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Theorem bnj1118 31503
Description: Technical lemma for bnj69 31529. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1118.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1118.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj1118.5 (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))
bnj1118.7 𝐷 = (ω ∖ {∅})
bnj1118.18 (𝜎 ↔ ((𝑗𝑛𝑗 E 𝑖) → 𝜂′))
bnj1118.19 (𝜑0 ↔ (𝑖𝑛𝜎𝑓𝐾𝑖 ∈ dom 𝑓))
bnj1118.26 (𝜂′ ↔ ((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵))
Assertion
Ref Expression
bnj1118 𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑓𝑖) ⊆ 𝐵)
Distinct variable groups:   𝐴,𝑖,𝑗,𝑦   𝑦,𝐵   𝐷,𝑗   𝑅,𝑖,𝑗,𝑦   𝑓,𝑖,𝑗,𝑦   𝑖,𝑛,𝑗
Allowed substitution hints:   𝜑(𝑦,𝑓,𝑖,𝑗,𝑛)   𝜓(𝑦,𝑓,𝑖,𝑗,𝑛)   𝜒(𝑦,𝑓,𝑖,𝑗,𝑛)   𝜃(𝑦,𝑓,𝑖,𝑗,𝑛)   𝜏(𝑦,𝑓,𝑖,𝑗,𝑛)   𝜎(𝑦,𝑓,𝑖,𝑗,𝑛)   𝐴(𝑓,𝑛)   𝐵(𝑓,𝑖,𝑗,𝑛)   𝐷(𝑦,𝑓,𝑖,𝑛)   𝑅(𝑓,𝑛)   𝐾(𝑦,𝑓,𝑖,𝑗,𝑛)   𝑋(𝑦,𝑓,𝑖,𝑗,𝑛)   𝜂′(𝑦,𝑓,𝑖,𝑗,𝑛)   𝜑0(𝑦,𝑓,𝑖,𝑗,𝑛)

Proof of Theorem bnj1118
StepHypRef Expression
1 bnj1118.3 . . . 4 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
2 bnj1118.7 . . . 4 𝐷 = (ω ∖ {∅})
3 bnj1118.18 . . . 4 (𝜎 ↔ ((𝑗𝑛𝑗 E 𝑖) → 𝜂′))
4 bnj1118.19 . . . 4 (𝜑0 ↔ (𝑖𝑛𝜎𝑓𝐾𝑖 ∈ dom 𝑓))
5 bnj1118.26 . . . 4 (𝜂′ ↔ ((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵))
61, 2, 3, 4, 5bnj1110 31501 . . 3 𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵))
7 ancl 540 . . 3 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵))))
86, 7bnj101 31243 . 2 𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)))
9 simpr2 1250 . . . 4 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → 𝑖 = suc 𝑗)
101bnj1254 31331 . . . . . . 7 (𝜒𝜓)
11103ad2ant3 1165 . . . . . 6 ((𝜃𝜏𝜒) → 𝜓)
1211ad2antrl 719 . . . . 5 ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → 𝜓)
1312adantr 472 . . . 4 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → 𝜓)
141bnj1232 31325 . . . . . . . . 9 (𝜒𝑛𝐷)
15143ad2ant3 1165 . . . . . . . 8 ((𝜃𝜏𝜒) → 𝑛𝐷)
1615ad2antrl 719 . . . . . . 7 ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → 𝑛𝐷)
1716adantr 472 . . . . . 6 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → 𝑛𝐷)
18 simpr1 1248 . . . . . 6 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → 𝑗𝑛)
192bnj923 31289 . . . . . . . 8 (𝑛𝐷𝑛 ∈ ω)
2019anim1i 608 . . . . . . 7 ((𝑛𝐷𝑗𝑛) → (𝑛 ∈ ω ∧ 𝑗𝑛))
2120ancomd 453 . . . . . 6 ((𝑛𝐷𝑗𝑛) → (𝑗𝑛𝑛 ∈ ω))
2217, 18, 21syl2anc 579 . . . . 5 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → (𝑗𝑛𝑛 ∈ ω))
23 elnn 7275 . . . . 5 ((𝑗𝑛𝑛 ∈ ω) → 𝑗 ∈ ω)
2422, 23syl 17 . . . 4 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → 𝑗 ∈ ω)
254bnj1232 31325 . . . . . 6 (𝜑0𝑖𝑛)
2625adantl 473 . . . . 5 (((𝜃𝜏𝜒) ∧ 𝜑0) → 𝑖𝑛)
2726ad2antlr 718 . . . 4 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → 𝑖𝑛)
289, 13, 24, 27bnj951 31297 . . 3 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → (𝑖 = suc 𝑗𝜓𝑗 ∈ ω ∧ 𝑖𝑛))
29 bnj1118.5 . . . . . . 7 (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))
3029simp2bi 1176 . . . . . 6 (𝜏 → TrFo(𝐵, 𝐴, 𝑅))
31303ad2ant2 1164 . . . . 5 ((𝜃𝜏𝜒) → TrFo(𝐵, 𝐴, 𝑅))
3231ad2antrl 719 . . . 4 ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → TrFo(𝐵, 𝐴, 𝑅))
33 simp3 1168 . . . 4 ((𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵) → (𝑓𝑗) ⊆ 𝐵)
3432, 33anim12i 606 . . 3 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → ( TrFo(𝐵, 𝐴, 𝑅) ∧ (𝑓𝑗) ⊆ 𝐵))
35 bnj256 31226 . . . . 5 ((𝑖 = suc 𝑗𝜓𝑗 ∈ ω ∧ 𝑖𝑛) ↔ ((𝑖 = suc 𝑗𝜓) ∧ (𝑗 ∈ ω ∧ 𝑖𝑛)))
36 bnj1118.2 . . . . . . . . . 10 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
3736bnj1112 31502 . . . . . . . . 9 (𝜓 ↔ ∀𝑗((𝑗 ∈ ω ∧ suc 𝑗𝑛) → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)))
3837biimpi 207 . . . . . . . 8 (𝜓 → ∀𝑗((𝑗 ∈ ω ∧ suc 𝑗𝑛) → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)))
393819.21bi 2221 . . . . . . 7 (𝜓 → ((𝑗 ∈ ω ∧ suc 𝑗𝑛) → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)))
40 eleq1 2832 . . . . . . . . 9 (𝑖 = suc 𝑗 → (𝑖𝑛 ↔ suc 𝑗𝑛))
4140anbi2d 622 . . . . . . . 8 (𝑖 = suc 𝑗 → ((𝑗 ∈ ω ∧ 𝑖𝑛) ↔ (𝑗 ∈ ω ∧ suc 𝑗𝑛)))
42 fveqeq2 6386 . . . . . . . 8 (𝑖 = suc 𝑗 → ((𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ↔ (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)))
4341, 42imbi12d 335 . . . . . . 7 (𝑖 = suc 𝑗 → (((𝑗 ∈ ω ∧ 𝑖𝑛) → (𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)) ↔ ((𝑗 ∈ ω ∧ suc 𝑗𝑛) → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅))))
4439, 43syl5ibr 237 . . . . . 6 (𝑖 = suc 𝑗 → (𝜓 → ((𝑗 ∈ ω ∧ 𝑖𝑛) → (𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅))))
4544imp31 408 . . . . 5 (((𝑖 = suc 𝑗𝜓) ∧ (𝑗 ∈ ω ∧ 𝑖𝑛)) → (𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅))
4635, 45sylbi 208 . . . 4 ((𝑖 = suc 𝑗𝜓𝑗 ∈ ω ∧ 𝑖𝑛) → (𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅))
47 df-bnj19 31217 . . . . . . 7 ( TrFo(𝐵, 𝐴, 𝑅) ↔ ∀𝑦𝐵 pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵)
48 ssralv 3828 . . . . . . 7 ((𝑓𝑗) ⊆ 𝐵 → (∀𝑦𝐵 pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵 → ∀𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵))
4947, 48syl5bi 233 . . . . . 6 ((𝑓𝑗) ⊆ 𝐵 → ( TrFo(𝐵, 𝐴, 𝑅) → ∀𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵))
5049impcom 396 . . . . 5 (( TrFo(𝐵, 𝐴, 𝑅) ∧ (𝑓𝑗) ⊆ 𝐵) → ∀𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵)
51 iunss 4719 . . . . 5 ( 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵 ↔ ∀𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵)
5250, 51sylibr 225 . . . 4 (( TrFo(𝐵, 𝐴, 𝑅) ∧ (𝑓𝑗) ⊆ 𝐵) → 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵)
53 sseq1 3788 . . . . 5 ((𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) → ((𝑓𝑖) ⊆ 𝐵 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵))
5453biimpar 469 . . . 4 (((𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ∧ 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵) → (𝑓𝑖) ⊆ 𝐵)
5546, 52, 54syl2an 589 . . 3 (((𝑖 = suc 𝑗𝜓𝑗 ∈ ω ∧ 𝑖𝑛) ∧ ( TrFo(𝐵, 𝐴, 𝑅) ∧ (𝑓𝑗) ⊆ 𝐵)) → (𝑓𝑖) ⊆ 𝐵)
5628, 34, 55syl2anc 579 . 2 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → (𝑓𝑖) ⊆ 𝐵)
578, 56bnj1023 31302 1 𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑓𝑖) ⊆ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107  wal 1650   = wceq 1652  wex 1874  wcel 2155  wne 2937  wral 3055  Vcvv 3350  cdif 3731  wss 3734  c0 4081  {csn 4336   ciun 4678   class class class wbr 4811   E cep 5191  dom cdm 5279  suc csuc 5912   Fn wfn 6065  cfv 6070  ωcom 7265  w-bnj17 31206   predc-bnj14 31208   TrFow-bnj19 31216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064  ax-un 7149
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-tr 4914  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fn 6073  df-fv 6078  df-om 7266  df-bnj17 31207  df-bnj19 31217
This theorem is referenced by:  bnj1030  31506
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