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Theorem bnj1118 30795
Description: Technical lemma for bnj69 30821. 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 30793 . . 3 𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵))
7 ancl 568 . . 3 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵))))
86, 7bnj101 30532 . 2 𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)))
9 simpr2 1066 . . . 4 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → 𝑖 = suc 𝑗)
101bnj1254 30623 . . . . . . 7 (𝜒𝜓)
11103ad2ant3 1082 . . . . . 6 ((𝜃𝜏𝜒) → 𝜓)
1211ad2antrl 763 . . . . 5 ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → 𝜓)
1312adantr 481 . . . 4 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → 𝜓)
141bnj1232 30617 . . . . . . . . 9 (𝜒𝑛𝐷)
15143ad2ant3 1082 . . . . . . . 8 ((𝜃𝜏𝜒) → 𝑛𝐷)
1615ad2antrl 763 . . . . . . 7 ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → 𝑛𝐷)
1716adantr 481 . . . . . 6 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → 𝑛𝐷)
18 simpr1 1065 . . . . . 6 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → 𝑗𝑛)
192bnj923 30581 . . . . . . . 8 (𝑛𝐷𝑛 ∈ ω)
2019anim1i 591 . . . . . . 7 ((𝑛𝐷𝑗𝑛) → (𝑛 ∈ ω ∧ 𝑗𝑛))
2120ancomd 467 . . . . . 6 ((𝑛𝐷𝑗𝑛) → (𝑗𝑛𝑛 ∈ ω))
2217, 18, 21syl2anc 692 . . . . 5 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → (𝑗𝑛𝑛 ∈ ω))
23 elnn 7029 . . . . 5 ((𝑗𝑛𝑛 ∈ ω) → 𝑗 ∈ ω)
2422, 23syl 17 . . . 4 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → 𝑗 ∈ ω)
254bnj1232 30617 . . . . . 6 (𝜑0𝑖𝑛)
2625adantl 482 . . . . 5 (((𝜃𝜏𝜒) ∧ 𝜑0) → 𝑖𝑛)
2726ad2antlr 762 . . . 4 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → 𝑖𝑛)
289, 13, 24, 27bnj951 30589 . . 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 1075 . . . . . 6 (𝜏 → TrFo(𝐵, 𝐴, 𝑅))
31303ad2ant2 1081 . . . . 5 ((𝜃𝜏𝜒) → TrFo(𝐵, 𝐴, 𝑅))
3231ad2antrl 763 . . . 4 ((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → TrFo(𝐵, 𝐴, 𝑅))
33 simp3 1061 . . . 4 ((𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵) → (𝑓𝑗) ⊆ 𝐵)
3432, 33anim12i 589 . . 3 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → ( TrFo(𝐵, 𝐴, 𝑅) ∧ (𝑓𝑗) ⊆ 𝐵))
35 bnj256 30514 . . . . 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 30794 . . . . . . . . 9 (𝜓 ↔ ∀𝑗((𝑗 ∈ ω ∧ suc 𝑗𝑛) → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)))
3837biimpi 206 . . . . . . . 8 (𝜓 → ∀𝑗((𝑗 ∈ ω ∧ suc 𝑗𝑛) → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)))
393819.21bi 2057 . . . . . . 7 (𝜓 → ((𝑗 ∈ ω ∧ suc 𝑗𝑛) → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)))
40 eleq1 2686 . . . . . . . . 9 (𝑖 = suc 𝑗 → (𝑖𝑛 ↔ suc 𝑗𝑛))
4140anbi2d 739 . . . . . . . 8 (𝑖 = suc 𝑗 → ((𝑗 ∈ ω ∧ 𝑖𝑛) ↔ (𝑗 ∈ ω ∧ suc 𝑗𝑛)))
42 fveq2 6153 . . . . . . . . 9 (𝑖 = suc 𝑗 → (𝑓𝑖) = (𝑓‘suc 𝑗))
4342eqeq1d 2623 . . . . . . . 8 (𝑖 = suc 𝑗 → ((𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ↔ (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)))
4441, 43imbi12d 334 . . . . . . 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))))
4539, 44syl5ibr 236 . . . . . 6 (𝑖 = suc 𝑗 → (𝜓 → ((𝑗 ∈ ω ∧ 𝑖𝑛) → (𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅))))
4645imp31 448 . . . . 5 (((𝑖 = suc 𝑗𝜓) ∧ (𝑗 ∈ ω ∧ 𝑖𝑛)) → (𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅))
4735, 46sylbi 207 . . . 4 ((𝑖 = suc 𝑗𝜓𝑗 ∈ ω ∧ 𝑖𝑛) → (𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅))
48 df-bnj19 30505 . . . . . . 7 ( TrFo(𝐵, 𝐴, 𝑅) ↔ ∀𝑦𝐵 pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵)
49 ssralv 3650 . . . . . . 7 ((𝑓𝑗) ⊆ 𝐵 → (∀𝑦𝐵 pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵 → ∀𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵))
5048, 49syl5bi 232 . . . . . 6 ((𝑓𝑗) ⊆ 𝐵 → ( TrFo(𝐵, 𝐴, 𝑅) → ∀𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵))
5150impcom 446 . . . . 5 (( TrFo(𝐵, 𝐴, 𝑅) ∧ (𝑓𝑗) ⊆ 𝐵) → ∀𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵)
52 iunss 4532 . . . . 5 ( 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵 ↔ ∀𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵)
5351, 52sylibr 224 . . . 4 (( TrFo(𝐵, 𝐴, 𝑅) ∧ (𝑓𝑗) ⊆ 𝐵) → 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵)
54 sseq1 3610 . . . . 5 ((𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) → ((𝑓𝑖) ⊆ 𝐵 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵))
5554biimpar 502 . . . 4 (((𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ∧ 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵) → (𝑓𝑖) ⊆ 𝐵)
5647, 53, 55syl2an 494 . . 3 (((𝑖 = suc 𝑗𝜓𝑗 ∈ ω ∧ 𝑖𝑛) ∧ ( TrFo(𝐵, 𝐴, 𝑅) ∧ (𝑓𝑗) ⊆ 𝐵)) → (𝑓𝑖) ⊆ 𝐵)
5728, 34, 56syl2anc 692 . 2 (((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) ∧ (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵)) → (𝑓𝑖) ⊆ 𝐵)
588, 57bnj1023 30594 1 𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑓𝑖) ⊆ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wal 1478   = wceq 1480  wex 1701  wcel 1987  wne 2790  wral 2907  Vcvv 3189  cdif 3556  wss 3559  c0 3896  {csn 4153   ciun 4490   class class class wbr 4618   E cep 4988  dom cdm 5079  suc csuc 5689   Fn wfn 5847  cfv 5852  ωcom 7019  w-bnj17 30494   predc-bnj14 30496   TrFow-bnj19 30504
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 4746  ax-nul 4754  ax-pr 4872  ax-un 6909
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 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-tr 4718  df-eprel 4990  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fn 5855  df-fv 5860  df-om 7020  df-bnj17 30495  df-bnj19 30505
This theorem is referenced by:  bnj1030  30798
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