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Theorem bnj1118 32484
 Description: Technical lemma for bnj69 32510. 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 32482 . . 3 𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵))
7 ancl 548 . . 3 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵))))
86, 7bnj101 32221 . 2 𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)))
9 simpr2 1192 . . . 4 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → 𝑖 = suc 𝑗)
101bnj1254 32309 . . . . . . 7 (𝜒𝜓)
11103ad2ant3 1132 . . . . . 6 ((𝜃𝜏𝜒) → 𝜓)
1211ad2antrl 727 . . . . 5 ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → 𝜓)
1312adantr 484 . . . 4 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → 𝜓)
141bnj1232 32303 . . . . . . . . 9 (𝜒𝑛𝐷)
15143ad2ant3 1132 . . . . . . . 8 ((𝜃𝜏𝜒) → 𝑛𝐷)
1615ad2antrl 727 . . . . . . 7 ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → 𝑛𝐷)
1716adantr 484 . . . . . 6 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → 𝑛𝐷)
18 simpr1 1191 . . . . . 6 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → 𝑗𝑛)
192bnj923 32267 . . . . . . . 8 (𝑛𝐷𝑛 ∈ ω)
2019anim1i 617 . . . . . . 7 ((𝑛𝐷𝑗𝑛) → (𝑛 ∈ ω ∧ 𝑗𝑛))
2120ancomd 465 . . . . . 6 ((𝑛𝐷𝑗𝑛) → (𝑗𝑛𝑛 ∈ ω))
2217, 18, 21syl2anc 587 . . . . 5 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → (𝑗𝑛𝑛 ∈ ω))
23 elnn 7589 . . . . 5 ((𝑗𝑛𝑛 ∈ ω) → 𝑗 ∈ ω)
2422, 23syl 17 . . . 4 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → 𝑗 ∈ ω)
254bnj1232 32303 . . . . . 6 (𝜑0𝑖𝑛)
2625adantl 485 . . . . 5 (((𝜃𝜏𝜒) ∧ 𝜑0) → 𝑖𝑛)
2726ad2antlr 726 . . . 4 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → 𝑖𝑛)
289, 13, 24, 27bnj951 32275 . . 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 1143 . . . . . 6 (𝜏 → TrFo(𝐵, 𝐴, 𝑅))
31303ad2ant2 1131 . . . . 5 ((𝜃𝜏𝜒) → TrFo(𝐵, 𝐴, 𝑅))
3231ad2antrl 727 . . . 4 ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → TrFo(𝐵, 𝐴, 𝑅))
33 simp3 1135 . . . 4 ((𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵) → (𝑓𝑗) ⊆ 𝐵)
3432, 33anim12i 615 . . 3 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → ( TrFo(𝐵, 𝐴, 𝑅) ∧ (𝑓𝑗) ⊆ 𝐵))
35 bnj256 32204 . . . . 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 32483 . . . . . . . . 9 (𝜓 ↔ ∀𝑗((𝑗 ∈ ω ∧ suc 𝑗𝑛) → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)))
3837biimpi 219 . . . . . . . 8 (𝜓 → ∀𝑗((𝑗 ∈ ω ∧ suc 𝑗𝑛) → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)))
393819.21bi 2186 . . . . . . 7 (𝜓 → ((𝑗 ∈ ω ∧ suc 𝑗𝑛) → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)))
40 eleq1 2839 . . . . . . . . 9 (𝑖 = suc 𝑗 → (𝑖𝑛 ↔ suc 𝑗𝑛))
4140anbi2d 631 . . . . . . . 8 (𝑖 = suc 𝑗 → ((𝑗 ∈ ω ∧ 𝑖𝑛) ↔ (𝑗 ∈ ω ∧ suc 𝑗𝑛)))
42 fveqeq2 6667 . . . . . . . 8 (𝑖 = suc 𝑗 → ((𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ↔ (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)))
4341, 42imbi12d 348 . . . . . . 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 249 . . . . . 6 (𝑖 = suc 𝑗 → (𝜓 → ((𝑗 ∈ ω ∧ 𝑖𝑛) → (𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅))))
4544imp31 421 . . . . 5 (((𝑖 = suc 𝑗𝜓) ∧ (𝑗 ∈ ω ∧ 𝑖𝑛)) → (𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅))
4635, 45sylbi 220 . . . 4 ((𝑖 = suc 𝑗𝜓𝑗 ∈ ω ∧ 𝑖𝑛) → (𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅))
47 df-bnj19 32195 . . . . . . 7 ( TrFo(𝐵, 𝐴, 𝑅) ↔ ∀𝑦𝐵 pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵)
48 ssralv 3958 . . . . . . 7 ((𝑓𝑗) ⊆ 𝐵 → (∀𝑦𝐵 pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵 → ∀𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵))
4947, 48syl5bi 245 . . . . . 6 ((𝑓𝑗) ⊆ 𝐵 → ( TrFo(𝐵, 𝐴, 𝑅) → ∀𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵))
5049impcom 411 . . . . 5 (( TrFo(𝐵, 𝐴, 𝑅) ∧ (𝑓𝑗) ⊆ 𝐵) → ∀𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵)
51 iunss 4934 . . . . 5 ( 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵 ↔ ∀𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵)
5250, 51sylibr 237 . . . 4 (( TrFo(𝐵, 𝐴, 𝑅) ∧ (𝑓𝑗) ⊆ 𝐵) → 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵)
53 sseq1 3917 . . . . 5 ((𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) → ((𝑓𝑖) ⊆ 𝐵 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵))
5453biimpar 481 . . . 4 (((𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ∧ 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵) → (𝑓𝑖) ⊆ 𝐵)
5546, 52, 54syl2an 598 . . 3 (((𝑖 = suc 𝑗𝜓𝑗 ∈ ω ∧ 𝑖𝑛) ∧ ( TrFo(𝐵, 𝐴, 𝑅) ∧ (𝑓𝑗) ⊆ 𝐵)) → (𝑓𝑖) ⊆ 𝐵)
5628, 34, 55syl2anc 587 . 2 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → (𝑓𝑖) ⊆ 𝐵)
578, 56bnj1023 32280 1 𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑓𝑖) ⊆ 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ≠ wne 2951  ∀wral 3070  Vcvv 3409   ∖ cdif 3855   ⊆ wss 3858  ∅c0 4225  {csn 4522  ∪ ciun 4883   class class class wbr 5032   E cep 5434  dom cdm 5524  suc csuc 6171   Fn wfn 6330  ‘cfv 6335  ωcom 7579   ∧ w-bnj17 32184   predc-bnj14 32186   TrFow-bnj19 32194 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-tr 5139  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fn 6338  df-fv 6343  df-om 7580  df-bnj17 32185  df-bnj19 32195 This theorem is referenced by:  bnj1030  32487
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