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Theorem bnj1110 31382
Description: Technical lemma for bnj69 31410. 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
bnj1110.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj1110.7 𝐷 = (ω ∖ {∅})
bnj1110.18 (𝜎 ↔ ((𝑗𝑛𝑗 E 𝑖) → 𝜂′))
bnj1110.19 (𝜑0 ↔ (𝑖𝑛𝜎𝑓𝐾𝑖 ∈ dom 𝑓))
bnj1110.26 (𝜂′ ↔ ((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵))
Assertion
Ref Expression
bnj1110 𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵))
Distinct variable groups:   𝐷,𝑗   𝑖,𝑗   𝑗,𝑛
Allowed substitution hints:   𝜑(𝑓,𝑖,𝑗,𝑛)   𝜓(𝑓,𝑖,𝑗,𝑛)   𝜒(𝑓,𝑖,𝑗,𝑛)   𝜃(𝑓,𝑖,𝑗,𝑛)   𝜏(𝑓,𝑖,𝑗,𝑛)   𝜎(𝑓,𝑖,𝑗,𝑛)   𝐵(𝑓,𝑖,𝑗,𝑛)   𝐷(𝑓,𝑖,𝑛)   𝐾(𝑓,𝑖,𝑗,𝑛)   𝜂′(𝑓,𝑖,𝑗,𝑛)   𝜑0(𝑓,𝑖,𝑗,𝑛)

Proof of Theorem bnj1110
StepHypRef Expression
1 bnj1110.7 . . . . . . . . 9 𝐷 = (ω ∖ {∅})
21bnj1098 31186 . . . . . . . 8 𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑗𝑛𝑖 = suc 𝑗))
3 bnj219 31133 . . . . . . . . . . 11 (𝑖 = suc 𝑗𝑗 E 𝑖)
43adantl 467 . . . . . . . . . 10 ((𝑗𝑛𝑖 = suc 𝑗) → 𝑗 E 𝑖)
54ancli 530 . . . . . . . . 9 ((𝑗𝑛𝑖 = suc 𝑗) → ((𝑗𝑛𝑖 = suc 𝑗) ∧ 𝑗 E 𝑖))
6 df-3an 1072 . . . . . . . . 9 ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ↔ ((𝑗𝑛𝑖 = suc 𝑗) ∧ 𝑗 E 𝑖))
75, 6sylibr 224 . . . . . . . 8 ((𝑗𝑛𝑖 = suc 𝑗) → (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖))
82, 7bnj1023 31183 . . . . . . 7 𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖))
9 bnj1110.3 . . . . . . . . . . . 12 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
109bnj1232 31206 . . . . . . . . . . 11 (𝜒𝑛𝐷)
11103ad2ant3 1128 . . . . . . . . . 10 ((𝜃𝜏𝜒) → 𝑛𝐷)
12 bnj1110.19 . . . . . . . . . . 11 (𝜑0 ↔ (𝑖𝑛𝜎𝑓𝐾𝑖 ∈ dom 𝑓))
1312bnj1232 31206 . . . . . . . . . 10 (𝜑0𝑖𝑛)
1411, 13anim12ci 593 . . . . . . . . 9 (((𝜃𝜏𝜒) ∧ 𝜑0) → (𝑖𝑛𝑛𝐷))
1514anim2i 595 . . . . . . . 8 ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑖 ≠ ∅ ∧ (𝑖𝑛𝑛𝐷)))
16 3anass 1079 . . . . . . . 8 ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) ↔ (𝑖 ≠ ∅ ∧ (𝑖𝑛𝑛𝐷)))
1715, 16sylibr 224 . . . . . . 7 ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷))
188, 17bnj1101 31187 . . . . . 6 𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖))
19 3simpb 1143 . . . . . . . . 9 ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) → (𝑗𝑛𝑗 E 𝑖))
2012bnj1235 31207 . . . . . . . . . . 11 (𝜑0𝜎)
2120ad2antll 700 . . . . . . . . . 10 ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → 𝜎)
22 bnj1110.18 . . . . . . . . . 10 (𝜎 ↔ ((𝑗𝑛𝑗 E 𝑖) → 𝜂′))
2321, 22sylib 208 . . . . . . . . 9 ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → ((𝑗𝑛𝑗 E 𝑖) → 𝜂′))
2419, 23syl5 34 . . . . . . . 8 ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) → 𝜂′))
2524a2i 14 . . . . . . 7 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖)) → ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → 𝜂′))
26 pm3.43 451 . . . . . . 7 ((((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖)) ∧ ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → 𝜂′)) → ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝜂′)))
2725, 26mpdan 659 . . . . . 6 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖)) → ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝜂′)))
2818, 27bnj101 31123 . . . . 5 𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝜂′))
2912bnj1247 31211 . . . . . . 7 (𝜑0𝑓𝐾)
3029ad2antll 700 . . . . . 6 ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → 𝑓𝐾)
31 pm3.43i 450 . . . . . 6 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → 𝑓𝐾) → (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝜂′)) → ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑓𝐾 ∧ ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝜂′)))))
3230, 31ax-mp 5 . . . . 5 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝜂′)) → ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑓𝐾 ∧ ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝜂′))))
3328, 32bnj101 31123 . . . 4 𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑓𝐾 ∧ ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝜂′)))
34 fndm 6130 . . . . . . . . 9 (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
359, 34bnj770 31165 . . . . . . . 8 (𝜒 → dom 𝑓 = 𝑛)
36353ad2ant3 1128 . . . . . . 7 ((𝜃𝜏𝜒) → dom 𝑓 = 𝑛)
3736ad2antrl 699 . . . . . 6 ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → dom 𝑓 = 𝑛)
3837eleq2d 2835 . . . . 5 ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑗 ∈ dom 𝑓𝑗𝑛))
39 pm3.43i 450 . . . . 5 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑗 ∈ dom 𝑓𝑗𝑛)) → (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑓𝐾 ∧ ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝜂′))) → ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → ((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑓𝐾 ∧ ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝜂′))))))
4038, 39ax-mp 5 . . . 4 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑓𝐾 ∧ ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝜂′))) → ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → ((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑓𝐾 ∧ ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝜂′)))))
4133, 40bnj101 31123 . . 3 𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → ((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑓𝐾 ∧ ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝜂′))))
42 bnj268 31109 . . . . . 6 (((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ 𝑓𝐾 ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝜂′) ↔ ((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝑓𝐾𝜂′))
43 bnj251 31102 . . . . . 6 (((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ 𝑓𝐾 ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝜂′) ↔ ((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑓𝐾 ∧ ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝜂′))))
4442, 43bitr3i 266 . . . . 5 (((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝑓𝐾𝜂′) ↔ ((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑓𝐾 ∧ ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝜂′))))
4544imbi2i 325 . . . 4 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → ((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝑓𝐾𝜂′)) ↔ ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → ((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑓𝐾 ∧ ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝜂′)))))
4645exbii 1923 . . 3 (∃𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → ((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝑓𝐾𝜂′)) ↔ ∃𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → ((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑓𝐾 ∧ ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝜂′)))))
4741, 46mpbir 221 . 2 𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → ((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝑓𝐾𝜂′))
48 simp1 1129 . . . 4 ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) → 𝑗𝑛)
4948bnj706 31156 . . 3 (((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝑓𝐾𝜂′) → 𝑗𝑛)
50 simp2 1130 . . . 4 ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) → 𝑖 = suc 𝑗)
5150bnj706 31156 . . 3 (((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝑓𝐾𝜂′) → 𝑖 = suc 𝑗)
52 bnj258 31108 . . . . 5 (((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝑓𝐾𝜂′) ↔ (((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝜂′) ∧ 𝑓𝐾))
5352simprbi 478 . . . 4 (((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝑓𝐾𝜂′) → 𝑓𝐾)
54 bnj642 31150 . . . . 5 (((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝑓𝐾𝜂′) → (𝑗 ∈ dom 𝑓𝑗𝑛))
5549, 54mpbird 247 . . . 4 (((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝑓𝐾𝜂′) → 𝑗 ∈ dom 𝑓)
56 bnj645 31152 . . . . 5 (((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝑓𝐾𝜂′) → 𝜂′)
57 bnj1110.26 . . . . 5 (𝜂′ ↔ ((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵))
5856, 57sylib 208 . . . 4 (((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝑓𝐾𝜂′) → ((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵))
5953, 55, 58mp2and 671 . . 3 (((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝑓𝐾𝜂′) → (𝑓𝑗) ⊆ 𝐵)
6049, 51, 593jca 1121 . 2 (((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝑓𝐾𝜂′) → (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵))
6147, 60bnj1023 31183 1 𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1070   = wceq 1630  wex 1851  wcel 2144  wne 2942  cdif 3718  wss 3721  c0 4061  {csn 4314   class class class wbr 4784   E cep 5161  dom cdm 5249  suc csuc 5868   Fn wfn 6026  cfv 6031  ωcom 7211  w-bnj17 31086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-tr 4885  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-fn 6034  df-om 7212  df-bnj17 31087
This theorem is referenced by:  bnj1118  31384
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