Theorem bnj1311 33004

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
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 260	. . . . . . . 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 32783	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)) → 𝑅 FrSe 𝐴)
3		ssrab2 4013	. . . . . . . 8 ⊢ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} ⊆ 𝐷
4		bnj1311.4	. . . . . . . . 9 ⊢ 𝐷 = (dom 𝑔 ∩ dom ℎ)
5	1	bnj1235 32784	. . . . . . . . . . 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 2738	. . . . . . . . . . . 12 ⊢ ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩
9		eqid 2738	. . . . . . . . . . . 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 32993	. . . . . . . . . . 11 ⊢ 𝐶 = {𝑔 ∣ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))}
11	5, 10	eleqtrdi 2849	. . . . . . . . . 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 2719	. . . . . . . . . . . . . 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 32786	. . . . . . . . . . . . 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 32774	. . . . . . . . . . . 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	abeq2i 2875	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ 𝐵 ↔ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
17	16	simplbi 498	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ 𝐵 → 𝑑 ⊆ 𝐴)
18		fndm 6536	. . . . . . . . . . . . 13 ⊢ (𝑔 Fn 𝑑 → dom 𝑔 = 𝑑)
19	17, 18	bnj1241 32787	. . . . . . . . . . . 12 ⊢ ((𝑑 ∈ 𝐵 ∧ 𝑔 Fn 𝑑) → dom 𝑔 ⊆ 𝐴)
20	14, 19	bnj593 32725	. . . . . . . . . . 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 32751	. . . . . . . . . 10 ⊢ (𝑔 ∈ {𝑔 ∣ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))} → dom 𝑔 ⊆ 𝐴)
22		ssinss1 4171	. . . . . . . . . 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 3969	. . . . . . . 8 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)) → 𝐷 ⊆ 𝐴)
25	3, 24	sstrid 3932	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)) → {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} ⊆ 𝐴)
26		eqid 2738	. . . . . . . 8 ⊢ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} = {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)}
27		biid 260	. . . . . . . 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 32997	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)) → {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} ≠ ∅)
29		nfrab1 3317	. . . . . . . . 9 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)}
30	29	nfcrii 2899	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} → ∀𝑥 𝑧 ∈ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)})
31	30	bnj1228 32991	. . . . . . 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 1370	. . . . . 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 1913	. . . . . . 7 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 𝑅 FrSe 𝐴)
34	15	bnj1309 33002	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝐵 → ∀𝑥 𝑤 ∈ 𝐵)
35	7, 34	bnj1307 33003	. . . . . . . 8 ⊢ (𝑤 ∈ 𝐶 → ∀𝑥 𝑤 ∈ 𝐶)
36	35	hblem 2870	. . . . . . 7 ⊢ (𝑔 ∈ 𝐶 → ∀𝑥 𝑔 ∈ 𝐶)
37	35	hblem 2870	. . . . . . 7 ⊢ (ℎ ∈ 𝐶 → ∀𝑥 ℎ ∈ 𝐶)
38		ax-5 1913	. . . . . . 7 ⊢ ((𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷) → ∀𝑥(𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷))
39	33, 36, 37, 38	bnj982 32758	. . . . . 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 32831	. . . . 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 1136	. . . . 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 32998	. . . . . . . . 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 1129	. . . . . . . 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 33000	. . . . . . 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 2738	. . . . . . 7 ⊢ ⟨𝑥, (ℎ ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, (ℎ ↾ pred(𝑥, 𝐴, 𝑅))⟩
46		eqid 2738	. . . . . . 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 33001	. . . . . 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 32835	. . . . . . 7 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} → (𝑔‘𝑥) ≠ (ℎ‘𝑥))
49	48	necon2bi 2974	. . . . . 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 32799	. . . 4 ⊢ ¬ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷))
52		df-bnj17 32666	. . . 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 401	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶) → ¬ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷))
55		nne 2947	. 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 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  {cab 2715   ≠ wne 2943  ∀wral 3064  ∃wrex 3065  {crab 3068   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  ⟨cop 4567   class class class wbr 5074  dom cdm 5589   ↾ cres 5591   Fn wfn 6428  ‘cfv 6433   ∧ w-bnj17 32665   predc-bnj14 32667   FrSe w-bnj15 32671

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-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-reg 9351  ax-inf2 9399

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  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-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-om 7713  df-1o 8297  df-bnj17 32666  df-bnj14 32668  df-bnj13 32670  df-bnj15 32672  df-bnj18 32674  df-bnj19 32676

This theorem is referenced by:  bnj1326  33006  bnj60  33042