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Theorem bnj1450 32540
 Description: Technical lemma for bnj60 32552. 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
bnj1450.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1450.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1450.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1450.4 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1450.5 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1450.6 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
bnj1450.7 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
bnj1450.8 (𝜏′[𝑦 / 𝑥]𝜏)
bnj1450.9 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1450.10 𝑃 = 𝐻
bnj1450.11 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1450.12 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
bnj1450.13 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
bnj1450.14 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
bnj1450.15 (𝜒𝑃 Fn trCl(𝑥, 𝐴, 𝑅))
bnj1450.16 (𝜒𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
bnj1450.17 (𝜃 ↔ (𝜒𝑧𝐸))
bnj1450.18 (𝜂 ↔ (𝜃𝑧 ∈ {𝑥}))
bnj1450.19 (𝜁 ↔ (𝜃𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))
bnj1450.20 (𝜌 ↔ (𝜁𝑓𝐻𝑧 ∈ dom 𝑓))
bnj1450.21 (𝜎 ↔ (𝜌𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
bnj1450.22 (𝜑 ↔ (𝜎𝑑𝐵𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
bnj1450.23 𝑋 = ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩
Assertion
Ref Expression
bnj1450 (𝜁 → (𝑄𝑧) = (𝐺𝑊))
Distinct variable groups:   𝐴,𝑑,𝑓,𝑥,𝑦,𝑧   𝐵,𝑓   𝑦,𝐷   𝐸,𝑑,𝑓,𝑦   𝐺,𝑑,𝑓,𝑥,𝑦,𝑧   𝑅,𝑑,𝑓,𝑥,𝑦,𝑧   𝑥,𝑋   𝑧,𝑌   𝜓,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜓(𝑥,𝑧,𝑓,𝑑)   𝜒(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜃(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜏(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜂(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜁(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜎(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜌(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐵(𝑥,𝑦,𝑧,𝑑)   𝐶(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐷(𝑥,𝑧,𝑓,𝑑)   𝑃(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑄(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐸(𝑥,𝑧)   𝐻(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑊(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑋(𝑦,𝑧,𝑓,𝑑)   𝑌(𝑥,𝑦,𝑓,𝑑)   𝑍(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜏′(𝑥,𝑦,𝑧,𝑓,𝑑)

Proof of Theorem bnj1450
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 bnj1450.19 . . . . . . . . 9 (𝜁 ↔ (𝜃𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))
21simprbi 501 . . . . . . . 8 (𝜁𝑧 ∈ trCl(𝑥, 𝐴, 𝑅))
3 bnj1450.17 . . . . . . . . . 10 (𝜃 ↔ (𝜒𝑧𝐸))
4 bnj1450.15 . . . . . . . . . . 11 (𝜒𝑃 Fn trCl(𝑥, 𝐴, 𝑅))
54fndmd 6436 . . . . . . . . . 10 (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
63, 5bnj832 32247 . . . . . . . . 9 (𝜃 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
71, 6bnj832 32247 . . . . . . . 8 (𝜁 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
82, 7eleqtrrd 2856 . . . . . . 7 (𝜁𝑧 ∈ dom 𝑃)
9 bnj1450.10 . . . . . . . 8 𝑃 = 𝐻
109dmeqi 5742 . . . . . . 7 dom 𝑃 = dom 𝐻
118, 10eleqtrdi 2863 . . . . . 6 (𝜁𝑧 ∈ dom 𝐻)
12 bnj1450.9 . . . . . . . 8 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
1312bnj1317 32311 . . . . . . 7 (𝑤𝐻 → ∀𝑓 𝑤𝐻)
1413bnj1400 32325 . . . . . 6 dom 𝐻 = 𝑓𝐻 dom 𝑓
1511, 14eleqtrdi 2863 . . . . 5 (𝜁𝑧 𝑓𝐻 dom 𝑓)
1615bnj1405 32326 . . . 4 (𝜁 → ∃𝑓𝐻 𝑧 ∈ dom 𝑓)
17 bnj1450.20 . . . 4 (𝜌 ↔ (𝜁𝑓𝐻𝑧 ∈ dom 𝑓))
18 bnj1450.1 . . . . 5 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
19 bnj1450.2 . . . . 5 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
20 bnj1450.3 . . . . 5 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
21 bnj1450.4 . . . . 5 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
22 bnj1450.5 . . . . 5 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
23 bnj1450.6 . . . . 5 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
24 bnj1450.7 . . . . 5 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
25 bnj1450.8 . . . . 5 (𝜏′[𝑦 / 𝑥]𝜏)
26 bnj1450.11 . . . . 5 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
27 bnj1450.12 . . . . 5 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
28 bnj1450.13 . . . . 5 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
29 bnj1450.14 . . . . 5 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
30 bnj1450.16 . . . . 5 (𝜒𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
31 bnj1450.18 . . . . 5 (𝜂 ↔ (𝜃𝑧 ∈ {𝑥}))
3218, 19, 20, 21, 22, 23, 24, 25, 12, 9, 26, 27, 28, 29, 4, 30, 3, 31, 1bnj1449 32538 . . . 4 (𝜁 → ∀𝑓𝜁)
3316, 17, 32bnj1521 32341 . . 3 (𝜁 → ∃𝑓𝜌)
3412bnj1436 32329 . . . . . . . . . 10 (𝑓𝐻 → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′)
3517, 34bnj836 32249 . . . . . . . . 9 (𝜌 → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′)
3618, 19, 20, 21, 25bnj1373 32520 . . . . . . . . . 10 (𝜏′ ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
3736rexbii 3176 . . . . . . . . 9 (∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′ ↔ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)(𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
3835, 37sylib 221 . . . . . . . 8 (𝜌 → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)(𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
3938bnj1196 32284 . . . . . . 7 (𝜌 → ∃𝑦(𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))))
40 3anass 1093 . . . . . . 7 ((𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) ↔ (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))))
4139, 40bnj1198 32285 . . . . . 6 (𝜌 → ∃𝑦(𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
42 bnj1450.21 . . . . . . 7 (𝜎 ↔ (𝜌𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
43 bnj252 32191 . . . . . . 7 ((𝜌𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) ↔ (𝜌 ∧ (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))))
4442, 43bitri 278 . . . . . 6 (𝜎 ↔ (𝜌 ∧ (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))))
4518, 19, 20, 21, 22, 23, 24, 25, 12, 9, 26, 27, 28, 29, 4, 30, 3, 31, 1, 17bnj1444 32533 . . . . . 6 (𝜌 → ∀𝑦𝜌)
4641, 44, 45bnj1340 32313 . . . . 5 (𝜌 → ∃𝑦𝜎)
4720bnj1436 32329 . . . . . . . . . . 11 (𝑓𝐶 → ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
4842, 47bnj771 32253 . . . . . . . . . 10 (𝜎 → ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
4948bnj1196 32284 . . . . . . . . 9 (𝜎 → ∃𝑑(𝑑𝐵 ∧ (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))))
50 3anass 1093 . . . . . . . . 9 ((𝑑𝐵𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)) ↔ (𝑑𝐵 ∧ (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))))
5149, 50bnj1198 32285 . . . . . . . 8 (𝜎 → ∃𝑑(𝑑𝐵𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
52 bnj1450.22 . . . . . . . . 9 (𝜑 ↔ (𝜎𝑑𝐵𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
53 bnj252 32191 . . . . . . . . 9 ((𝜎𝑑𝐵𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)) ↔ (𝜎 ∧ (𝑑𝐵𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))))
5452, 53bitri 278 . . . . . . . 8 (𝜑 ↔ (𝜎 ∧ (𝑑𝐵𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))))
55 bnj1450.23 . . . . . . . . 9 𝑋 = ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩
5618, 19, 20, 21, 22, 23, 24, 25, 12, 9, 26, 27, 28, 29, 4, 30, 3, 31, 1, 17, 42, 52, 55bnj1445 32534 . . . . . . . 8 (𝜎 → ∀𝑑𝜎)
5751, 54, 56bnj1340 32313 . . . . . . 7 (𝜎 → ∃𝑑𝜑)
58 fveq2 6656 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑓𝑥) = (𝑓𝑧))
59 id 22 . . . . . . . . . . . . 13 (𝑥 = 𝑧𝑥 = 𝑧)
60 bnj602 32405 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → pred(𝑥, 𝐴, 𝑅) = pred(𝑧, 𝐴, 𝑅))
6160reseq2d 5821 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (𝑓 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑓 ↾ pred(𝑧, 𝐴, 𝑅)))
6259, 61opeq12d 4769 . . . . . . . . . . . 12 (𝑥 = 𝑧 → ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)
6362, 19, 553eqtr4g 2819 . . . . . . . . . . 11 (𝑥 = 𝑧𝑌 = 𝑋)
6463fveq2d 6660 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝐺𝑌) = (𝐺𝑋))
6558, 64eqeq12d 2775 . . . . . . . . 9 (𝑥 = 𝑧 → ((𝑓𝑥) = (𝐺𝑌) ↔ (𝑓𝑧) = (𝐺𝑋)))
6652bnj1254 32299 . . . . . . . . 9 (𝜑 → ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))
6717simp3bi 1145 . . . . . . . . . . . 12 (𝜌𝑧 ∈ dom 𝑓)
6842, 67bnj769 32251 . . . . . . . . . . 11 (𝜎𝑧 ∈ dom 𝑓)
6952, 68bnj769 32251 . . . . . . . . . 10 (𝜑𝑧 ∈ dom 𝑓)
70 fndm 6434 . . . . . . . . . . 11 (𝑓 Fn 𝑑 → dom 𝑓 = 𝑑)
7152, 70bnj771 32253 . . . . . . . . . 10 (𝜑 → dom 𝑓 = 𝑑)
7269, 71eleqtrd 2855 . . . . . . . . 9 (𝜑𝑧𝑑)
7365, 66, 72rspcdva 3544 . . . . . . . 8 (𝜑 → (𝑓𝑧) = (𝐺𝑋))
7430bnj930 32259 . . . . . . . . . . . . . 14 (𝜒 → Fun 𝑄)
753, 74bnj832 32247 . . . . . . . . . . . . 13 (𝜃 → Fun 𝑄)
761, 75bnj832 32247 . . . . . . . . . . . 12 (𝜁 → Fun 𝑄)
7717, 76bnj835 32248 . . . . . . . . . . 11 (𝜌 → Fun 𝑄)
7842, 77bnj769 32251 . . . . . . . . . 10 (𝜎 → Fun 𝑄)
7952, 78bnj769 32251 . . . . . . . . 9 (𝜑 → Fun 𝑄)
8017simp2bi 1144 . . . . . . . . . . . 12 (𝜌𝑓𝐻)
8142, 80bnj769 32251 . . . . . . . . . . 11 (𝜎𝑓𝐻)
8252, 81bnj769 32251 . . . . . . . . . 10 (𝜑𝑓𝐻)
83 elssuni 4828 . . . . . . . . . . 11 (𝑓𝐻𝑓 𝐻)
8483, 9sseqtrrdi 3944 . . . . . . . . . 10 (𝑓𝐻𝑓𝑃)
85 ssun3 4080 . . . . . . . . . . 11 (𝑓𝑃𝑓 ⊆ (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩}))
8685, 27sseqtrrdi 3944 . . . . . . . . . 10 (𝑓𝑃𝑓𝑄)
8782, 84, 863syl 18 . . . . . . . . 9 (𝜑𝑓𝑄)
8879, 87, 69bnj1502 32338 . . . . . . . 8 (𝜑 → (𝑄𝑧) = (𝑓𝑧))
8960sseq1d 3924 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → ( pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑 ↔ pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑))
9018bnj1517 32340 . . . . . . . . . . . . . . 15 (𝑑𝐵 → ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
9152, 90bnj770 32252 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
9289, 91, 72rspcdva 3544 . . . . . . . . . . . . 13 (𝜑 → pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑)
9392, 71sseqtrrd 3934 . . . . . . . . . . . 12 (𝜑 → pred(𝑧, 𝐴, 𝑅) ⊆ dom 𝑓)
9479, 87, 93bnj1503 32339 . . . . . . . . . . 11 (𝜑 → (𝑄 ↾ pred(𝑧, 𝐴, 𝑅)) = (𝑓 ↾ pred(𝑧, 𝐴, 𝑅)))
9594opeq2d 4768 . . . . . . . . . 10 (𝜑 → ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩ = ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)
9695, 28, 553eqtr4g 2819 . . . . . . . . 9 (𝜑𝑊 = 𝑋)
9796fveq2d 6660 . . . . . . . 8 (𝜑 → (𝐺𝑊) = (𝐺𝑋))
9873, 88, 973eqtr4d 2804 . . . . . . 7 (𝜑 → (𝑄𝑧) = (𝐺𝑊))
9957, 98bnj593 32234 . . . . . 6 (𝜎 → ∃𝑑(𝑄𝑧) = (𝐺𝑊))
10018, 19, 20, 21, 22, 23, 24, 25, 12, 9, 26, 27, 28bnj1446 32535 . . . . . 6 ((𝑄𝑧) = (𝐺𝑊) → ∀𝑑(𝑄𝑧) = (𝐺𝑊))
10199, 100bnj1397 32324 . . . . 5 (𝜎 → (𝑄𝑧) = (𝐺𝑊))
10246, 101bnj593 32234 . . . 4 (𝜌 → ∃𝑦(𝑄𝑧) = (𝐺𝑊))
10318, 19, 20, 21, 22, 23, 24, 25, 12, 9, 26, 27, 28bnj1447 32536 . . . 4 ((𝑄𝑧) = (𝐺𝑊) → ∀𝑦(𝑄𝑧) = (𝐺𝑊))
104102, 103bnj1397 32324 . . 3 (𝜌 → (𝑄𝑧) = (𝐺𝑊))
10533, 104bnj593 32234 . 2 (𝜁 → ∃𝑓(𝑄𝑧) = (𝐺𝑊))
10618, 19, 20, 21, 22, 23, 24, 25, 12, 9, 26, 27, 28bnj1448 32537 . 2 ((𝑄𝑧) = (𝐺𝑊) → ∀𝑓(𝑄𝑧) = (𝐺𝑊))
107105, 106bnj1397 32324 1 (𝜁 → (𝑄𝑧) = (𝐺𝑊))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1085   = wceq 1539  ∃wex 1782   ∈ wcel 2112  {cab 2736   ≠ wne 2952  ∀wral 3071  ∃wrex 3072  {crab 3075  [wsbc 3697   ∪ cun 3857   ⊆ wss 3859  ∅c0 4226  {csn 4520  ⟨cop 4526  ∪ cuni 4796  ∪ ciun 4881   class class class wbr 5030  dom cdm 5522   ↾ cres 5524  Fun wfun 6327   Fn wfn 6328  ‘cfv 6333   ∧ w-bnj17 32174   predc-bnj14 32176   FrSe w-bnj15 32180   trClc-bnj18 32182 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5167  ax-nul 5174  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4419  df-sn 4521  df-pr 4523  df-op 4527  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-id 5428  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-res 5534  df-iota 6292  df-fun 6335  df-fn 6336  df-fv 6341  df-bnj17 32175  df-bnj14 32177  df-bnj18 32183 This theorem is referenced by:  bnj1423  32541
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