Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1145 Structured version   Visualization version   GIF version

Theorem bnj1145 31187
Description: Technical lemma for bnj69 31204. 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
bnj1145.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj1145.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1145.3 𝐷 = (ω ∖ {∅})
bnj1145.4 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
bnj1145.5 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj1145.6 (𝜃 ↔ ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗)))
Assertion
Ref Expression
bnj1145 trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐴
Distinct variable groups:   𝐴,𝑓,𝑖,𝑗,𝑛,𝑦   𝐷,𝑖,𝑗   𝑅,𝑓,𝑖,𝑗,𝑛,𝑦   𝑓,𝑋,𝑖,𝑛,𝑦   𝜒,𝑗   𝜑,𝑖
Allowed substitution hints:   𝜑(𝑦,𝑓,𝑗,𝑛)   𝜓(𝑦,𝑓,𝑖,𝑗,𝑛)   𝜒(𝑦,𝑓,𝑖,𝑛)   𝜃(𝑦,𝑓,𝑖,𝑗,𝑛)   𝐵(𝑦,𝑓,𝑖,𝑗,𝑛)   𝐷(𝑦,𝑓,𝑛)   𝑋(𝑗)

Proof of Theorem bnj1145
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 bnj1145.1 . . 3 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
2 bnj1145.2 . . 3 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
3 bnj1145.3 . . 3 𝐷 = (ω ∖ {∅})
4 bnj1145.4 . . 3 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
51, 2, 3, 4bnj882 31122 . 2 trCl(𝑋, 𝐴, 𝑅) = 𝑓𝐵 𝑖 ∈ dom 𝑓(𝑓𝑖)
6 ss2iun 4568 . . . 4 (∀𝑓𝐵 𝑖 ∈ dom 𝑓(𝑓𝑖) ⊆ 𝐴 𝑓𝐵 𝑖 ∈ dom 𝑓(𝑓𝑖) ⊆ 𝑓𝐵 𝐴)
7 bnj1145.5 . . . . . . 7 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
87, 4bnj1083 31172 . . . . . 6 (𝑓𝐵 ↔ ∃𝑛𝜒)
92bnj1095 30978 . . . . . . . . 9 (𝜓 → ∀𝑖𝜓)
109, 7bnj1096 30979 . . . . . . . 8 (𝜒 → ∀𝑖𝜒)
113bnj1098 30980 . . . . . . . . . . . . . . . . 17 𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑗𝑛𝑖 = suc 𝑗))
127bnj1232 31000 . . . . . . . . . . . . . . . . . 18 (𝜒𝑛𝐷)
13123anim3i 1269 . . . . . . . . . . . . . . . . 17 ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝜒) → (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷))
1411, 13bnj1101 30981 . . . . . . . . . . . . . . . 16 𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝜒) → (𝑗𝑛𝑖 = suc 𝑗))
15 ancl 568 . . . . . . . . . . . . . . . 16 (((𝑖 ≠ ∅ ∧ 𝑖𝑛𝜒) → (𝑗𝑛𝑖 = suc 𝑗)) → ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝜒) → ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗))))
1614, 15bnj101 30917 . . . . . . . . . . . . . . 15 𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝜒) → ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗)))
17 bnj1145.6 . . . . . . . . . . . . . . . . 17 (𝜃 ↔ ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗)))
1817imbi2i 325 . . . . . . . . . . . . . . . 16 (((𝑖 ≠ ∅ ∧ 𝑖𝑛𝜒) → 𝜃) ↔ ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝜒) → ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗))))
1918exbii 1814 . . . . . . . . . . . . . . 15 (∃𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝜒) → 𝜃) ↔ ∃𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝜒) → ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗))))
2016, 19mpbir 221 . . . . . . . . . . . . . 14 𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝜒) → 𝜃)
21 bnj213 31078 . . . . . . . . . . . . . . . 16 pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
2221bnj226 30928 . . . . . . . . . . . . . . 15 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
23 simpr 476 . . . . . . . . . . . . . . . . . . 19 ((𝑗𝑛𝑖 = suc 𝑗) → 𝑖 = suc 𝑗)
2417, 23simplbiim 659 . . . . . . . . . . . . . . . . . 18 (𝜃𝑖 = suc 𝑗)
25 simp2 1082 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝜒) → 𝑖𝑛)
26123ad2ant3 1104 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝜒) → 𝑛𝐷)
273bnj923 30964 . . . . . . . . . . . . . . . . . . . . 21 (𝑛𝐷𝑛 ∈ ω)
28 elnn 7117 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖𝑛𝑛 ∈ ω) → 𝑖 ∈ ω)
2927, 28sylan2 490 . . . . . . . . . . . . . . . . . . . 20 ((𝑖𝑛𝑛𝐷) → 𝑖 ∈ ω)
3025, 26, 29syl2anc 694 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝜒) → 𝑖 ∈ ω)
3117, 30bnj832 30954 . . . . . . . . . . . . . . . . . 18 (𝜃𝑖 ∈ ω)
32 vex 3234 . . . . . . . . . . . . . . . . . . . 20 𝑗 ∈ V
3332bnj216 30926 . . . . . . . . . . . . . . . . . . 19 (𝑖 = suc 𝑗𝑗𝑖)
34 elnn 7117 . . . . . . . . . . . . . . . . . . 19 ((𝑗𝑖𝑖 ∈ ω) → 𝑗 ∈ ω)
3533, 34sylan 487 . . . . . . . . . . . . . . . . . 18 ((𝑖 = suc 𝑗𝑖 ∈ ω) → 𝑗 ∈ ω)
3624, 31, 35syl2anc 694 . . . . . . . . . . . . . . . . 17 (𝜃𝑗 ∈ ω)
3717, 25bnj832 30954 . . . . . . . . . . . . . . . . . 18 (𝜃𝑖𝑛)
3824, 37eqeltrrd 2731 . . . . . . . . . . . . . . . . 17 (𝜃 → suc 𝑗𝑛)
392bnj589 31105 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜓 ↔ ∀𝑗 ∈ ω (suc 𝑗𝑛 → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)))
4039biimpi 206 . . . . . . . . . . . . . . . . . . . . . 22 (𝜓 → ∀𝑗 ∈ ω (suc 𝑗𝑛 → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)))
4140bnj708 30952 . . . . . . . . . . . . . . . . . . . . 21 ((𝑛𝐷𝑓 Fn 𝑛𝜑𝜓) → ∀𝑗 ∈ ω (suc 𝑗𝑛 → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)))
42 rsp 2958 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑗 ∈ ω (suc 𝑗𝑛 → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)) → (𝑗 ∈ ω → (suc 𝑗𝑛 → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅))))
4341, 42syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝑛𝐷𝑓 Fn 𝑛𝜑𝜓) → (𝑗 ∈ ω → (suc 𝑗𝑛 → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅))))
447, 43sylbi 207 . . . . . . . . . . . . . . . . . . 19 (𝜒 → (𝑗 ∈ ω → (suc 𝑗𝑛 → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅))))
45443ad2ant3 1104 . . . . . . . . . . . . . . . . . 18 ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝜒) → (𝑗 ∈ ω → (suc 𝑗𝑛 → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅))))
4617, 45bnj832 30954 . . . . . . . . . . . . . . . . 17 (𝜃 → (𝑗 ∈ ω → (suc 𝑗𝑛 → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅))))
4736, 38, 46mp2d 49 . . . . . . . . . . . . . . . 16 (𝜃 → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅))
48 fveq2 6229 . . . . . . . . . . . . . . . . . 18 (𝑖 = suc 𝑗 → (𝑓𝑖) = (𝑓‘suc 𝑗))
4948eqeq1d 2653 . . . . . . . . . . . . . . . . 17 (𝑖 = suc 𝑗 → ((𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ↔ (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)))
5024, 49syl 17 . . . . . . . . . . . . . . . 16 (𝜃 → ((𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ↔ (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)))
5147, 50mpbird 247 . . . . . . . . . . . . . . 15 (𝜃 → (𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅))
5222, 51bnj1262 31007 . . . . . . . . . . . . . 14 (𝜃 → (𝑓𝑖) ⊆ 𝐴)
5320, 52bnj1023 30977 . . . . . . . . . . . . 13 𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝜒) → (𝑓𝑖) ⊆ 𝐴)
54 3anass 1059 . . . . . . . . . . . . . . 15 ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝜒) ↔ (𝑖 ≠ ∅ ∧ (𝑖𝑛𝜒)))
5554imbi1i 338 . . . . . . . . . . . . . 14 (((𝑖 ≠ ∅ ∧ 𝑖𝑛𝜒) → (𝑓𝑖) ⊆ 𝐴) ↔ ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜒)) → (𝑓𝑖) ⊆ 𝐴))
5655exbii 1814 . . . . . . . . . . . . 13 (∃𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝜒) → (𝑓𝑖) ⊆ 𝐴) ↔ ∃𝑗((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜒)) → (𝑓𝑖) ⊆ 𝐴))
5753, 56mpbi 220 . . . . . . . . . . . 12 𝑗((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜒)) → (𝑓𝑖) ⊆ 𝐴)
581biimpi 206 . . . . . . . . . . . . . . 15 (𝜑 → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
597, 58bnj771 30960 . . . . . . . . . . . . . 14 (𝜒 → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
60 fveq2 6229 . . . . . . . . . . . . . . 15 (𝑖 = ∅ → (𝑓𝑖) = (𝑓‘∅))
61 bnj213 31078 . . . . . . . . . . . . . . . 16 pred(𝑋, 𝐴, 𝑅) ⊆ 𝐴
62 sseq1 3659 . . . . . . . . . . . . . . . 16 ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) → ((𝑓‘∅) ⊆ 𝐴 ↔ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐴))
6361, 62mpbiri 248 . . . . . . . . . . . . . . 15 ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) → (𝑓‘∅) ⊆ 𝐴)
64 sseq1 3659 . . . . . . . . . . . . . . . 16 ((𝑓𝑖) = (𝑓‘∅) → ((𝑓𝑖) ⊆ 𝐴 ↔ (𝑓‘∅) ⊆ 𝐴))
6564biimpar 501 . . . . . . . . . . . . . . 15 (((𝑓𝑖) = (𝑓‘∅) ∧ (𝑓‘∅) ⊆ 𝐴) → (𝑓𝑖) ⊆ 𝐴)
6660, 63, 65syl2an 493 . . . . . . . . . . . . . 14 ((𝑖 = ∅ ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) → (𝑓𝑖) ⊆ 𝐴)
6759, 66sylan2 490 . . . . . . . . . . . . 13 ((𝑖 = ∅ ∧ 𝜒) → (𝑓𝑖) ⊆ 𝐴)
6867adantrl 752 . . . . . . . . . . . 12 ((𝑖 = ∅ ∧ (𝑖𝑛𝜒)) → (𝑓𝑖) ⊆ 𝐴)
6957, 68bnj1109 30983 . . . . . . . . . . 11 𝑗((𝑖𝑛𝜒) → (𝑓𝑖) ⊆ 𝐴)
70 19.9v 1953 . . . . . . . . . . 11 (∃𝑗((𝑖𝑛𝜒) → (𝑓𝑖) ⊆ 𝐴) ↔ ((𝑖𝑛𝜒) → (𝑓𝑖) ⊆ 𝐴))
7169, 70mpbi 220 . . . . . . . . . 10 ((𝑖𝑛𝜒) → (𝑓𝑖) ⊆ 𝐴)
7271expcom 450 . . . . . . . . 9 (𝜒 → (𝑖𝑛 → (𝑓𝑖) ⊆ 𝐴))
73 fndm 6028 . . . . . . . . . . 11 (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
747, 73bnj770 30959 . . . . . . . . . 10 (𝜒 → dom 𝑓 = 𝑛)
75 eleq2 2719 . . . . . . . . . . 11 (dom 𝑓 = 𝑛 → (𝑖 ∈ dom 𝑓𝑖𝑛))
7675imbi1d 330 . . . . . . . . . 10 (dom 𝑓 = 𝑛 → ((𝑖 ∈ dom 𝑓 → (𝑓𝑖) ⊆ 𝐴) ↔ (𝑖𝑛 → (𝑓𝑖) ⊆ 𝐴)))
7774, 76syl 17 . . . . . . . . 9 (𝜒 → ((𝑖 ∈ dom 𝑓 → (𝑓𝑖) ⊆ 𝐴) ↔ (𝑖𝑛 → (𝑓𝑖) ⊆ 𝐴)))
7872, 77mpbird 247 . . . . . . . 8 (𝜒 → (𝑖 ∈ dom 𝑓 → (𝑓𝑖) ⊆ 𝐴))
7910, 78hbralrimi 2983 . . . . . . 7 (𝜒 → ∀𝑖 ∈ dom 𝑓(𝑓𝑖) ⊆ 𝐴)
8079exlimiv 1898 . . . . . 6 (∃𝑛𝜒 → ∀𝑖 ∈ dom 𝑓(𝑓𝑖) ⊆ 𝐴)
818, 80sylbi 207 . . . . 5 (𝑓𝐵 → ∀𝑖 ∈ dom 𝑓(𝑓𝑖) ⊆ 𝐴)
82 ss2iun 4568 . . . . . 6 (∀𝑖 ∈ dom 𝑓(𝑓𝑖) ⊆ 𝐴 𝑖 ∈ dom 𝑓(𝑓𝑖) ⊆ 𝑖 ∈ dom 𝑓 𝐴)
83 bnj1143 30987 . . . . . 6 𝑖 ∈ dom 𝑓 𝐴𝐴
8482, 83syl6ss 3648 . . . . 5 (∀𝑖 ∈ dom 𝑓(𝑓𝑖) ⊆ 𝐴 𝑖 ∈ dom 𝑓(𝑓𝑖) ⊆ 𝐴)
8581, 84syl 17 . . . 4 (𝑓𝐵 𝑖 ∈ dom 𝑓(𝑓𝑖) ⊆ 𝐴)
866, 85mprg 2955 . . 3 𝑓𝐵 𝑖 ∈ dom 𝑓(𝑓𝑖) ⊆ 𝑓𝐵 𝐴
874bnj1317 31018 . . . 4 (𝑤𝐵 → ∀𝑓 𝑤𝐵)
8887bnj1146 30988 . . 3 𝑓𝐵 𝐴𝐴
8986, 88sstri 3645 . 2 𝑓𝐵 𝑖 ∈ dom 𝑓(𝑓𝑖) ⊆ 𝐴
905, 89eqsstri 3668 1 trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐴
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wne 2823  wral 2941  wrex 2942  cdif 3604  wss 3607  c0 3948  {csn 4210   ciun 4552  dom cdm 5143  suc csuc 5763   Fn wfn 5921  cfv 5926  ωcom 7107  w-bnj17 30880   predc-bnj14 30882   trClc-bnj18 30888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fn 5929  df-fv 5934  df-om 7108  df-bnj17 30881  df-bnj14 30883  df-bnj18 30889
This theorem is referenced by:  bnj1147  31188
  Copyright terms: Public domain W3C validator