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Theorem bnj1110 30811
Description: Technical lemma for bnj69 30839. 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 30615 . . . . . . . 8 𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑗𝑛𝑖 = suc 𝑗))
3 bnj219 30562 . . . . . . . . . . 11 (𝑖 = suc 𝑗𝑗 E 𝑖)
43adantl 482 . . . . . . . . . 10 ((𝑗𝑛𝑖 = suc 𝑗) → 𝑗 E 𝑖)
54ancli 573 . . . . . . . . 9 ((𝑗𝑛𝑖 = suc 𝑗) → ((𝑗𝑛𝑖 = suc 𝑗) ∧ 𝑗 E 𝑖))
6 df-3an 1038 . . . . . . . . 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 30612 . . . . . . 7 𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖))
9 bnj1110.3 . . . . . . . . . . . 12 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
109bnj1232 30635 . . . . . . . . . . 11 (𝜒𝑛𝐷)
11103ad2ant3 1082 . . . . . . . . . 10 ((𝜃𝜏𝜒) → 𝑛𝐷)
12 bnj1110.19 . . . . . . . . . . 11 (𝜑0 ↔ (𝑖𝑛𝜎𝑓𝐾𝑖 ∈ dom 𝑓))
1312bnj1232 30635 . . . . . . . . . 10 (𝜑0𝑖𝑛)
1411, 13anim12ci 590 . . . . . . . . 9 (((𝜃𝜏𝜒) ∧ 𝜑0) → (𝑖𝑛𝑛𝐷))
1514anim2i 592 . . . . . . . 8 ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑖 ≠ ∅ ∧ (𝑖𝑛𝑛𝐷)))
16 3anass 1040 . . . . . . . 8 ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) ↔ (𝑖 ≠ ∅ ∧ (𝑖𝑛𝑛𝐷)))
1715, 16sylibr 224 . . . . . . 7 ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷))
188, 17bnj1101 30616 . . . . . 6 𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖))
19 3simpb 1057 . . . . . . . . 9 ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) → (𝑗𝑛𝑗 E 𝑖))
2012bnj1235 30636 . . . . . . . . . . 11 (𝜑0𝜎)
2120ad2antll 764 . . . . . . . . . 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 pm3.43 905 . . . . . . 7 ((((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖)) ∧ ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → 𝜂′)) → ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝜂′)))
2725, 26mpdan 701 . . . . . 6 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖)) → ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝜂′)))
2818, 27bnj101 30550 . . . . 5 𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝜂′))
2912bnj1247 30640 . . . . . . 7 (𝜑0𝑓𝐾)
3029ad2antll 764 . . . . . 6 ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → 𝑓𝐾)
31 pm3.43i 472 . . . . . 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 30550 . . . 4 𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑓𝐾 ∧ ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝜂′)))
34 fndm 5958 . . . . . . . . 9 (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
359, 34bnj770 30594 . . . . . . . 8 (𝜒 → dom 𝑓 = 𝑛)
36353ad2ant3 1082 . . . . . . 7 ((𝜃𝜏𝜒) → dom 𝑓 = 𝑛)
3736ad2antrl 763 . . . . . 6 ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → dom 𝑓 = 𝑛)
3837eleq2d 2684 . . . . 5 ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑗 ∈ dom 𝑓𝑗𝑛))
39 pm3.43i 472 . . . . 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 30550 . . 3 𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → ((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑓𝐾 ∧ ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝜂′))))
42 bnj268 30535 . . . . . 6 (((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ 𝑓𝐾 ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝜂′) ↔ ((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝑓𝐾𝜂′))
43 bnj251 30528 . . . . . 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 326 . . . 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 1771 . . 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 1059 . . . 4 ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) → 𝑗𝑛)
4948bnj706 30585 . . 3 (((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝑓𝐾𝜂′) → 𝑗𝑛)
50 simp2 1060 . . . 4 ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) → 𝑖 = suc 𝑗)
5150bnj706 30585 . . 3 (((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝑓𝐾𝜂′) → 𝑖 = suc 𝑗)
52 bnj258 30534 . . . . 5 (((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝑓𝐾𝜂′) ↔ (((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝜂′) ∧ 𝑓𝐾))
5352simprbi 480 . . . 4 (((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝑓𝐾𝜂′) → 𝑓𝐾)
54 bnj642 30579 . . . . 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 30581 . . . . 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 714 . . 3 (((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝑓𝐾𝜂′) → (𝑓𝑗) ⊆ 𝐵)
6049, 51, 593jca 1240 . 2 (((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝑓𝐾𝜂′) → (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵))
6147, 60bnj1023 30612 1 𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  wne 2790  cdif 3557  wss 3560  c0 3897  {csn 4155   class class class wbr 4623   E cep 4993  dom cdm 5084  suc csuc 5694   Fn wfn 5852  cfv 5857  ωcom 7027  w-bnj17 30512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-tr 4723  df-eprel 4995  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-fn 5860  df-om 7028  df-bnj17 30513
This theorem is referenced by:  bnj1118  30813
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