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Theorem bnj1118 33596
Description: Technical lemma for bnj69 33622. 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 33594 . . 3 𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵))
7 ancl 545 . . 3 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵))))
86, 7bnj101 33335 . 2 𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)))
9 simpr2 1195 . . . 4 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → 𝑖 = suc 𝑗)
101bnj1254 33421 . . . . . . 7 (𝜒𝜓)
11103ad2ant3 1135 . . . . . 6 ((𝜃𝜏𝜒) → 𝜓)
1211ad2antrl 726 . . . . 5 ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → 𝜓)
1312adantr 481 . . . 4 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → 𝜓)
141bnj1232 33415 . . . . . . . . 9 (𝜒𝑛𝐷)
15143ad2ant3 1135 . . . . . . . 8 ((𝜃𝜏𝜒) → 𝑛𝐷)
1615ad2antrl 726 . . . . . . 7 ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → 𝑛𝐷)
1716adantr 481 . . . . . 6 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → 𝑛𝐷)
18 simpr1 1194 . . . . . 6 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → 𝑗𝑛)
192bnj923 33380 . . . . . . . 8 (𝑛𝐷𝑛 ∈ ω)
2019anim1i 615 . . . . . . 7 ((𝑛𝐷𝑗𝑛) → (𝑛 ∈ ω ∧ 𝑗𝑛))
2120ancomd 462 . . . . . 6 ((𝑛𝐷𝑗𝑛) → (𝑗𝑛𝑛 ∈ ω))
2217, 18, 21syl2anc 584 . . . . 5 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → (𝑗𝑛𝑛 ∈ ω))
23 elnn 7813 . . . . 5 ((𝑗𝑛𝑛 ∈ ω) → 𝑗 ∈ ω)
2422, 23syl 17 . . . 4 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → 𝑗 ∈ ω)
254bnj1232 33415 . . . . . 6 (𝜑0𝑖𝑛)
2625adantl 482 . . . . 5 (((𝜃𝜏𝜒) ∧ 𝜑0) → 𝑖𝑛)
2726ad2antlr 725 . . . 4 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → 𝑖𝑛)
289, 13, 24, 27bnj951 33387 . . 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 1146 . . . . . 6 (𝜏 → TrFo(𝐵, 𝐴, 𝑅))
31303ad2ant2 1134 . . . . 5 ((𝜃𝜏𝜒) → TrFo(𝐵, 𝐴, 𝑅))
3231ad2antrl 726 . . . 4 ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → TrFo(𝐵, 𝐴, 𝑅))
33 simp3 1138 . . . 4 ((𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵) → (𝑓𝑗) ⊆ 𝐵)
3432, 33anim12i 613 . . 3 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → ( TrFo(𝐵, 𝐴, 𝑅) ∧ (𝑓𝑗) ⊆ 𝐵))
35 bnj256 33318 . . . . 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 33595 . . . . . . . . 9 (𝜓 ↔ ∀𝑗((𝑗 ∈ ω ∧ suc 𝑗𝑛) → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)))
3837biimpi 215 . . . . . . . 8 (𝜓 → ∀𝑗((𝑗 ∈ ω ∧ suc 𝑗𝑛) → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)))
393819.21bi 2182 . . . . . . 7 (𝜓 → ((𝑗 ∈ ω ∧ suc 𝑗𝑛) → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)))
40 eleq1 2825 . . . . . . . . 9 (𝑖 = suc 𝑗 → (𝑖𝑛 ↔ suc 𝑗𝑛))
4140anbi2d 629 . . . . . . . 8 (𝑖 = suc 𝑗 → ((𝑗 ∈ ω ∧ 𝑖𝑛) ↔ (𝑗 ∈ ω ∧ suc 𝑗𝑛)))
42 fveqeq2 6851 . . . . . . . 8 (𝑖 = suc 𝑗 → ((𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ↔ (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)))
4341, 42imbi12d 344 . . . . . . 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 245 . . . . . 6 (𝑖 = suc 𝑗 → (𝜓 → ((𝑗 ∈ ω ∧ 𝑖𝑛) → (𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅))))
4544imp31 418 . . . . 5 (((𝑖 = suc 𝑗𝜓) ∧ (𝑗 ∈ ω ∧ 𝑖𝑛)) → (𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅))
4635, 45sylbi 216 . . . 4 ((𝑖 = suc 𝑗𝜓𝑗 ∈ ω ∧ 𝑖𝑛) → (𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅))
47 df-bnj19 33309 . . . . . . 7 ( TrFo(𝐵, 𝐴, 𝑅) ↔ ∀𝑦𝐵 pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵)
48 ssralv 4010 . . . . . . 7 ((𝑓𝑗) ⊆ 𝐵 → (∀𝑦𝐵 pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵 → ∀𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵))
4947, 48biimtrid 241 . . . . . 6 ((𝑓𝑗) ⊆ 𝐵 → ( TrFo(𝐵, 𝐴, 𝑅) → ∀𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵))
5049impcom 408 . . . . 5 (( TrFo(𝐵, 𝐴, 𝑅) ∧ (𝑓𝑗) ⊆ 𝐵) → ∀𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵)
51 iunss 5005 . . . . 5 ( 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵 ↔ ∀𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵)
5250, 51sylibr 233 . . . 4 (( TrFo(𝐵, 𝐴, 𝑅) ∧ (𝑓𝑗) ⊆ 𝐵) → 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵)
53 sseq1 3969 . . . . 5 ((𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) → ((𝑓𝑖) ⊆ 𝐵 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵))
5453biimpar 478 . . . 4 (((𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ∧ 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵) → (𝑓𝑖) ⊆ 𝐵)
5546, 52, 54syl2an 596 . . 3 (((𝑖 = suc 𝑗𝜓𝑗 ∈ ω ∧ 𝑖𝑛) ∧ ( TrFo(𝐵, 𝐴, 𝑅) ∧ (𝑓𝑗) ⊆ 𝐵)) → (𝑓𝑖) ⊆ 𝐵)
5628, 34, 55syl2anc 584 . 2 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → (𝑓𝑖) ⊆ 𝐵)
578, 56bnj1023 33392 1 𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑓𝑖) ⊆ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087  wal 1539   = wceq 1541  wex 1781  wcel 2106  wne 2943  wral 3064  Vcvv 3445  cdif 3907  wss 3910  c0 4282  {csn 4586   ciun 4954   class class class wbr 5105   E cep 5536  dom cdm 5633  suc csuc 6319   Fn wfn 6491  cfv 6496  ωcom 7802  w-bnj17 33298   predc-bnj14 33300   TrFow-bnj19 33308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-tr 5223  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fn 6499  df-fv 6504  df-om 7803  df-bnj17 33299  df-bnj19 33309
This theorem is referenced by:  bnj1030  33599
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