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Theorem bnj1450 30861
 Description: Technical lemma for bnj60 30873. 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 480 . . . . . . . 8 (𝜁𝑧 ∈ trCl(𝑥, 𝐴, 𝑅))
3 bnj1450.17 . . . . . . . . . 10 (𝜃 ↔ (𝜒𝑧𝐸))
4 bnj1450.15 . . . . . . . . . . 11 (𝜒𝑃 Fn trCl(𝑥, 𝐴, 𝑅))
5 fndm 5953 . . . . . . . . . . 11 (𝑃 Fn trCl(𝑥, 𝐴, 𝑅) → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
64, 5syl 17 . . . . . . . . . 10 (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
73, 6bnj832 30571 . . . . . . . . 9 (𝜃 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
81, 7bnj832 30571 . . . . . . . 8 (𝜁 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
92, 8eleqtrrd 2701 . . . . . . 7 (𝜁𝑧 ∈ dom 𝑃)
10 bnj1450.10 . . . . . . . 8 𝑃 = 𝐻
1110dmeqi 5290 . . . . . . 7 dom 𝑃 = dom 𝐻
129, 11syl6eleq 2708 . . . . . 6 (𝜁𝑧 ∈ dom 𝐻)
13 bnj1450.9 . . . . . . . 8 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
1413bnj1317 30635 . . . . . . 7 (𝑤𝐻 → ∀𝑓 𝑤𝐻)
1514bnj1400 30649 . . . . . 6 dom 𝐻 = 𝑓𝐻 dom 𝑓
1612, 15syl6eleq 2708 . . . . 5 (𝜁𝑧 𝑓𝐻 dom 𝑓)
1716bnj1405 30650 . . . 4 (𝜁 → ∃𝑓𝐻 𝑧 ∈ dom 𝑓)
18 bnj1450.20 . . . 4 (𝜌 ↔ (𝜁𝑓𝐻𝑧 ∈ dom 𝑓))
19 bnj1450.1 . . . . 5 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
20 bnj1450.2 . . . . 5 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
21 bnj1450.3 . . . . 5 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
22 bnj1450.4 . . . . 5 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
23 bnj1450.5 . . . . 5 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
24 bnj1450.6 . . . . 5 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
25 bnj1450.7 . . . . 5 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
26 bnj1450.8 . . . . 5 (𝜏′[𝑦 / 𝑥]𝜏)
27 bnj1450.11 . . . . 5 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
28 bnj1450.12 . . . . 5 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
29 bnj1450.13 . . . . 5 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
30 bnj1450.14 . . . . 5 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
31 bnj1450.16 . . . . 5 (𝜒𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
32 bnj1450.18 . . . . 5 (𝜂 ↔ (𝜃𝑧 ∈ {𝑥}))
3319, 20, 21, 22, 23, 24, 25, 26, 13, 10, 27, 28, 29, 30, 4, 31, 3, 32, 1bnj1449 30859 . . . 4 (𝜁 → ∀𝑓𝜁)
3417, 18, 33bnj1521 30664 . . 3 (𝜁 → ∃𝑓𝜌)
3513bnj1436 30653 . . . . . . . . . 10 (𝑓𝐻 → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′)
3618, 35bnj836 30573 . . . . . . . . 9 (𝜌 → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′)
3719, 20, 21, 22, 26bnj1373 30841 . . . . . . . . . 10 (𝜏′ ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
3837rexbii 3035 . . . . . . . . 9 (∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′ ↔ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)(𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
3936, 38sylib 208 . . . . . . . 8 (𝜌 → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)(𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
4039bnj1196 30608 . . . . . . 7 (𝜌 → ∃𝑦(𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))))
41 3anass 1040 . . . . . . 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)))))
4240, 41bnj1198 30609 . . . . . 6 (𝜌 → ∃𝑦(𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
43 bnj1450.21 . . . . . . 7 (𝜎 ↔ (𝜌𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
44 bnj252 30511 . . . . . . 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)))))
4543, 44bitri 264 . . . . . 6 (𝜎 ↔ (𝜌 ∧ (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))))
4619, 20, 21, 22, 23, 24, 25, 26, 13, 10, 27, 28, 29, 30, 4, 31, 3, 32, 1, 18bnj1444 30854 . . . . . 6 (𝜌 → ∀𝑦𝜌)
4742, 45, 46bnj1340 30637 . . . . 5 (𝜌 → ∃𝑦𝜎)
4821bnj1436 30653 . . . . . . . . . . 11 (𝑓𝐶 → ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
4943, 48bnj771 30577 . . . . . . . . . 10 (𝜎 → ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
5049bnj1196 30608 . . . . . . . . 9 (𝜎 → ∃𝑑(𝑑𝐵 ∧ (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))))
51 3anass 1040 . . . . . . . . 9 ((𝑑𝐵𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)) ↔ (𝑑𝐵 ∧ (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))))
5250, 51bnj1198 30609 . . . . . . . 8 (𝜎 → ∃𝑑(𝑑𝐵𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
53 bnj1450.22 . . . . . . . . 9 (𝜑 ↔ (𝜎𝑑𝐵𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
54 bnj252 30511 . . . . . . . . 9 ((𝜎𝑑𝐵𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)) ↔ (𝜎 ∧ (𝑑𝐵𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))))
5553, 54bitri 264 . . . . . . . 8 (𝜑 ↔ (𝜎 ∧ (𝑑𝐵𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))))
56 bnj1450.23 . . . . . . . . 9 𝑋 = ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩
5719, 20, 21, 22, 23, 24, 25, 26, 13, 10, 27, 28, 29, 30, 4, 31, 3, 32, 1, 18, 43, 53, 56bnj1445 30855 . . . . . . . 8 (𝜎 → ∀𝑑𝜎)
5852, 55, 57bnj1340 30637 . . . . . . 7 (𝜎 → ∃𝑑𝜑)
5953bnj1254 30623 . . . . . . . . . 10 (𝜑 → ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))
60 fveq2 6153 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑓𝑥) = (𝑓𝑧))
61 id 22 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧𝑥 = 𝑧)
62 bnj602 30728 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → pred(𝑥, 𝐴, 𝑅) = pred(𝑧, 𝐴, 𝑅))
6362reseq2d 5361 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (𝑓 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑓 ↾ pred(𝑧, 𝐴, 𝑅)))
6461, 63opeq12d 4383 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)
6564, 20, 563eqtr4g 2680 . . . . . . . . . . . . 13 (𝑥 = 𝑧𝑌 = 𝑋)
6665fveq2d 6157 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝐺𝑌) = (𝐺𝑋))
6760, 66eqeq12d 2636 . . . . . . . . . . 11 (𝑥 = 𝑧 → ((𝑓𝑥) = (𝐺𝑌) ↔ (𝑓𝑧) = (𝐺𝑋)))
6867cbvralv 3162 . . . . . . . . . 10 (∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌) ↔ ∀𝑧𝑑 (𝑓𝑧) = (𝐺𝑋))
6959, 68sylib 208 . . . . . . . . 9 (𝜑 → ∀𝑧𝑑 (𝑓𝑧) = (𝐺𝑋))
7018simp3bi 1076 . . . . . . . . . . . 12 (𝜌𝑧 ∈ dom 𝑓)
7143, 70bnj769 30575 . . . . . . . . . . 11 (𝜎𝑧 ∈ dom 𝑓)
7253, 71bnj769 30575 . . . . . . . . . 10 (𝜑𝑧 ∈ dom 𝑓)
73 fndm 5953 . . . . . . . . . . 11 (𝑓 Fn 𝑑 → dom 𝑓 = 𝑑)
7453, 73bnj771 30577 . . . . . . . . . 10 (𝜑 → dom 𝑓 = 𝑑)
7572, 74eleqtrd 2700 . . . . . . . . 9 (𝜑𝑧𝑑)
7669, 75bnj1294 30631 . . . . . . . 8 (𝜑 → (𝑓𝑧) = (𝐺𝑋))
7731bnj930 30583 . . . . . . . . . . . . . 14 (𝜒 → Fun 𝑄)
783, 77bnj832 30571 . . . . . . . . . . . . 13 (𝜃 → Fun 𝑄)
791, 78bnj832 30571 . . . . . . . . . . . 12 (𝜁 → Fun 𝑄)
8018, 79bnj835 30572 . . . . . . . . . . 11 (𝜌 → Fun 𝑄)
8143, 80bnj769 30575 . . . . . . . . . 10 (𝜎 → Fun 𝑄)
8253, 81bnj769 30575 . . . . . . . . 9 (𝜑 → Fun 𝑄)
8318simp2bi 1075 . . . . . . . . . . . 12 (𝜌𝑓𝐻)
8443, 83bnj769 30575 . . . . . . . . . . 11 (𝜎𝑓𝐻)
8553, 84bnj769 30575 . . . . . . . . . 10 (𝜑𝑓𝐻)
86 elssuni 4438 . . . . . . . . . . 11 (𝑓𝐻𝑓 𝐻)
8786, 10syl6sseqr 3636 . . . . . . . . . 10 (𝑓𝐻𝑓𝑃)
88 ssun3 3761 . . . . . . . . . . 11 (𝑓𝑃𝑓 ⊆ (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩}))
8988, 28syl6sseqr 3636 . . . . . . . . . 10 (𝑓𝑃𝑓𝑄)
9085, 87, 893syl 18 . . . . . . . . 9 (𝜑𝑓𝑄)
9182, 90, 72bnj1502 30661 . . . . . . . 8 (𝜑 → (𝑄𝑧) = (𝑓𝑧))
9219bnj1517 30663 . . . . . . . . . . . . . . . 16 (𝑑𝐵 → ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
9353, 92bnj770 30576 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
9462sseq1d 3616 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → ( pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑 ↔ pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑))
9594cbvralv 3162 . . . . . . . . . . . . . . 15 (∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑 ↔ ∀𝑧𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑)
9693, 95sylib 208 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑧𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑)
9796, 75bnj1294 30631 . . . . . . . . . . . . 13 (𝜑 → pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑)
9897, 74sseqtr4d 3626 . . . . . . . . . . . 12 (𝜑 → pred(𝑧, 𝐴, 𝑅) ⊆ dom 𝑓)
9982, 90, 98bnj1503 30662 . . . . . . . . . . 11 (𝜑 → (𝑄 ↾ pred(𝑧, 𝐴, 𝑅)) = (𝑓 ↾ pred(𝑧, 𝐴, 𝑅)))
10099opeq2d 4382 . . . . . . . . . 10 (𝜑 → ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩ = ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)
101100, 29, 563eqtr4g 2680 . . . . . . . . 9 (𝜑𝑊 = 𝑋)
102101fveq2d 6157 . . . . . . . 8 (𝜑 → (𝐺𝑊) = (𝐺𝑋))
10376, 91, 1023eqtr4d 2665 . . . . . . 7 (𝜑 → (𝑄𝑧) = (𝐺𝑊))
10458, 103bnj593 30558 . . . . . 6 (𝜎 → ∃𝑑(𝑄𝑧) = (𝐺𝑊))
10519, 20, 21, 22, 23, 24, 25, 26, 13, 10, 27, 28, 29bnj1446 30856 . . . . . 6 ((𝑄𝑧) = (𝐺𝑊) → ∀𝑑(𝑄𝑧) = (𝐺𝑊))
106104, 105bnj1397 30648 . . . . 5 (𝜎 → (𝑄𝑧) = (𝐺𝑊))
10747, 106bnj593 30558 . . . 4 (𝜌 → ∃𝑦(𝑄𝑧) = (𝐺𝑊))
10819, 20, 21, 22, 23, 24, 25, 26, 13, 10, 27, 28, 29bnj1447 30857 . . . 4 ((𝑄𝑧) = (𝐺𝑊) → ∀𝑦(𝑄𝑧) = (𝐺𝑊))
109107, 108bnj1397 30648 . . 3 (𝜌 → (𝑄𝑧) = (𝐺𝑊))
11034, 109bnj593 30558 . 2 (𝜁 → ∃𝑓(𝑄𝑧) = (𝐺𝑊))
11119, 20, 21, 22, 23, 24, 25, 26, 13, 10, 27, 28, 29bnj1448 30858 . 2 ((𝑄𝑧) = (𝐺𝑊) → ∀𝑓(𝑄𝑧) = (𝐺𝑊))
112110, 111bnj1397 30648 1 (𝜁 → (𝑄𝑧) = (𝐺𝑊))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036   = wceq 1480  ∃wex 1701   ∈ wcel 1987  {cab 2607   ≠ wne 2790  ∀wral 2907  ∃wrex 2908  {crab 2911  [wsbc 3421   ∪ cun 3557   ⊆ wss 3559  ∅c0 3896  {csn 4153  ⟨cop 4159  ∪ cuni 4407  ∪ ciun 4490   class class class wbr 4618  dom cdm 5079   ↾ cres 5081  Fun wfun 5846   Fn wfn 5847  ‘cfv 5852   ∧ w-bnj17 30494   predc-bnj14 30496   FrSe w-bnj15 30500   trClc-bnj18 30502 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-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 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-res 5091  df-iota 5815  df-fun 5854  df-fn 5855  df-fv 5860  df-bnj17 30495  df-bnj14 30497  df-bnj18 30503 This theorem is referenced by:  bnj1423  30862
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