Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1110 Structured version   Visualization version   GIF version

Theorem bnj1110 33634
Description: Technical lemma for bnj69 33662. 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 33435 . . . . . . . 8 𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑗𝑛𝑖 = suc 𝑗))
3 bnj219 33385 . . . . . . . . . . 11 (𝑖 = suc 𝑗𝑗 E 𝑖)
43adantl 483 . . . . . . . . . 10 ((𝑗𝑛𝑖 = suc 𝑗) → 𝑗 E 𝑖)
54ancli 550 . . . . . . . . 9 ((𝑗𝑛𝑖 = suc 𝑗) → ((𝑗𝑛𝑖 = suc 𝑗) ∧ 𝑗 E 𝑖))
6 df-3an 1090 . . . . . . . . 9 ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ↔ ((𝑗𝑛𝑖 = suc 𝑗) ∧ 𝑗 E 𝑖))
75, 6sylibr 233 . . . . . . . 8 ((𝑗𝑛𝑖 = suc 𝑗) → (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖))
82, 7bnj1023 33432 . . . . . . 7 𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖))
9 bnj1110.3 . . . . . . . . . . . 12 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
109bnj1232 33455 . . . . . . . . . . 11 (𝜒𝑛𝐷)
11103ad2ant3 1136 . . . . . . . . . 10 ((𝜃𝜏𝜒) → 𝑛𝐷)
12 bnj1110.19 . . . . . . . . . . 11 (𝜑0 ↔ (𝑖𝑛𝜎𝑓𝐾𝑖 ∈ dom 𝑓))
1312bnj1232 33455 . . . . . . . . . 10 (𝜑0𝑖𝑛)
1411, 13anim12ci 615 . . . . . . . . 9 (((𝜃𝜏𝜒) ∧ 𝜑0) → (𝑖𝑛𝑛𝐷))
1514anim2i 618 . . . . . . . 8 ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑖 ≠ ∅ ∧ (𝑖𝑛𝑛𝐷)))
16 3anass 1096 . . . . . . . 8 ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) ↔ (𝑖 ≠ ∅ ∧ (𝑖𝑛𝑛𝐷)))
1715, 16sylibr 233 . . . . . . 7 ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷))
188, 17bnj1101 33436 . . . . . 6 𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖))
19 3simpb 1150 . . . . . . . . 9 ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) → (𝑗𝑛𝑗 E 𝑖))
2012bnj1235 33456 . . . . . . . . . . 11 (𝜑0𝜎)
2120ad2antll 728 . . . . . . . . . 10 ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → 𝜎)
22 bnj1110.18 . . . . . . . . . 10 (𝜎 ↔ ((𝑗𝑛𝑗 E 𝑖) → 𝜂′))
2321, 22sylib 217 . . . . . . . . 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 pm3.43 475 . . . . . . 7 ((((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖)) ∧ ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → 𝜂′)) → ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝜂′)))
2725, 26mpdan 686 . . . . . 6 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖)) → ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝜂′)))
2818, 27bnj101 33375 . . . . 5 𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝜂′))
2912bnj1247 33460 . . . . . . 7 (𝜑0𝑓𝐾)
3029ad2antll 728 . . . . . 6 ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → 𝑓𝐾)
31 pm3.43i 474 . . . . . 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 33375 . . . 4 𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑓𝐾 ∧ ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝜂′)))
34 fndm 6610 . . . . . . . . 9 (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
359, 34bnj770 33415 . . . . . . . 8 (𝜒 → dom 𝑓 = 𝑛)
36353ad2ant3 1136 . . . . . . 7 ((𝜃𝜏𝜒) → dom 𝑓 = 𝑛)
3736ad2antrl 727 . . . . . 6 ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → dom 𝑓 = 𝑛)
3837eleq2d 2824 . . . . 5 ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑗 ∈ dom 𝑓𝑗𝑛))
39 pm3.43i 474 . . . . 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 33375 . . 3 𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → ((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑓𝐾 ∧ ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝜂′))))
42 bnj268 33361 . . . . . 6 (((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ 𝑓𝐾 ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝜂′) ↔ ((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝑓𝐾𝜂′))
43 bnj251 33354 . . . . . 6 (((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ 𝑓𝐾 ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝜂′) ↔ ((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑓𝐾 ∧ ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝜂′))))
4442, 43bitr3i 277 . . . . 5 (((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝑓𝐾𝜂′) ↔ ((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑓𝐾 ∧ ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝜂′))))
4544imbi2i 336 . . . 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 1851 . . 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 230 . 2 𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → ((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝑓𝐾𝜂′))
48 simp1 1137 . . . 4 ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) → 𝑗𝑛)
4948bnj706 33406 . . 3 (((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝑓𝐾𝜂′) → 𝑗𝑛)
50 simp2 1138 . . . 4 ((𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) → 𝑖 = suc 𝑗)
5150bnj706 33406 . . 3 (((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝑓𝐾𝜂′) → 𝑖 = suc 𝑗)
52 bnj258 33360 . . . . 5 (((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝑓𝐾𝜂′) ↔ (((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝜂′) ∧ 𝑓𝐾))
5352simprbi 498 . . . 4 (((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝑓𝐾𝜂′) → 𝑓𝐾)
54 bnj642 33400 . . . . 5 (((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝑓𝐾𝜂′) → (𝑗 ∈ dom 𝑓𝑗𝑛))
5549, 54mpbird 257 . . . 4 (((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝑓𝐾𝜂′) → 𝑗 ∈ dom 𝑓)
56 bnj645 33402 . . . . 5 (((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝑓𝐾𝜂′) → 𝜂′)
57 bnj1110.26 . . . . 5 (𝜂′ ↔ ((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵))
5856, 57sylib 217 . . . 4 (((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝑓𝐾𝜂′) → ((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵))
5953, 55, 58mp2and 698 . . 3 (((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝑓𝐾𝜂′) → (𝑓𝑗) ⊆ 𝐵)
6049, 51, 593jca 1129 . 2 (((𝑗 ∈ dom 𝑓𝑗𝑛) ∧ (𝑗𝑛𝑖 = suc 𝑗𝑗 E 𝑖) ∧ 𝑓𝐾𝜂′) → (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵))
6147, 60bnj1023 33432 1 𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wex 1782  wcel 2107  wne 2944  cdif 3912  wss 3915  c0 4287  {csn 4591   class class class wbr 5110   E cep 5541  dom cdm 5638  suc csuc 6324   Fn wfn 6496  cfv 6501  ωcom 7807  w-bnj17 33338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-fn 6504  df-om 7808  df-bnj17 33339
This theorem is referenced by:  bnj1118  33636
  Copyright terms: Public domain W3C validator