Theorem bnj1450 30372

Description: Technical lemma for bnj60 30384. 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 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1450.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1450.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1450.4	⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1450.5	⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1450.6	⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
bnj1450.7	⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 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, (𝐺‘𝑍)⟩})
bnj1450.13	⊢ 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
bnj1450.14	⊢ 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
bnj1450.15	⊢ (𝜒 → 𝑃 Fn trCl(𝑥, 𝐴, 𝑅))
bnj1450.16	⊢ (𝜒 → 𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
bnj1450.17	⊢ (𝜃 ↔ (𝜒 ∧ 𝑧 ∈ 𝐸))
bnj1450.18	⊢ (𝜂 ↔ (𝜃 ∧ 𝑧 ∈ {𝑥}))
bnj1450.19	⊢ (𝜁 ↔ (𝜃 ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))
bnj1450.20	⊢ (𝜌 ↔ (𝜁 ∧ 𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓))
bnj1450.21	⊢ (𝜎 ↔ (𝜌 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
bnj1450.22	⊢ (𝜑 ↔ (𝜎 ∧ 𝑑 ∈ 𝐵 ∧ 𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)))
bnj1450.23	⊢ 𝑋 = ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩

Assertion

Ref	Expression
bnj1450	⊢ (𝜁 → (𝑄‘𝑧) = (𝐺‘𝑊))

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.

Step	Hyp	Ref	Expression
1		bnj1450.19	. . . . . . . . 9 ⊢ (𝜁 ↔ (𝜃 ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))
2	1	simprbi 479	. . . . . . . 8 ⊢ (𝜁 → 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅))
3		bnj1450.17	. . . . . . . . . 10 ⊢ (𝜃 ↔ (𝜒 ∧ 𝑧 ∈ 𝐸))
4		bnj1450.15	. . . . . . . . . . 11 ⊢ (𝜒 → 𝑃 Fn trCl(𝑥, 𝐴, 𝑅))
5		fndm 5904	. . . . . . . . . . 11 ⊢ (𝑃 Fn trCl(𝑥, 𝐴, 𝑅) → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
6	4, 5	syl 17	. . . . . . . . . 10 ⊢ (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
7	3, 6	bnj832 30082	. . . . . . . . 9 ⊢ (𝜃 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
8	1, 7	bnj832 30082	. . . . . . . 8 ⊢ (𝜁 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
9	2, 8	eleqtrrd 2691	. . . . . . 7 ⊢ (𝜁 → 𝑧 ∈ dom 𝑃)
10		bnj1450.10	. . . . . . . 8 ⊢ 𝑃 = ∪ 𝐻
11	10	dmeqi 5247	. . . . . . 7 ⊢ dom 𝑃 = dom ∪ 𝐻
12	9, 11	syl6eleq 2698	. . . . . 6 ⊢ (𝜁 → 𝑧 ∈ dom ∪ 𝐻)
13		bnj1450.9	. . . . . . . 8 ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
14	13	bnj1317 30146	. . . . . . 7 ⊢ (𝑤 ∈ 𝐻 → ∀𝑓 𝑤 ∈ 𝐻)
15	14	bnj1400 30160	. . . . . 6 ⊢ dom ∪ 𝐻 = ∪ 𝑓 ∈ 𝐻 dom 𝑓
16	12, 15	syl6eleq 2698	. . . . 5 ⊢ (𝜁 → 𝑧 ∈ ∪ 𝑓 ∈ 𝐻 dom 𝑓)
17	16	bnj1405 30161	. . . 4 ⊢ (𝜁 → ∃𝑓 ∈ 𝐻 𝑧 ∈ dom 𝑓)
18		bnj1450.20	. . . 4 ⊢ (𝜌 ↔ (𝜁 ∧ 𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓))
19		bnj1450.1	. . . . 5 ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
20		bnj1450.2	. . . . 5 ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
21		bnj1450.3	. . . . 5 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
22		bnj1450.4	. . . . 5 ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
23		bnj1450.5	. . . . 5 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
24		bnj1450.6	. . . . 5 ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
25		bnj1450.7	. . . . 5 ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 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, (𝐺‘𝑍)⟩})
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 ⊢ (𝜒 → 𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
32		bnj1450.18	. . . . 5 ⊢ (𝜂 ↔ (𝜃 ∧ 𝑧 ∈ {𝑥}))
33	19, 20, 21, 22, 23, 24, 25, 26, 13, 10, 27, 28, 29, 30, 4, 31, 3, 32, 1	bnj1449 30370	. . . 4 ⊢ (𝜁 → ∀𝑓𝜁)
34	17, 18, 33	bnj1521 30175	. . 3 ⊢ (𝜁 → ∃𝑓𝜌)
35	13	bnj1436 30164	. . . . . . . . . 10 ⊢ (𝑓 ∈ 𝐻 → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′)
36	18, 35	bnj836 30084	. . . . . . . . 9 ⊢ (𝜌 → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′)
37	19, 20, 21, 22, 26	bnj1373 30352	. . . . . . . . . 10 ⊢ (𝜏′ ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
38	37	rexbii 3023	. . . . . . . . 9 ⊢ (∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′ ↔ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
39	36, 38	sylib 207	. . . . . . . 8 ⊢ (𝜌 → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
40	39	bnj1196 30119	. . . . . . 7 ⊢ (𝜌 → ∃𝑦(𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))))
41		3anass 1035	. . . . . . 7 ⊢ ((𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) ↔ (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))))
42	40, 41	bnj1198 30120	. . . . . 6 ⊢ (𝜌 → ∃𝑦(𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
43		bnj1450.21	. . . . . . 7 ⊢ (𝜎 ↔ (𝜌 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
44		bnj252 30022	. . . . . . 7 ⊢ ((𝜌 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) ↔ (𝜌 ∧ (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))))
45	43, 44	bitri 263	. . . . . 6 ⊢ (𝜎 ↔ (𝜌 ∧ (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))))
46	19, 20, 21, 22, 23, 24, 25, 26, 13, 10, 27, 28, 29, 30, 4, 31, 3, 32, 1, 18	bnj1444 30365	. . . . . 6 ⊢ (𝜌 → ∀𝑦𝜌)
47	42, 45, 46	bnj1340 30148	. . . . 5 ⊢ (𝜌 → ∃𝑦𝜎)
48	21	bnj1436 30164	. . . . . . . . . . 11 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)))
49	43, 48	bnj771 30088	. . . . . . . . . 10 ⊢ (𝜎 → ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)))
50	49	bnj1196 30119	. . . . . . . . 9 ⊢ (𝜎 → ∃𝑑(𝑑 ∈ 𝐵 ∧ (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))))
51		3anass 1035	. . . . . . . . 9 ⊢ ((𝑑 ∈ 𝐵 ∧ 𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)) ↔ (𝑑 ∈ 𝐵 ∧ (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))))
52	50, 51	bnj1198 30120	. . . . . . . 8 ⊢ (𝜎 → ∃𝑑(𝑑 ∈ 𝐵 ∧ 𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)))
53		bnj1450.22	. . . . . . . . 9 ⊢ (𝜑 ↔ (𝜎 ∧ 𝑑 ∈ 𝐵 ∧ 𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)))
54		bnj252 30022	. . . . . . . . 9 ⊢ ((𝜎 ∧ 𝑑 ∈ 𝐵 ∧ 𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)) ↔ (𝜎 ∧ (𝑑 ∈ 𝐵 ∧ 𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))))
55	53, 54	bitri 263	. . . . . . . 8 ⊢ (𝜑 ↔ (𝜎 ∧ (𝑑 ∈ 𝐵 ∧ 𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))))
56		bnj1450.23	. . . . . . . . 9 ⊢ 𝑋 = ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩
57	19, 20, 21, 22, 23, 24, 25, 26, 13, 10, 27, 28, 29, 30, 4, 31, 3, 32, 1, 18, 43, 53, 56	bnj1445 30366	. . . . . . . 8 ⊢ (𝜎 → ∀𝑑𝜎)
58	52, 55, 57	bnj1340 30148	. . . . . . 7 ⊢ (𝜎 → ∃𝑑𝜑)
59	53	bnj1254 30134	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))
60		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑓‘𝑥) = (𝑓‘𝑧))
61		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
62		bnj602 30239	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → pred(𝑥, 𝐴, 𝑅) = pred(𝑧, 𝐴, 𝑅))
63	62	reseq2d 5317	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → (𝑓 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑓 ↾ pred(𝑧, 𝐴, 𝑅)))
64	61, 63	opeq12d 4348	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)
65	64, 20, 56	3eqtr4g 2669	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → 𝑌 = 𝑋)
66	65	fveq2d 6107	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝐺‘𝑌) = (𝐺‘𝑋))
67	60, 66	eqeq12d 2625	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → ((𝑓‘𝑥) = (𝐺‘𝑌) ↔ (𝑓‘𝑧) = (𝐺‘𝑋)))
68	67	cbvralv 3147	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌) ↔ ∀𝑧 ∈ 𝑑 (𝑓‘𝑧) = (𝐺‘𝑋))
69	59, 68	sylib 207	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑧 ∈ 𝑑 (𝑓‘𝑧) = (𝐺‘𝑋))
70	18	simp3bi 1071	. . . . . . . . . . . 12 ⊢ (𝜌 → 𝑧 ∈ dom 𝑓)
71	43, 70	bnj769 30086	. . . . . . . . . . 11 ⊢ (𝜎 → 𝑧 ∈ dom 𝑓)
72	53, 71	bnj769 30086	. . . . . . . . . 10 ⊢ (𝜑 → 𝑧 ∈ dom 𝑓)
73		fndm 5904	. . . . . . . . . . 11 ⊢ (𝑓 Fn 𝑑 → dom 𝑓 = 𝑑)
74	53, 73	bnj771 30088	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝑓 = 𝑑)
75	72, 74	eleqtrd 2690	. . . . . . . . 9 ⊢ (𝜑 → 𝑧 ∈ 𝑑)
76	69, 75	bnj1294 30142	. . . . . . . 8 ⊢ (𝜑 → (𝑓‘𝑧) = (𝐺‘𝑋))
77	31	bnj930 30094	. . . . . . . . . . . . . 14 ⊢ (𝜒 → Fun 𝑄)
78	3, 77	bnj832 30082	. . . . . . . . . . . . 13 ⊢ (𝜃 → Fun 𝑄)
79	1, 78	bnj832 30082	. . . . . . . . . . . 12 ⊢ (𝜁 → Fun 𝑄)
80	18, 79	bnj835 30083	. . . . . . . . . . 11 ⊢ (𝜌 → Fun 𝑄)
81	43, 80	bnj769 30086	. . . . . . . . . 10 ⊢ (𝜎 → Fun 𝑄)
82	53, 81	bnj769 30086	. . . . . . . . 9 ⊢ (𝜑 → Fun 𝑄)
83	18	simp2bi 1070	. . . . . . . . . . . 12 ⊢ (𝜌 → 𝑓 ∈ 𝐻)
84	43, 83	bnj769 30086	. . . . . . . . . . 11 ⊢ (𝜎 → 𝑓 ∈ 𝐻)
85	53, 84	bnj769 30086	. . . . . . . . . 10 ⊢ (𝜑 → 𝑓 ∈ 𝐻)
86		elssuni 4403	. . . . . . . . . . 11 ⊢ (𝑓 ∈ 𝐻 → 𝑓 ⊆ ∪ 𝐻)
87	86, 10	syl6sseqr 3615	. . . . . . . . . 10 ⊢ (𝑓 ∈ 𝐻 → 𝑓 ⊆ 𝑃)
88		ssun3 3740	. . . . . . . . . . 11 ⊢ (𝑓 ⊆ 𝑃 → 𝑓 ⊆ (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩}))
89	88, 28	syl6sseqr 3615	. . . . . . . . . 10 ⊢ (𝑓 ⊆ 𝑃 → 𝑓 ⊆ 𝑄)
90	85, 87, 89	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑓 ⊆ 𝑄)
91	82, 90, 72	bnj1502 30172	. . . . . . . 8 ⊢ (𝜑 → (𝑄‘𝑧) = (𝑓‘𝑧))
92	19	bnj1517 30174	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 ∈ 𝐵 → ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
93	53, 92	bnj770 30087	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
94	62	sseq1d 3595	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → ( pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑 ↔ pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑))
95	94	cbvralv 3147	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑 ↔ ∀𝑧 ∈ 𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑)
96	93, 95	sylib 207	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑧 ∈ 𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑)
97	96, 75	bnj1294 30142	. . . . . . . . . . . . 13 ⊢ (𝜑 → pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑)
98	97, 74	sseqtr4d 3605	. . . . . . . . . . . 12 ⊢ (𝜑 → pred(𝑧, 𝐴, 𝑅) ⊆ dom 𝑓)
99	82, 90, 98	bnj1503 30173	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑄 ↾ pred(𝑧, 𝐴, 𝑅)) = (𝑓 ↾ pred(𝑧, 𝐴, 𝑅)))
100	99	opeq2d 4347	. . . . . . . . . 10 ⊢ (𝜑 → ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩ = ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)
101	100, 29, 56	3eqtr4g 2669	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 = 𝑋)
102	101	fveq2d 6107	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘𝑊) = (𝐺‘𝑋))
103	76, 91, 102	3eqtr4d 2654	. . . . . . 7 ⊢ (𝜑 → (𝑄‘𝑧) = (𝐺‘𝑊))
104	58, 103	bnj593 30069	. . . . . 6 ⊢ (𝜎 → ∃𝑑(𝑄‘𝑧) = (𝐺‘𝑊))
105	19, 20, 21, 22, 23, 24, 25, 26, 13, 10, 27, 28, 29	bnj1446 30367	. . . . . 6 ⊢ ((𝑄‘𝑧) = (𝐺‘𝑊) → ∀𝑑(𝑄‘𝑧) = (𝐺‘𝑊))
106	104, 105	bnj1397 30159	. . . . 5 ⊢ (𝜎 → (𝑄‘𝑧) = (𝐺‘𝑊))
107	47, 106	bnj593 30069	. . . 4 ⊢ (𝜌 → ∃𝑦(𝑄‘𝑧) = (𝐺‘𝑊))
108	19, 20, 21, 22, 23, 24, 25, 26, 13, 10, 27, 28, 29	bnj1447 30368	. . . 4 ⊢ ((𝑄‘𝑧) = (𝐺‘𝑊) → ∀𝑦(𝑄‘𝑧) = (𝐺‘𝑊))
109	107, 108	bnj1397 30159	. . 3 ⊢ (𝜌 → (𝑄‘𝑧) = (𝐺‘𝑊))
110	34, 109	bnj593 30069	. 2 ⊢ (𝜁 → ∃𝑓(𝑄‘𝑧) = (𝐺‘𝑊))
111	19, 20, 21, 22, 23, 24, 25, 26, 13, 10, 27, 28, 29	bnj1448 30369	. 2 ⊢ ((𝑄‘𝑧) = (𝐺‘𝑊) → ∀𝑓(𝑄‘𝑧) = (𝐺‘𝑊))
112	110, 111	bnj1397 30159	1 ⊢ (𝜁 → (𝑄‘𝑧) = (𝐺‘𝑊))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  [wsbc 3402   ∪ cun 3538   ⊆ wss 3540  ∅c0 3874  {csn 4125  ⟨cop 4131  ∪ cuni 4372  ∪ ciun 4455   class class class wbr 4583  dom cdm 5038   ↾ cres 5040  Fun wfun 5798   Fn wfn 5799  ‘cfv 5804   ∧ w-bnj17 30005   predc-bnj14 30007   FrSe w-bnj15 30011   trClc-bnj18 30013

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-res 5050  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-bnj17 30006  df-bnj14 30008  df-bnj18 30014

This theorem is referenced by:  bnj1423  30373