Theorem bnj1450 33030

Description: Technical lemma for bnj60 33042. 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 497	. . . . . . . 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	4	fndmd 6538	. . . . . . . . . 10 ⊢ (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
6	3, 5	bnj832 32738	. . . . . . . . 9 ⊢ (𝜃 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
7	1, 6	bnj832 32738	. . . . . . . 8 ⊢ (𝜁 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
8	2, 7	eleqtrrd 2842	. . . . . . 7 ⊢ (𝜁 → 𝑧 ∈ dom 𝑃)
9		bnj1450.10	. . . . . . . 8 ⊢ 𝑃 = ∪ 𝐻
10	9	dmeqi 5813	. . . . . . 7 ⊢ dom 𝑃 = dom ∪ 𝐻
11	8, 10	eleqtrdi 2849	. . . . . 6 ⊢ (𝜁 → 𝑧 ∈ dom ∪ 𝐻)
12		bnj1450.9	. . . . . . . 8 ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
13	12	bnj1317 32801	. . . . . . 7 ⊢ (𝑤 ∈ 𝐻 → ∀𝑓 𝑤 ∈ 𝐻)
14	13	bnj1400 32815	. . . . . 6 ⊢ dom ∪ 𝐻 = ∪ 𝑓 ∈ 𝐻 dom 𝑓
15	11, 14	eleqtrdi 2849	. . . . 5 ⊢ (𝜁 → 𝑧 ∈ ∪ 𝑓 ∈ 𝐻 dom 𝑓)
16	15	bnj1405 32816	. . . 4 ⊢ (𝜁 → ∃𝑓 ∈ 𝐻 𝑧 ∈ dom 𝑓)
17		bnj1450.20	. . . 4 ⊢ (𝜌 ↔ (𝜁 ∧ 𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓))
18		bnj1450.1	. . . . 5 ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
19		bnj1450.2	. . . . 5 ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
20		bnj1450.3	. . . . 5 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
21		bnj1450.4	. . . . 5 ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
22		bnj1450.5	. . . . 5 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
23		bnj1450.6	. . . . 5 ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
24		bnj1450.7	. . . . 5 ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 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, (𝐺‘𝑍)⟩})
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 ⊢ (𝜒 → 𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
31		bnj1450.18	. . . . 5 ⊢ (𝜂 ↔ (𝜃 ∧ 𝑧 ∈ {𝑥}))
32	18, 19, 20, 21, 22, 23, 24, 25, 12, 9, 26, 27, 28, 29, 4, 30, 3, 31, 1	bnj1449 33028	. . . 4 ⊢ (𝜁 → ∀𝑓𝜁)
33	16, 17, 32	bnj1521 32831	. . 3 ⊢ (𝜁 → ∃𝑓𝜌)
34	12	bnj1436 32819	. . . . . . . . . 10 ⊢ (𝑓 ∈ 𝐻 → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′)
35	17, 34	bnj836 32740	. . . . . . . . 9 ⊢ (𝜌 → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′)
36	18, 19, 20, 21, 25	bnj1373 33010	. . . . . . . . . 10 ⊢ (𝜏′ ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
37	36	rexbii 3181	. . . . . . . . 9 ⊢ (∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′ ↔ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
38	35, 37	sylib 217	. . . . . . . 8 ⊢ (𝜌 → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
39	38	bnj1196 32774	. . . . . . 7 ⊢ (𝜌 → ∃𝑦(𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))))
40		3anass 1094	. . . . . . 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)))))
41	39, 40	bnj1198 32775	. . . . . 6 ⊢ (𝜌 → ∃𝑦(𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
42		bnj1450.21	. . . . . . 7 ⊢ (𝜎 ↔ (𝜌 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
43		bnj252 32682	. . . . . . 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)))))
44	42, 43	bitri 274	. . . . . 6 ⊢ (𝜎 ↔ (𝜌 ∧ (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))))
45	18, 19, 20, 21, 22, 23, 24, 25, 12, 9, 26, 27, 28, 29, 4, 30, 3, 31, 1, 17	bnj1444 33023	. . . . . 6 ⊢ (𝜌 → ∀𝑦𝜌)
46	41, 44, 45	bnj1340 32803	. . . . 5 ⊢ (𝜌 → ∃𝑦𝜎)
47	20	bnj1436 32819	. . . . . . . . . . 11 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)))
48	42, 47	bnj771 32744	. . . . . . . . . 10 ⊢ (𝜎 → ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)))
49	48	bnj1196 32774	. . . . . . . . 9 ⊢ (𝜎 → ∃𝑑(𝑑 ∈ 𝐵 ∧ (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))))
50		3anass 1094	. . . . . . . . 9 ⊢ ((𝑑 ∈ 𝐵 ∧ 𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)) ↔ (𝑑 ∈ 𝐵 ∧ (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))))
51	49, 50	bnj1198 32775	. . . . . . . 8 ⊢ (𝜎 → ∃𝑑(𝑑 ∈ 𝐵 ∧ 𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)))
52		bnj1450.22	. . . . . . . . 9 ⊢ (𝜑 ↔ (𝜎 ∧ 𝑑 ∈ 𝐵 ∧ 𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)))
53		bnj252 32682	. . . . . . . . 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))))
54	52, 53	bitri 274	. . . . . . . 8 ⊢ (𝜑 ↔ (𝜎 ∧ (𝑑 ∈ 𝐵 ∧ 𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))))
55		bnj1450.23	. . . . . . . . 9 ⊢ 𝑋 = ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩
56	18, 19, 20, 21, 22, 23, 24, 25, 12, 9, 26, 27, 28, 29, 4, 30, 3, 31, 1, 17, 42, 52, 55	bnj1445 33024	. . . . . . . 8 ⊢ (𝜎 → ∀𝑑𝜎)
57	51, 54, 56	bnj1340 32803	. . . . . . 7 ⊢ (𝜎 → ∃𝑑𝜑)
58		fveq2 6774	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑓‘𝑥) = (𝑓‘𝑧))
59		id 22	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
60		bnj602 32895	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → pred(𝑥, 𝐴, 𝑅) = pred(𝑧, 𝐴, 𝑅))
61	60	reseq2d 5891	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑓 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑓 ↾ pred(𝑧, 𝐴, 𝑅)))
62	59, 61	opeq12d 4812	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)
63	62, 19, 55	3eqtr4g 2803	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → 𝑌 = 𝑋)
64	63	fveq2d 6778	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝐺‘𝑌) = (𝐺‘𝑋))
65	58, 64	eqeq12d 2754	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ((𝑓‘𝑥) = (𝐺‘𝑌) ↔ (𝑓‘𝑧) = (𝐺‘𝑋)))
66	52	bnj1254 32789	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))
67	17	simp3bi 1146	. . . . . . . . . . . 12 ⊢ (𝜌 → 𝑧 ∈ dom 𝑓)
68	42, 67	bnj769 32742	. . . . . . . . . . 11 ⊢ (𝜎 → 𝑧 ∈ dom 𝑓)
69	52, 68	bnj769 32742	. . . . . . . . . 10 ⊢ (𝜑 → 𝑧 ∈ dom 𝑓)
70		fndm 6536	. . . . . . . . . . 11 ⊢ (𝑓 Fn 𝑑 → dom 𝑓 = 𝑑)
71	52, 70	bnj771 32744	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝑓 = 𝑑)
72	69, 71	eleqtrd 2841	. . . . . . . . 9 ⊢ (𝜑 → 𝑧 ∈ 𝑑)
73	65, 66, 72	rspcdva 3562	. . . . . . . 8 ⊢ (𝜑 → (𝑓‘𝑧) = (𝐺‘𝑋))
74	30	fnfund 6534	. . . . . . . . . . . . . 14 ⊢ (𝜒 → Fun 𝑄)
75	3, 74	bnj832 32738	. . . . . . . . . . . . 13 ⊢ (𝜃 → Fun 𝑄)
76	1, 75	bnj832 32738	. . . . . . . . . . . 12 ⊢ (𝜁 → Fun 𝑄)
77	17, 76	bnj835 32739	. . . . . . . . . . 11 ⊢ (𝜌 → Fun 𝑄)
78	42, 77	bnj769 32742	. . . . . . . . . 10 ⊢ (𝜎 → Fun 𝑄)
79	52, 78	bnj769 32742	. . . . . . . . 9 ⊢ (𝜑 → Fun 𝑄)
80	17	simp2bi 1145	. . . . . . . . . . . 12 ⊢ (𝜌 → 𝑓 ∈ 𝐻)
81	42, 80	bnj769 32742	. . . . . . . . . . 11 ⊢ (𝜎 → 𝑓 ∈ 𝐻)
82	52, 81	bnj769 32742	. . . . . . . . . 10 ⊢ (𝜑 → 𝑓 ∈ 𝐻)
83		elssuni 4871	. . . . . . . . . . 11 ⊢ (𝑓 ∈ 𝐻 → 𝑓 ⊆ ∪ 𝐻)
84	83, 9	sseqtrrdi 3972	. . . . . . . . . 10 ⊢ (𝑓 ∈ 𝐻 → 𝑓 ⊆ 𝑃)
85		ssun3 4108	. . . . . . . . . . 11 ⊢ (𝑓 ⊆ 𝑃 → 𝑓 ⊆ (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩}))
86	85, 27	sseqtrrdi 3972	. . . . . . . . . 10 ⊢ (𝑓 ⊆ 𝑃 → 𝑓 ⊆ 𝑄)
87	82, 84, 86	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑓 ⊆ 𝑄)
88	79, 87, 69	bnj1502 32828	. . . . . . . 8 ⊢ (𝜑 → (𝑄‘𝑧) = (𝑓‘𝑧))
89	60	sseq1d 3952	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → ( pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑 ↔ pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑))
90	18	bnj1517 32830	. . . . . . . . . . . . . . 15 ⊢ (𝑑 ∈ 𝐵 → ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
91	52, 90	bnj770 32743	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
92	89, 91, 72	rspcdva 3562	. . . . . . . . . . . . 13 ⊢ (𝜑 → pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑)
93	92, 71	sseqtrrd 3962	. . . . . . . . . . . 12 ⊢ (𝜑 → pred(𝑧, 𝐴, 𝑅) ⊆ dom 𝑓)
94	79, 87, 93	bnj1503 32829	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑄 ↾ pred(𝑧, 𝐴, 𝑅)) = (𝑓 ↾ pred(𝑧, 𝐴, 𝑅)))
95	94	opeq2d 4811	. . . . . . . . . 10 ⊢ (𝜑 → ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩ = ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)
96	95, 28, 55	3eqtr4g 2803	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 = 𝑋)
97	96	fveq2d 6778	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘𝑊) = (𝐺‘𝑋))
98	73, 88, 97	3eqtr4d 2788	. . . . . . 7 ⊢ (𝜑 → (𝑄‘𝑧) = (𝐺‘𝑊))
99	57, 98	bnj593 32725	. . . . . 6 ⊢ (𝜎 → ∃𝑑(𝑄‘𝑧) = (𝐺‘𝑊))
100	18, 19, 20, 21, 22, 23, 24, 25, 12, 9, 26, 27, 28	bnj1446 33025	. . . . . 6 ⊢ ((𝑄‘𝑧) = (𝐺‘𝑊) → ∀𝑑(𝑄‘𝑧) = (𝐺‘𝑊))
101	99, 100	bnj1397 32814	. . . . 5 ⊢ (𝜎 → (𝑄‘𝑧) = (𝐺‘𝑊))
102	46, 101	bnj593 32725	. . . 4 ⊢ (𝜌 → ∃𝑦(𝑄‘𝑧) = (𝐺‘𝑊))
103	18, 19, 20, 21, 22, 23, 24, 25, 12, 9, 26, 27, 28	bnj1447 33026	. . . 4 ⊢ ((𝑄‘𝑧) = (𝐺‘𝑊) → ∀𝑦(𝑄‘𝑧) = (𝐺‘𝑊))
104	102, 103	bnj1397 32814	. . 3 ⊢ (𝜌 → (𝑄‘𝑧) = (𝐺‘𝑊))
105	33, 104	bnj593 32725	. 2 ⊢ (𝜁 → ∃𝑓(𝑄‘𝑧) = (𝐺‘𝑊))
106	18, 19, 20, 21, 22, 23, 24, 25, 12, 9, 26, 27, 28	bnj1448 33027	. 2 ⊢ ((𝑄‘𝑧) = (𝐺‘𝑊) → ∀𝑓(𝑄‘𝑧) = (𝐺‘𝑊))
107	105, 106	bnj1397 32814	1 ⊢ (𝜁 → (𝑄‘𝑧) = (𝐺‘𝑊))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539  ∃wex 1782   ∈ wcel 2106  {cab 2715   ≠ wne 2943  ∀wral 3064  ∃wrex 3065  {crab 3068  [wsbc 3716   ∪ cun 3885   ⊆ wss 3887  ∅c0 4256  {csn 4561  ⟨cop 4567  ∪ cuni 4839  ∪ ciun 4924   class class class wbr 5074  dom cdm 5589   ↾ cres 5591  Fun wfun 6427   Fn wfn 6428  ‘cfv 6433   ∧ w-bnj17 32665   predc-bnj14 32667   FrSe w-bnj15 32671   trClc-bnj18 32673

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-res 5601  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441  df-bnj17 32666  df-bnj14 32668  df-bnj18 32674

This theorem is referenced by:  bnj1423  33031