V A 𝑉 B 𝜔) → {x ∣ (𝑚 𝜔 (dom (recs(𝐺) ↾ suc B) = suc 𝑚 x (𝐹‘((recs(𝐺) ↾ suc B)‘𝑚))) (dom (recs(𝐺) ↾ suc B) = ∅ x A))} V)
15 dmeq 4478 . . . . . . . . . . . . . . . . 17 (g = (recs(𝐺) ↾ suc B) → dom g = dom (recs(𝐺) ↾ suc B))
1615eqeq1d 2045 . . . . . . . . . . . . . . . 16 (g = (recs(𝐺) ↾ suc B) → (dom g = suc 𝑚 ↔ dom (recs(𝐺) ↾ suc B) = suc 𝑚))
17 fveq1 5120 . . . . . . . . . . . . . . . . . 18 (g = (recs(𝐺) ↾ suc B) → (g𝑚) = ((recs(𝐺) ↾ suc B)‘𝑚))
1817fveq2d 5125 . . . . . . . . . . . . . . . . 17 (g = (recs(𝐺) ↾ suc B) → (𝐹‘(g𝑚)) = (𝐹‘((recs(𝐺) ↾ suc B)‘𝑚)))
1918eleq2d 2104 . . . . . . . . . . . . . . . 16 (g = (recs(𝐺) ↾ suc B) → (x (𝐹‘(g𝑚)) ↔ x (𝐹‘((recs(𝐺) ↾ suc B)‘𝑚))))
2016, 19anbi12d 442 . . . . . . . . . . . . . . 15 (g = (recs(𝐺) ↾ suc B) → ((dom g = suc 𝑚 x (𝐹‘(g𝑚))) ↔ (dom (recs(𝐺) ↾ suc B) = suc 𝑚 x (𝐹‘((recs(𝐺) ↾ suc B)‘𝑚)))))
2120rexbidv 2321 . . . . . . . . . . . . . 14 (g = (recs(𝐺) ↾ suc B) → (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) ↔ 𝑚 𝜔 (dom (recs(𝐺) ↾ suc B) = suc 𝑚 x (𝐹‘((recs(𝐺) ↾ suc B)‘𝑚)))))
2215eqeq1d 2045 . . . . . . . . . . . . . . 15 (g = (recs(𝐺) ↾ suc B) → (dom g = ∅ ↔ dom (recs(𝐺) ↾ suc B) = ∅))
2322anbi1d 438 . . . . . . . . . . . . . 14 (g = (recs(𝐺) ↾ suc B) → ((dom g = ∅ x A) ↔ (dom (recs(𝐺) ↾ suc B) = ∅ x A)))
2421, 23orbi12d 706 . . . . . . . . . . . . 13 (g = (recs(𝐺) ↾ suc B) → ((𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A)) ↔ (𝑚 𝜔 (dom (recs(𝐺) ↾ suc B) = suc 𝑚 x (𝐹‘((recs(𝐺) ↾ suc B)‘𝑚))) (dom (recs(𝐺) ↾ suc B) = ∅ x A))))
2524abbidv 2152 . . . . . . . . . . . 12 (g = (recs(𝐺) ↾ suc B) → {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))} = {x ∣ (𝑚 𝜔 (dom (recs(𝐺) ↾ suc B) = suc 𝑚 x (𝐹‘((recs(𝐺) ↾ suc B)‘𝑚))) (dom (recs(𝐺) ↾ suc B) = ∅ x A))})
2625, 2fvmptg 5191 . . . . . . . . . . 11 (((recs(𝐺) ↾ suc B) V {x ∣ (𝑚 𝜔 (dom (recs(𝐺) ↾ suc B) = suc 𝑚 x (𝐹‘((recs(𝐺) ↾ suc B)‘𝑚))) (dom (recs(𝐺) ↾ suc B) = ∅ x A))} V) → (𝐺‘(recs(𝐺) ↾ suc B)) = {x ∣ (𝑚 𝜔 (dom (recs(𝐺) ↾ suc B) = suc 𝑚 x (𝐹‘((recs(𝐺) ↾ suc B)‘𝑚))) (dom (recs(𝐺) ↾ suc B) = ∅ x A))})
2726ex 108 . . . . . . . . . 10 ((recs(𝐺) ↾ suc B) V → ({x ∣ (𝑚 𝜔 (dom (recs(𝐺) ↾ suc B) = suc 𝑚 x (𝐹‘((recs(𝐺) ↾ suc B)‘𝑚))) (dom (recs(𝐺) ↾ suc B) = ∅ x A))} V → (𝐺‘(recs(𝐺) ↾ suc B)) = {x ∣ (𝑚 𝜔 (dom (recs(𝐺) ↾ suc B) = suc 𝑚 x (𝐹‘((recs(𝐺) ↾ suc B)‘𝑚))) (dom (recs(𝐺) ↾ suc B) = ∅ x A))}))
2811, 27syl 14 . . . . . . . . 9 ((z(𝐹z) V A 𝑉 B 𝜔) → ({x ∣ (𝑚 𝜔 (dom (recs(𝐺) ↾ suc B) = suc 𝑚 x (𝐹‘((recs(𝐺) ↾ suc B)‘𝑚))) (dom (recs(𝐺) ↾ suc B) = ∅ x A))} V → (𝐺‘(recs(𝐺) ↾ suc B)) = {x ∣ (𝑚 𝜔 (dom (recs(𝐺) ↾ suc B) = suc 𝑚 x (𝐹‘((recs(𝐺) ↾ suc B)‘𝑚))) (dom (recs(𝐺) ↾ suc B) = ∅ x A))}))
292frecsuclem1 5926 . . . . . . . . . 10 ((z(𝐹z) V A 𝑉 B 𝜔) → (frec(𝐹, A)‘suc B) = (𝐺‘(recs(𝐺) ↾ suc B)))
3029eqeq1d 2045 . . . . . . . . 9 ((z(𝐹z) V A 𝑉 B 𝜔) → ((frec(𝐹, A)‘suc B) = {x ∣ (𝑚 𝜔 (dom (recs(𝐺) ↾ suc B) = suc 𝑚 x (𝐹‘((recs(𝐺) ↾ suc B)‘𝑚))) (dom (recs(𝐺) ↾ suc B) = ∅ x A))} ↔ (𝐺‘(recs(𝐺) ↾ suc B)) = {x ∣ (𝑚 𝜔 (dom (recs(𝐺) ↾ suc B) = suc 𝑚 x (𝐹‘((recs(𝐺) ↾ suc B)‘𝑚))) (dom (recs(𝐺) ↾ suc B) = ∅ x A))}))
3128, 30sylibrd 158 . . . . . . . 8 ((z(𝐹z) V A 𝑉 B 𝜔) → ({x ∣ (𝑚 𝜔 (dom (recs(𝐺) ↾ suc B) = suc 𝑚 x (𝐹‘((recs(𝐺) ↾ suc B)‘𝑚))) (dom (recs(𝐺) ↾ suc B) = ∅ x A))} V → (frec(𝐹, A)‘suc B) = {x ∣ (𝑚 𝜔 (dom (recs(𝐺) ↾ suc B) = suc 𝑚 x (𝐹‘((recs(𝐺) ↾ suc B)‘𝑚))) (dom (recs(𝐺) ↾ suc B) = ∅ x A))}))
3214, 31mpd 13 . . . . . . 7 ((z(𝐹z) V A 𝑉 B 𝜔) → (frec(𝐹, A)‘suc B) = {x ∣ (𝑚 𝜔 (dom (recs(𝐺) ↾ suc B) = suc 𝑚 x (𝐹‘((recs(𝐺) ↾ suc B)‘𝑚))) (dom (recs(𝐺) ↾ suc B) = ∅ x A))})
3332abeq2d 2147 . . . . . 6 ((z(𝐹z) V A 𝑉 B 𝜔) → (x (frec(𝐹, A)‘suc B) ↔ (𝑚 𝜔 (dom (recs(𝐺) ↾ suc B) = suc 𝑚 x (𝐹‘((recs(𝐺) ↾ suc B)‘𝑚))) (dom (recs(𝐺) ↾ suc B) = ∅ x A))))
342frecsuclemdm 5927 . . . . . . . . . . 11 ((z(𝐹z) V A 𝑉 B 𝜔) → dom (recs(𝐺) ↾ suc B) = suc B)
35 peano3 4262 . . . . . . . . . . . 12 (B 𝜔 → suc B ≠ ∅)
36353ad2ant3 926 . . . . . . . . . . 11 ((z(𝐹z) V A 𝑉 B 𝜔) → suc B ≠ ∅)
3734, 36eqnetrd 2223 . . . . . . . . . 10 ((z(𝐹z) V A 𝑉 B 𝜔) → dom (recs(𝐺) ↾ suc B) ≠ ∅)
3837neneqd 2221 . . . . . . . . 9 ((z(𝐹z) V A 𝑉 B 𝜔) → ¬ dom (recs(𝐺) ↾ suc B) = ∅)
3938intnanrd 840 . . . . . . . 8 ((z(𝐹z) V A 𝑉 B 𝜔) → ¬ (dom (recs(𝐺) ↾ suc B) = ∅ x A))
40 biorf 662 . . . . . . . 8 (¬ (dom (recs(𝐺) ↾ suc B) = ∅ x A) → (𝑚 𝜔 (dom (recs(𝐺) ↾ suc B) = suc 𝑚 x (𝐹‘((recs(𝐺) ↾ suc B)‘𝑚))) ↔ ((dom (recs(𝐺) ↾ suc B) = ∅ x A) 𝑚 𝜔 (dom (recs(𝐺) ↾ suc B) = suc 𝑚 x (𝐹‘((recs(𝐺) ↾ suc B)‘𝑚))))))
4139, 40syl 14 . . . . . . 7 ((z(𝐹z) V A 𝑉 B 𝜔) → (𝑚 𝜔 (dom (recs(𝐺) ↾ suc B) = suc 𝑚 x (𝐹‘((recs(𝐺) ↾ suc B)‘𝑚))) ↔ ((dom (recs(𝐺) ↾ suc B) = ∅ x A) 𝑚 𝜔 (dom (recs(𝐺) ↾ suc B) = suc 𝑚 x (𝐹‘((recs(𝐺) ↾ suc B)‘𝑚))))))
42 orcom 646 . . . . . . 7 (((dom (recs(𝐺) ↾ suc B) = ∅ x A) 𝑚 𝜔 (dom (recs(𝐺) ↾ suc B) = suc 𝑚 x (𝐹‘((recs(𝐺) ↾ suc B)‘𝑚)))) ↔ (𝑚 𝜔 (dom (recs(𝐺) ↾ suc B) = suc 𝑚 x (𝐹‘((recs(𝐺) ↾ suc B)‘𝑚))) (dom (recs(𝐺) ↾ suc B) = ∅ x A)))
4341, 42syl6bb 185 . . . . . 6 ((z(𝐹z) V A 𝑉 B 𝜔) → (𝑚 𝜔 (dom (recs(𝐺) ↾ suc B) = suc 𝑚 x (𝐹‘((recs(𝐺) ↾ suc B)‘𝑚))) ↔ (𝑚 𝜔 (dom (recs(𝐺) ↾ suc B) = suc 𝑚 x (𝐹‘((recs(𝐺) ↾ suc B)‘𝑚))) (dom (recs(𝐺) ↾ suc B) = ∅ x A))))
4434eqeq1d 2045 . . . . . . . . . 10 ((z(𝐹z) V A 𝑉 B 𝜔) → (dom (recs(𝐺) ↾ suc B) = suc 𝑚 ↔ suc B = suc 𝑚))
45 vex 2554 . . . . . . . . . . . 12 𝑚 V
46 suc11g 4235 . . . . . . . . . . . 12 ((B 𝜔 𝑚 V) → (suc B = suc 𝑚B = 𝑚))
4745, 46mpan2 401 . . . . . . . . . . 11 (B 𝜔 → (suc B = suc 𝑚B = 𝑚))
48473ad2ant3 926 . . . . . . . . . 10 ((z(𝐹z) V A 𝑉 B 𝜔) → (suc B = suc 𝑚B = 𝑚))
4944, 48bitrd 177 . . . . . . . . 9 ((z(𝐹z) V A 𝑉 B 𝜔) → (dom (recs(𝐺) ↾ suc B) = suc 𝑚B = 𝑚))
50 eqcom 2039 . . . . . . . . 9 (B = 𝑚𝑚 = B)
5149, 50syl6bb 185 . . . . . . . 8 ((z(𝐹z) V A 𝑉 B 𝜔) → (dom (recs(𝐺) ↾ suc B) = suc 𝑚𝑚 = B))
5251anbi1d 438 . . . . . . 7 ((z(𝐹z) V A 𝑉 B 𝜔) → ((dom (recs(𝐺) ↾ suc B) = suc 𝑚 x (𝐹‘((recs(𝐺) ↾ suc B)‘𝑚))) ↔ (𝑚 = B x (𝐹‘((recs(𝐺) ↾ suc B)‘𝑚)))))
5352rexbidv 2321 . . . . . 6 ((z(𝐹z) V A 𝑉 B 𝜔) → (𝑚 𝜔 (dom (recs(𝐺) ↾ suc B) = suc 𝑚 x (𝐹‘((recs(𝐺) ↾ suc B)‘𝑚))) ↔ 𝑚 𝜔 (𝑚 = B x (𝐹‘((recs(𝐺) ↾ suc B)‘𝑚)))))
5433, 43, 533bitr2d 205 . . . . 5 ((z(𝐹z) V A 𝑉 B 𝜔) → (x (frec(𝐹, A)‘suc B) ↔ 𝑚 𝜔 (𝑚 = B x (𝐹‘((recs(𝐺) ↾ suc B)‘𝑚)))))
55 fveq2 5121 . . . . . . . 8 (𝑚 = B → ((recs(𝐺) ↾ suc B)‘𝑚) = ((recs(𝐺) ↾ suc B)‘B))
5655fveq2d 5125 . . . . . . 7 (𝑚 = B → (𝐹‘((recs(𝐺) ↾ suc B)‘𝑚)) = (𝐹‘((recs(𝐺) ↾ suc B)‘B)))
5756eleq2d 2104 . . . . . 6 (𝑚 = B → (x (𝐹‘((recs(𝐺) ↾ suc B)‘𝑚)) ↔ x (𝐹‘((recs(𝐺) ↾ suc B)‘B))))
5857ceqsrexbv 2669 . . . . 5 (𝑚 𝜔 (𝑚 = B x (𝐹‘((recs(𝐺) ↾ suc B)‘𝑚))) ↔ (B 𝜔 x (𝐹‘((recs(𝐺) ↾ suc B)‘B))))
5954, 58syl6bb 185 . . . 4 ((z(𝐹z) V A 𝑉 B 𝜔) → (x (frec(𝐹, A)‘suc B) ↔ (B 𝜔 x (𝐹‘((recs(𝐺) ↾ suc B)‘B)))))
60593anibar 1071 . . 3 ((z(𝐹z) V A 𝑉 B 𝜔) → (x (frec(𝐹, A)‘suc B) ↔ x (𝐹‘((recs(𝐺) ↾ suc B)‘B))))
6160eqrdv 2035 . 2 ((z(𝐹z) V A 𝑉 B 𝜔) → (frec(𝐹, A)‘suc B) = (𝐹‘((recs(𝐺) ↾ suc B)‘B)))
622frecsuclem2 5928 . . 3 ((z(𝐹z) V A 𝑉 B 𝜔) → ((recs(𝐺) ↾ suc B)‘B) = (frec(𝐹, A)‘B))
6362fveq2d 5125 . 2 ((z(𝐹z) V A 𝑉 B 𝜔) → (𝐹‘((recs(𝐺) ↾ suc B)‘B)) = (𝐹‘(frec(𝐹, A)‘B)))
6461, 63eqtrd 2069 1 ((z(𝐹z) V A 𝑉 B 𝜔) → (frec(𝐹, A)‘suc B) = (𝐹‘(frec(𝐹, A)‘B)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 628   w3a 884  wal 1240   = wceq 1242   wcel 1390  {cab 2023  wne 2201  wrex 2301  Vcvv 2551  c0 3218  cmpt 3809  Oncon0 4066  suc csuc 4068  𝜔com 4256  dom cdm 4288  cres 4290  Fun wfun 4839   Fn wfn 4840  cfv 4845  recscrecs 5860  freccfrec 5917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-recs 5861  df-frec 5918
This theorem is referenced by:  frecsuc  5930
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