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Theorem bnj1450 31244
Description: Technical lemma for bnj60 31256. 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 479 . . . . . . . 8 (𝜁𝑧 ∈ trCl(𝑥, 𝐴, 𝑅))
3 bnj1450.17 . . . . . . . . . 10 (𝜃 ↔ (𝜒𝑧𝐸))
4 bnj1450.15 . . . . . . . . . . 11 (𝜒𝑃 Fn trCl(𝑥, 𝐴, 𝑅))
5 fndm 6028 . . . . . . . . . . 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 30954 . . . . . . . . 9 (𝜃 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
81, 7bnj832 30954 . . . . . . . 8 (𝜁 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
92, 8eleqtrrd 2733 . . . . . . 7 (𝜁𝑧 ∈ dom 𝑃)
10 bnj1450.10 . . . . . . . 8 𝑃 = 𝐻
1110dmeqi 5357 . . . . . . 7 dom 𝑃 = dom 𝐻
129, 11syl6eleq 2740 . . . . . 6 (𝜁𝑧 ∈ dom 𝐻)
13 bnj1450.9 . . . . . . . 8 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
1413bnj1317 31018 . . . . . . 7 (𝑤𝐻 → ∀𝑓 𝑤𝐻)
1514bnj1400 31032 . . . . . 6 dom 𝐻 = 𝑓𝐻 dom 𝑓
1612, 15syl6eleq 2740 . . . . 5 (𝜁𝑧 𝑓𝐻 dom 𝑓)
1716bnj1405 31033 . . . 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 31242 . . . 4 (𝜁 → ∀𝑓𝜁)
3417, 18, 33bnj1521 31047 . . 3 (𝜁 → ∃𝑓𝜌)
3513bnj1436 31036 . . . . . . . . . 10 (𝑓𝐻 → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′)
3618, 35bnj836 30956 . . . . . . . . 9 (𝜌 → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′)
3719, 20, 21, 22, 26bnj1373 31224 . . . . . . . . . 10 (𝜏′ ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
3837rexbii 3070 . . . . . . . . 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 30991 . . . . . . 7 (𝜌 → ∃𝑦(𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))))
41 3anass 1059 . . . . . . 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 30992 . . . . . 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 30897 . . . . . . 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 31237 . . . . . 6 (𝜌 → ∀𝑦𝜌)
4742, 45, 46bnj1340 31020 . . . . 5 (𝜌 → ∃𝑦𝜎)
4821bnj1436 31036 . . . . . . . . . . 11 (𝑓𝐶 → ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
4943, 48bnj771 30960 . . . . . . . . . 10 (𝜎 → ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
5049bnj1196 30991 . . . . . . . . 9 (𝜎 → ∃𝑑(𝑑𝐵 ∧ (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))))
51 3anass 1059 . . . . . . . . 9 ((𝑑𝐵𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)) ↔ (𝑑𝐵 ∧ (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))))
5250, 51bnj1198 30992 . . . . . . . 8 (𝜎 → ∃𝑑(𝑑𝐵𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
53 bnj1450.22 . . . . . . . . 9 (𝜑 ↔ (𝜎𝑑𝐵𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
54 bnj252 30897 . . . . . . . . 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 31238 . . . . . . . 8 (𝜎 → ∀𝑑𝜎)
5852, 55, 57bnj1340 31020 . . . . . . 7 (𝜎 → ∃𝑑𝜑)
5953bnj1254 31006 . . . . . . . . . 10 (𝜑 → ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))
60 fveq2 6229 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑓𝑥) = (𝑓𝑧))
61 id 22 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧𝑥 = 𝑧)
62 bnj602 31111 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → pred(𝑥, 𝐴, 𝑅) = pred(𝑧, 𝐴, 𝑅))
6362reseq2d 5428 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (𝑓 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑓 ↾ pred(𝑧, 𝐴, 𝑅)))
6461, 63opeq12d 4441 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)
6564, 20, 563eqtr4g 2710 . . . . . . . . . . . . 13 (𝑥 = 𝑧𝑌 = 𝑋)
6665fveq2d 6233 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝐺𝑌) = (𝐺𝑋))
6760, 66eqeq12d 2666 . . . . . . . . . . 11 (𝑥 = 𝑧 → ((𝑓𝑥) = (𝐺𝑌) ↔ (𝑓𝑧) = (𝐺𝑋)))
6867cbvralv 3201 . . . . . . . . . 10 (∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌) ↔ ∀𝑧𝑑 (𝑓𝑧) = (𝐺𝑋))
6959, 68sylib 208 . . . . . . . . 9 (𝜑 → ∀𝑧𝑑 (𝑓𝑧) = (𝐺𝑋))
7018simp3bi 1098 . . . . . . . . . . . 12 (𝜌𝑧 ∈ dom 𝑓)
7143, 70bnj769 30958 . . . . . . . . . . 11 (𝜎𝑧 ∈ dom 𝑓)
7253, 71bnj769 30958 . . . . . . . . . 10 (𝜑𝑧 ∈ dom 𝑓)
73 fndm 6028 . . . . . . . . . . 11 (𝑓 Fn 𝑑 → dom 𝑓 = 𝑑)
7453, 73bnj771 30960 . . . . . . . . . 10 (𝜑 → dom 𝑓 = 𝑑)
7572, 74eleqtrd 2732 . . . . . . . . 9 (𝜑𝑧𝑑)
7669, 75bnj1294 31014 . . . . . . . 8 (𝜑 → (𝑓𝑧) = (𝐺𝑋))
7731bnj930 30966 . . . . . . . . . . . . . 14 (𝜒 → Fun 𝑄)
783, 77bnj832 30954 . . . . . . . . . . . . 13 (𝜃 → Fun 𝑄)
791, 78bnj832 30954 . . . . . . . . . . . 12 (𝜁 → Fun 𝑄)
8018, 79bnj835 30955 . . . . . . . . . . 11 (𝜌 → Fun 𝑄)
8143, 80bnj769 30958 . . . . . . . . . 10 (𝜎 → Fun 𝑄)
8253, 81bnj769 30958 . . . . . . . . 9 (𝜑 → Fun 𝑄)
8318simp2bi 1097 . . . . . . . . . . . 12 (𝜌𝑓𝐻)
8443, 83bnj769 30958 . . . . . . . . . . 11 (𝜎𝑓𝐻)
8553, 84bnj769 30958 . . . . . . . . . 10 (𝜑𝑓𝐻)
86 elssuni 4499 . . . . . . . . . . 11 (𝑓𝐻𝑓 𝐻)
8786, 10syl6sseqr 3685 . . . . . . . . . 10 (𝑓𝐻𝑓𝑃)
88 ssun3 3811 . . . . . . . . . . 11 (𝑓𝑃𝑓 ⊆ (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩}))
8988, 28syl6sseqr 3685 . . . . . . . . . 10 (𝑓𝑃𝑓𝑄)
9085, 87, 893syl 18 . . . . . . . . 9 (𝜑𝑓𝑄)
9182, 90, 72bnj1502 31044 . . . . . . . 8 (𝜑 → (𝑄𝑧) = (𝑓𝑧))
9219bnj1517 31046 . . . . . . . . . . . . . . . 16 (𝑑𝐵 → ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
9353, 92bnj770 30959 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
9462sseq1d 3665 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → ( pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑 ↔ pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑))
9594cbvralv 3201 . . . . . . . . . . . . . . 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 31014 . . . . . . . . . . . . 13 (𝜑 → pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑)
9897, 74sseqtr4d 3675 . . . . . . . . . . . 12 (𝜑 → pred(𝑧, 𝐴, 𝑅) ⊆ dom 𝑓)
9982, 90, 98bnj1503 31045 . . . . . . . . . . 11 (𝜑 → (𝑄 ↾ pred(𝑧, 𝐴, 𝑅)) = (𝑓 ↾ pred(𝑧, 𝐴, 𝑅)))
10099opeq2d 4440 . . . . . . . . . 10 (𝜑 → ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩ = ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)
101100, 29, 563eqtr4g 2710 . . . . . . . . 9 (𝜑𝑊 = 𝑋)
102101fveq2d 6233 . . . . . . . 8 (𝜑 → (𝐺𝑊) = (𝐺𝑋))
10376, 91, 1023eqtr4d 2695 . . . . . . 7 (𝜑 → (𝑄𝑧) = (𝐺𝑊))
10458, 103bnj593 30941 . . . . . 6 (𝜎 → ∃𝑑(𝑄𝑧) = (𝐺𝑊))
10519, 20, 21, 22, 23, 24, 25, 26, 13, 10, 27, 28, 29bnj1446 31239 . . . . . 6 ((𝑄𝑧) = (𝐺𝑊) → ∀𝑑(𝑄𝑧) = (𝐺𝑊))
106104, 105bnj1397 31031 . . . . 5 (𝜎 → (𝑄𝑧) = (𝐺𝑊))
10747, 106bnj593 30941 . . . 4 (𝜌 → ∃𝑦(𝑄𝑧) = (𝐺𝑊))
10819, 20, 21, 22, 23, 24, 25, 26, 13, 10, 27, 28, 29bnj1447 31240 . . . 4 ((𝑄𝑧) = (𝐺𝑊) → ∀𝑦(𝑄𝑧) = (𝐺𝑊))
109107, 108bnj1397 31031 . . 3 (𝜌 → (𝑄𝑧) = (𝐺𝑊))
11034, 109bnj593 30941 . 2 (𝜁 → ∃𝑓(𝑄𝑧) = (𝐺𝑊))
11119, 20, 21, 22, 23, 24, 25, 26, 13, 10, 27, 28, 29bnj1448 31241 . 2 ((𝑄𝑧) = (𝐺𝑊) → ∀𝑓(𝑄𝑧) = (𝐺𝑊))
112110, 111bnj1397 31031 1 (𝜁 → (𝑄𝑧) = (𝐺𝑊))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wne 2823  wral 2941  wrex 2942  {crab 2945  [wsbc 3468  cun 3605  wss 3607  c0 3948  {csn 4210  cop 4216   cuni 4468   ciun 4552   class class class wbr 4685  dom cdm 5143  cres 5145  Fun wfun 5920   Fn wfn 5921  cfv 5926  w-bnj17 30880   predc-bnj14 30882   FrSe w-bnj15 30886   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-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
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  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-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-res 5155  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934  df-bnj17 30881  df-bnj14 30883  df-bnj18 30889
This theorem is referenced by:  bnj1423  31245
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