Theorem bnj1311 35654

Description: Technical lemma for bnj60 35692. 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 264	. . . . . . . 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 35433	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)) → 𝑅 FrSe 𝐴)
3		ssrab2 4005	. . . . . . . 8 ⊢ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} ⊆ 𝐷
4		bnj1311.4	. . . . . . . . 9 ⊢ 𝐷 = (dom 𝑔 ∩ dom ℎ)
5	1	bnj1235 35434	. . . . . . . . . . 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 2760	. . . . . . . . . . . 12 ⊢ ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩
9		eqid 2760	. . . . . . . . . . . 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 35643	. . . . . . . . . . 11 ⊢ 𝐶 = {𝑔 ∣ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))}
11	5, 10	eleqtrdi 2870	. . . . . . . . . 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 2742	. . . . . . . . . . . . . 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 35436	. . . . . . . . . . . . 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 35424	. . . . . . . . . . . 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 2902	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ 𝐵 ↔ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
17	16	simplbi 502	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ 𝐵 → 𝑑 ⊆ 𝐴)
18		fndm 6642	. . . . . . . . . . . . 13 ⊢ (𝑔 Fn 𝑑 → dom 𝑔 = 𝑑)
19	17, 18	bnj1241 35437	. . . . . . . . . . . 12 ⊢ ((𝑑 ∈ 𝐵 ∧ 𝑔 Fn 𝑑) → dom 𝑔 ⊆ 𝐴)
20	14, 19	bnj593 35376	. . . . . . . . . . 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 35402	. . . . . . . . . 10 ⊢ (𝑔 ∈ {𝑔 ∣ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))} → dom 𝑔 ⊆ 𝐴)
22		ssinss1 4142	. . . . . . . . . 10 ⊢ (dom 𝑔 ⊆ 𝐴 → (dom 𝑔 ∩ dom ℎ) ⊆ 𝐴)
23	11, 21, 22	3syl 19	. . . . . . . . 9 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)) → (dom 𝑔 ∩ dom ℎ) ⊆ 𝐴)
24	4, 23	eqsstrid 3946	. . . . . . . 8 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)) → 𝐷 ⊆ 𝐴)
25	3, 24	sstrid 3919	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)) → {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} ⊆ 𝐴)
26		eqid 2760	. . . . . . . 8 ⊢ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} = {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)}
27		biid 264	. . . . . . . 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 35647	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)) → {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} ≠ ∅)
29		nfrab1 3431	. . . . . . . . 9 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)}
30	29	nfcrii 2917	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} → ∀𝑥 𝑧 ∈ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)})
31	30	bnj1228 35641	. . . . . . 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 1398	. . . . . 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 1942	. . . . . . 7 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 𝑅 FrSe 𝐴)
34	15	bnj1309 35652	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝐵 → ∀𝑥 𝑤 ∈ 𝐵)
35	7, 34	bnj1307 35653	. . . . . . . 8 ⊢ (𝑤 ∈ 𝐶 → ∀𝑥 𝑤 ∈ 𝐶)
36	35	hblem 2891	. . . . . . 7 ⊢ (𝑔 ∈ 𝐶 → ∀𝑥 𝑔 ∈ 𝐶)
37	35	hblem 2891	. . . . . . 7 ⊢ (ℎ ∈ 𝐶 → ∀𝑥 ℎ ∈ 𝐶)
38		ax-5 1942	. . . . . . 7 ⊢ ((𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷) → ∀𝑥(𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷))
39	33, 36, 37, 38	bnj982 35409	. . . . . 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 35481	. . . . 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 1155	. . . . 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 35648	. . . . . . . . 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 1148	. . . . . . . 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 35650	. . . . . . 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 2760	. . . . . . 7 ⊢ ⟨𝑥, (ℎ ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, (ℎ ↾ pred(𝑥, 𝐴, 𝑅))⟩
46		eqid 2760	. . . . . . 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 35651	. . . . . 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 35485	. . . . . . 7 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} → (𝑔‘𝑥) ≠ (ℎ‘𝑥))
49	48	necon2bi 2985	. . . . . 6 ⊢ ((𝑔‘𝑥) = (ℎ‘𝑥) → ¬ 𝑥 ∈ {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)})
50	47, 49	syl 18	. . . . 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 35449	. . . 4 ⊢ ¬ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷))
52		df-bnj17 35318	. . . 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 325	. . 3 ⊢ ¬ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶) ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷))
54	53	imnani 406	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶) → ¬ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷))
55		nne 2959	. 2 ⊢ (¬ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷) ↔ (𝑔 ↾ 𝐷) = (ℎ ↾ 𝐷))
56	54, 55	sylib 221	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶) → (𝑔 ↾ 𝐷) = (ℎ ↾ 𝐷))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  {cab 2738   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  {crab 3412   ⊆ wss 3895   ∩ cin 4089  ∅c0 4278  ⟨cop 4589   class class class wbr 5102  dom cdm 5650   ↾ cres 5652   Fn wfn 6533  ‘cfv 6538   ∧ w-bnj17 35317   predc-bnj14 35319   FrSe w-bnj15 35323

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-reg 9586  ax-inf2 9642

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-om 7878  df-1o 8476  df-bnj17 35318  df-bnj14 35320  df-bnj13 35322  df-bnj15 35324  df-bnj18 35326  df-bnj19 35328

This theorem is used by:  bnj1326  35656  bnj60  35692