Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1450 Structured version   Visualization version   GIF version

Theorem bnj1450 31587
Description: Technical lemma for bnj60 31599. 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 490 . . . . . . . 8 (𝜁𝑧 ∈ trCl(𝑥, 𝐴, 𝑅))
3 bnj1450.17 . . . . . . . . . 10 (𝜃 ↔ (𝜒𝑧𝐸))
4 bnj1450.15 . . . . . . . . . . 11 (𝜒𝑃 Fn trCl(𝑥, 𝐴, 𝑅))
5 fndm 6170 . . . . . . . . . . 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 31297 . . . . . . . . 9 (𝜃 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
81, 7bnj832 31297 . . . . . . . 8 (𝜁 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
92, 8eleqtrrd 2847 . . . . . . 7 (𝜁𝑧 ∈ dom 𝑃)
10 bnj1450.10 . . . . . . . 8 𝑃 = 𝐻
1110dmeqi 5495 . . . . . . 7 dom 𝑃 = dom 𝐻
129, 11syl6eleq 2854 . . . . . 6 (𝜁𝑧 ∈ dom 𝐻)
13 bnj1450.9 . . . . . . . 8 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
1413bnj1317 31361 . . . . . . 7 (𝑤𝐻 → ∀𝑓 𝑤𝐻)
1514bnj1400 31375 . . . . . 6 dom 𝐻 = 𝑓𝐻 dom 𝑓
1612, 15syl6eleq 2854 . . . . 5 (𝜁𝑧 𝑓𝐻 dom 𝑓)
1716bnj1405 31376 . . . 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 31585 . . . 4 (𝜁 → ∀𝑓𝜁)
3417, 18, 33bnj1521 31390 . . 3 (𝜁 → ∃𝑓𝜌)
3513bnj1436 31379 . . . . . . . . . 10 (𝑓𝐻 → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′)
3618, 35bnj836 31299 . . . . . . . . 9 (𝜌 → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′)
3719, 20, 21, 22, 26bnj1373 31567 . . . . . . . . . 10 (𝜏′ ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
3837rexbii 3188 . . . . . . . . 9 (∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′ ↔ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)(𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
3936, 38sylib 209 . . . . . . . 8 (𝜌 → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)(𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
4039bnj1196 31334 . . . . . . 7 (𝜌 → ∃𝑦(𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))))
41 3anass 1116 . . . . . . 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 31335 . . . . . 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 31241 . . . . . . 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 266 . . . . . 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 31580 . . . . . 6 (𝜌 → ∀𝑦𝜌)
4742, 45, 46bnj1340 31363 . . . . 5 (𝜌 → ∃𝑦𝜎)
4821bnj1436 31379 . . . . . . . . . . 11 (𝑓𝐶 → ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
4943, 48bnj771 31303 . . . . . . . . . 10 (𝜎 → ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
5049bnj1196 31334 . . . . . . . . 9 (𝜎 → ∃𝑑(𝑑𝐵 ∧ (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))))
51 3anass 1116 . . . . . . . . 9 ((𝑑𝐵𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)) ↔ (𝑑𝐵 ∧ (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))))
5250, 51bnj1198 31335 . . . . . . . 8 (𝜎 → ∃𝑑(𝑑𝐵𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
53 bnj1450.22 . . . . . . . . 9 (𝜑 ↔ (𝜎𝑑𝐵𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
54 bnj252 31241 . . . . . . . . 9 ((𝜎𝑑𝐵𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)) ↔ (𝜎 ∧ (𝑑𝐵𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))))
5553, 54bitri 266 . . . . . . . 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 31581 . . . . . . . 8 (𝜎 → ∀𝑑𝜎)
5852, 55, 57bnj1340 31363 . . . . . . 7 (𝜎 → ∃𝑑𝜑)
5953bnj1254 31349 . . . . . . . . . 10 (𝜑 → ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))
60 fveq2 6379 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑓𝑥) = (𝑓𝑧))
61 id 22 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧𝑥 = 𝑧)
62 bnj602 31454 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → pred(𝑥, 𝐴, 𝑅) = pred(𝑧, 𝐴, 𝑅))
6362reseq2d 5567 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (𝑓 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑓 ↾ pred(𝑧, 𝐴, 𝑅)))
6461, 63opeq12d 4569 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)
6564, 20, 563eqtr4g 2824 . . . . . . . . . . . . 13 (𝑥 = 𝑧𝑌 = 𝑋)
6665fveq2d 6383 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝐺𝑌) = (𝐺𝑋))
6760, 66eqeq12d 2780 . . . . . . . . . . 11 (𝑥 = 𝑧 → ((𝑓𝑥) = (𝐺𝑌) ↔ (𝑓𝑧) = (𝐺𝑋)))
6867cbvralv 3319 . . . . . . . . . 10 (∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌) ↔ ∀𝑧𝑑 (𝑓𝑧) = (𝐺𝑋))
6959, 68sylib 209 . . . . . . . . 9 (𝜑 → ∀𝑧𝑑 (𝑓𝑧) = (𝐺𝑋))
7018simp3bi 1177 . . . . . . . . . . . 12 (𝜌𝑧 ∈ dom 𝑓)
7143, 70bnj769 31301 . . . . . . . . . . 11 (𝜎𝑧 ∈ dom 𝑓)
7253, 71bnj769 31301 . . . . . . . . . 10 (𝜑𝑧 ∈ dom 𝑓)
73 fndm 6170 . . . . . . . . . . 11 (𝑓 Fn 𝑑 → dom 𝑓 = 𝑑)
7453, 73bnj771 31303 . . . . . . . . . 10 (𝜑 → dom 𝑓 = 𝑑)
7572, 74eleqtrd 2846 . . . . . . . . 9 (𝜑𝑧𝑑)
7669, 75bnj1294 31357 . . . . . . . 8 (𝜑 → (𝑓𝑧) = (𝐺𝑋))
7731bnj930 31309 . . . . . . . . . . . . . 14 (𝜒 → Fun 𝑄)
783, 77bnj832 31297 . . . . . . . . . . . . 13 (𝜃 → Fun 𝑄)
791, 78bnj832 31297 . . . . . . . . . . . 12 (𝜁 → Fun 𝑄)
8018, 79bnj835 31298 . . . . . . . . . . 11 (𝜌 → Fun 𝑄)
8143, 80bnj769 31301 . . . . . . . . . 10 (𝜎 → Fun 𝑄)
8253, 81bnj769 31301 . . . . . . . . 9 (𝜑 → Fun 𝑄)
8318simp2bi 1176 . . . . . . . . . . . 12 (𝜌𝑓𝐻)
8443, 83bnj769 31301 . . . . . . . . . . 11 (𝜎𝑓𝐻)
8553, 84bnj769 31301 . . . . . . . . . 10 (𝜑𝑓𝐻)
86 elssuni 4627 . . . . . . . . . . 11 (𝑓𝐻𝑓 𝐻)
8786, 10syl6sseqr 3814 . . . . . . . . . 10 (𝑓𝐻𝑓𝑃)
88 ssun3 3942 . . . . . . . . . . 11 (𝑓𝑃𝑓 ⊆ (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩}))
8988, 28syl6sseqr 3814 . . . . . . . . . 10 (𝑓𝑃𝑓𝑄)
9085, 87, 893syl 18 . . . . . . . . 9 (𝜑𝑓𝑄)
9182, 90, 72bnj1502 31387 . . . . . . . 8 (𝜑 → (𝑄𝑧) = (𝑓𝑧))
9219bnj1517 31389 . . . . . . . . . . . . . . . 16 (𝑑𝐵 → ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
9353, 92bnj770 31302 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
9462sseq1d 3794 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → ( pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑 ↔ pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑))
9594cbvralv 3319 . . . . . . . . . . . . . . 15 (∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑 ↔ ∀𝑧𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑)
9693, 95sylib 209 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑧𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑)
9796, 75bnj1294 31357 . . . . . . . . . . . . 13 (𝜑 → pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑)
9897, 74sseqtr4d 3804 . . . . . . . . . . . 12 (𝜑 → pred(𝑧, 𝐴, 𝑅) ⊆ dom 𝑓)
9982, 90, 98bnj1503 31388 . . . . . . . . . . 11 (𝜑 → (𝑄 ↾ pred(𝑧, 𝐴, 𝑅)) = (𝑓 ↾ pred(𝑧, 𝐴, 𝑅)))
10099opeq2d 4568 . . . . . . . . . 10 (𝜑 → ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩ = ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)
101100, 29, 563eqtr4g 2824 . . . . . . . . 9 (𝜑𝑊 = 𝑋)
102101fveq2d 6383 . . . . . . . 8 (𝜑 → (𝐺𝑊) = (𝐺𝑋))
10376, 91, 1023eqtr4d 2809 . . . . . . 7 (𝜑 → (𝑄𝑧) = (𝐺𝑊))
10458, 103bnj593 31284 . . . . . 6 (𝜎 → ∃𝑑(𝑄𝑧) = (𝐺𝑊))
10519, 20, 21, 22, 23, 24, 25, 26, 13, 10, 27, 28, 29bnj1446 31582 . . . . . 6 ((𝑄𝑧) = (𝐺𝑊) → ∀𝑑(𝑄𝑧) = (𝐺𝑊))
106104, 105bnj1397 31374 . . . . 5 (𝜎 → (𝑄𝑧) = (𝐺𝑊))
10747, 106bnj593 31284 . . . 4 (𝜌 → ∃𝑦(𝑄𝑧) = (𝐺𝑊))
10819, 20, 21, 22, 23, 24, 25, 26, 13, 10, 27, 28, 29bnj1447 31583 . . . 4 ((𝑄𝑧) = (𝐺𝑊) → ∀𝑦(𝑄𝑧) = (𝐺𝑊))
109107, 108bnj1397 31374 . . 3 (𝜌 → (𝑄𝑧) = (𝐺𝑊))
11034, 109bnj593 31284 . 2 (𝜁 → ∃𝑓(𝑄𝑧) = (𝐺𝑊))
11119, 20, 21, 22, 23, 24, 25, 26, 13, 10, 27, 28, 29bnj1448 31584 . 2 ((𝑄𝑧) = (𝐺𝑊) → ∀𝑓(𝑄𝑧) = (𝐺𝑊))
112110, 111bnj1397 31374 1 (𝜁 → (𝑄𝑧) = (𝐺𝑊))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wex 1874  wcel 2155  {cab 2751  wne 2937  wral 3055  wrex 3056  {crab 3059  [wsbc 3598  cun 3732  wss 3734  c0 4081  {csn 4336  cop 4342   cuni 4596   ciun 4678   class class class wbr 4811  dom cdm 5279  cres 5281  Fun wfun 6064   Fn wfn 6065  cfv 6070  w-bnj17 31224   predc-bnj14 31226   FrSe w-bnj15 31230   trClc-bnj18 31232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-res 5291  df-iota 6033  df-fun 6072  df-fn 6073  df-fv 6078  df-bnj17 31225  df-bnj14 31227  df-bnj18 31233
This theorem is referenced by:  bnj1423  31588
  Copyright terms: Public domain W3C validator