Theorem bnj1311 34035

Description: Technical lemma for bnj60 34073. 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
bnj1311.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1311.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1311.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1311.4	⊢ 𝐷 = (dom 𝑔 ∩ dom ℎ)

Assertion

Ref	Expression
bnj1311	⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶) → (𝑔 ↾ 𝐷) = (ℎ ↾ 𝐷))

Distinct variable groups:   𝐴,𝑑,𝑓,𝑥   𝐵,𝑓,𝑔   𝐵,ℎ,𝑓   𝐷,𝑑,𝑥   𝐺,𝑑,𝑓,𝑔   ℎ,𝐺,𝑑   𝑅,𝑑,𝑓,𝑥   𝑔,𝑌   ℎ,𝑌   𝑥,𝑔   𝑥,ℎ

Allowed substitution hints:   𝐴(𝑔,ℎ)   𝐵(𝑥,𝑑)   𝐶(𝑥,𝑓,𝑔,ℎ,𝑑)   𝐷(𝑓,𝑔,ℎ)   𝑅(𝑔,ℎ)   𝐺(𝑥)   𝑌(𝑥,𝑓,𝑑)

Proof of Theorem bnj1311

Dummy variables 𝑤 𝑧 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		biid 261	. . . . . . . 8 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)) ↔ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)))
2	1	bnj1232 33814	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)) → 𝑅 FrSe 𝐴)
3		ssrab2 4078	. . . . . . . 8 ⊢ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} ⊆ 𝐷
4		bnj1311.4	. . . . . . . . 9 ⊢ 𝐷 = (dom 𝑔 ∩ dom ℎ)
5	1	bnj1235 33815	. . . . . . . . . . 11 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)) → 𝑔 ∈ 𝐶)
6		bnj1311.2	. . . . . . . . . . . 12 ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
7		bnj1311.3	. . . . . . . . . . . 12 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
8		eqid 2733	. . . . . . . . . . . 12 ⊢ ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩
9		eqid 2733	. . . . . . . . . . . 12 ⊢ {𝑔 ∣ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))} = {𝑔 ∣ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))}
10	6, 7, 8, 9	bnj1234 34024	. . . . . . . . . . 11 ⊢ 𝐶 = {𝑔 ∣ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))}
11	5, 10	eleqtrdi 2844	. . . . . . . . . 10 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)) → 𝑔 ∈ {𝑔 ∣ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))})
12		abid 2714	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ {𝑔 ∣ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))} ↔ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩)))
13	12	bnj1238 33817	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ {𝑔 ∣ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))} → ∃𝑑 ∈ 𝐵 𝑔 Fn 𝑑)
14	13	bnj1196 33805	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ {𝑔 ∣ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))} → ∃𝑑(𝑑 ∈ 𝐵 ∧ 𝑔 Fn 𝑑))
15		bnj1311.1	. . . . . . . . . . . . . . 15 ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
16	15	eqabri 2878	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ 𝐵 ↔ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
17	16	simplbi 499	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ 𝐵 → 𝑑 ⊆ 𝐴)
18		fndm 6653	. . . . . . . . . . . . 13 ⊢ (𝑔 Fn 𝑑 → dom 𝑔 = 𝑑)
19	17, 18	bnj1241 33818	. . . . . . . . . . . 12 ⊢ ((𝑑 ∈ 𝐵 ∧ 𝑔 Fn 𝑑) → dom 𝑔 ⊆ 𝐴)
20	14, 19	bnj593 33756	. . . . . . . . . . 11 ⊢ (𝑔 ∈ {𝑔 ∣ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))} → ∃𝑑dom 𝑔 ⊆ 𝐴)
21	20	bnj937 33782	. . . . . . . . . 10 ⊢ (𝑔 ∈ {𝑔 ∣ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))} → dom 𝑔 ⊆ 𝐴)
22		ssinss1 4238	. . . . . . . . . 10 ⊢ (dom 𝑔 ⊆ 𝐴 → (dom 𝑔 ∩ dom ℎ) ⊆ 𝐴)
23	11, 21, 22	3syl 18	. . . . . . . . 9 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)) → (dom 𝑔 ∩ dom ℎ) ⊆ 𝐴)
24	4, 23	eqsstrid 4031	. . . . . . . 8 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)) → 𝐷 ⊆ 𝐴)
25	3, 24	sstrid 3994	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)) → {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} ⊆ 𝐴)
26		eqid 2733	. . . . . . . 8 ⊢ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} = {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)}
27		biid 261	. . . . . . . 8 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)) ∧ 𝑥 ∈ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} ¬ 𝑦𝑅𝑥) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)) ∧ 𝑥 ∈ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} ¬ 𝑦𝑅𝑥))
28	15, 6, 7, 4, 26, 1, 27	bnj1253 34028	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)) → {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} ≠ ∅)
29		nfrab1 3452	. . . . . . . . 9 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)}
30	29	nfcrii 2896	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} → ∀𝑥 𝑧 ∈ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)})
31	30	bnj1228 34022	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} ⊆ 𝐴 ∧ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} ≠ ∅) → ∃𝑥 ∈ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)}∀𝑦 ∈ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} ¬ 𝑦𝑅𝑥)
32	2, 25, 28, 31	syl3anc 1372	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)) → ∃𝑥 ∈ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)}∀𝑦 ∈ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} ¬ 𝑦𝑅𝑥)
33		ax-5 1914	. . . . . . 7 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 𝑅 FrSe 𝐴)
34	15	bnj1309 34033	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝐵 → ∀𝑥 𝑤 ∈ 𝐵)
35	7, 34	bnj1307 34034	. . . . . . . 8 ⊢ (𝑤 ∈ 𝐶 → ∀𝑥 𝑤 ∈ 𝐶)
36	35	hblem 2866	. . . . . . 7 ⊢ (𝑔 ∈ 𝐶 → ∀𝑥 𝑔 ∈ 𝐶)
37	35	hblem 2866	. . . . . . 7 ⊢ (ℎ ∈ 𝐶 → ∀𝑥 ℎ ∈ 𝐶)
38		ax-5 1914	. . . . . . 7 ⊢ ((𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷) → ∀𝑥(𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷))
39	33, 36, 37, 38	bnj982 33789	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)) → ∀𝑥(𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)))
40	32, 27, 39	bnj1521 33862	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)) → ∃𝑥((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)) ∧ 𝑥 ∈ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} ¬ 𝑦𝑅𝑥))
41		simp2 1138	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)) ∧ 𝑥 ∈ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} ¬ 𝑦𝑅𝑥) → 𝑥 ∈ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)})
42	15, 6, 7, 4, 26, 1, 27	bnj1279 34029	. . . . . . . . 9 ⊢ ((𝑥 ∈ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} ¬ 𝑦𝑅𝑥) → ( pred(𝑥, 𝐴, 𝑅) ∩ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)}) = ∅)
43	42	3adant1 1131	. . . . . . . 8 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)) ∧ 𝑥 ∈ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} ¬ 𝑦𝑅𝑥) → ( pred(𝑥, 𝐴, 𝑅) ∩ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)}) = ∅)
44	15, 6, 7, 4, 26, 1, 27, 43	bnj1280 34031	. . . . . . 7 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)) ∧ 𝑥 ∈ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} ¬ 𝑦𝑅𝑥) → (𝑔 ↾ pred(𝑥, 𝐴, 𝑅)) = (ℎ ↾ pred(𝑥, 𝐴, 𝑅)))
45		eqid 2733	. . . . . . 7 ⊢ ⟨𝑥, (ℎ ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, (ℎ ↾ pred(𝑥, 𝐴, 𝑅))⟩
46		eqid 2733	. . . . . . 7 ⊢ {ℎ ∣ ∃𝑑 ∈ 𝐵 (ℎ Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (ℎ‘𝑥) = (𝐺‘⟨𝑥, (ℎ ↾ pred(𝑥, 𝐴, 𝑅))⟩))} = {ℎ ∣ ∃𝑑 ∈ 𝐵 (ℎ Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (ℎ‘𝑥) = (𝐺‘⟨𝑥, (ℎ ↾ pred(𝑥, 𝐴, 𝑅))⟩))}
47	15, 6, 7, 4, 26, 1, 27, 44, 8, 9, 45, 46	bnj1296 34032	. . . . . 6 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)) ∧ 𝑥 ∈ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} ¬ 𝑦𝑅𝑥) → (𝑔‘𝑥) = (ℎ‘𝑥))
48	26	bnj1538 33866	. . . . . . 7 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} → (𝑔‘𝑥) ≠ (ℎ‘𝑥))
49	48	necon2bi 2972	. . . . . 6 ⊢ ((𝑔‘𝑥) = (ℎ‘𝑥) → ¬ 𝑥 ∈ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)})
50	47, 49	syl 17	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)) ∧ 𝑥 ∈ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} ¬ 𝑦𝑅𝑥) → ¬ 𝑥 ∈ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)})
51	40, 41, 50	bnj1304 33830	. . . 4 ⊢ ¬ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷))
52		df-bnj17 33698	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶) ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)))
53	51, 52	mtbi 322	. . 3 ⊢ ¬ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶) ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷))
54	53	imnani 402	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶) → ¬ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷))
55		nne 2945	. 2 ⊢ (¬ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷) ↔ (𝑔 ↾ 𝐷) = (ℎ ↾ 𝐷))
56	54, 55	sylib 217	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶) → (𝑔 ↾ 𝐷) = (ℎ ↾ 𝐷))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {cab 2710   ≠ wne 2941  ∀wral 3062  ∃wrex 3071  {crab 3433   ∩ cin 3948   ⊆ wss 3949  ∅c0 4323  ⟨cop 4635   class class class wbr 5149  dom cdm 5677   ↾ cres 5679   Fn wfn 6539  ‘cfv 6544   ∧ w-bnj17 33697   predc-bnj14 33699   FrSe w-bnj15 33703

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-reg 9587  ax-inf2 9636

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-om 7856  df-1o 8466  df-bnj17 33698  df-bnj14 33700  df-bnj13 33702  df-bnj15 33704  df-bnj18 33706  df-bnj19 33708

This theorem is referenced by:  bnj1326  34037  bnj60  34073