Theorem fpropnf1 7019

Description: A function, given by an unordered pair of ordered pairs, which is not injective/one-to-one. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 8-Jan-2021.)

Hypothesis

Ref	Expression
fpropnf1.f	⊢ 𝐹 = {⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}

Assertion

Ref	Expression
fpropnf1	⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → (Fun 𝐹 ∧ ¬ Fun ◡𝐹))

Proof of Theorem fpropnf1

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉) → (𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉))
2	1	3adant3 1128	. . . . . 6 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉))
3	2	adantr 483	. . . . 5 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → (𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉))
4		id 22	. . . . . . . 8 ⊢ (𝑍 ∈ 𝑊 → 𝑍 ∈ 𝑊)
5	4, 4	jca 514	. . . . . . 7 ⊢ (𝑍 ∈ 𝑊 → (𝑍 ∈ 𝑊 ∧ 𝑍 ∈ 𝑊))
6	5	3ad2ant3 1131	. . . . . 6 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑍 ∈ 𝑊 ∧ 𝑍 ∈ 𝑊))
7	6	adantr 483	. . . . 5 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → (𝑍 ∈ 𝑊 ∧ 𝑍 ∈ 𝑊))
8		simpr 487	. . . . 5 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → 𝑋 ≠ 𝑌)
9	3, 7, 8	3jca 1124	. . . 4 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉) ∧ (𝑍 ∈ 𝑊 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌))
10		funprg 6403	. . . 4 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉) ∧ (𝑍 ∈ 𝑊 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → Fun {⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩})
11	9, 10	syl 17	. . 3 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → Fun {⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩})
12		fpropnf1.f	. . . 4 ⊢ 𝐹 = {⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}
13	12	funeqi 6371	. . 3 ⊢ (Fun 𝐹 ↔ Fun {⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩})
14	11, 13	sylibr 236	. 2 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → Fun 𝐹)
15		neneq 3022	. . . 4 ⊢ (𝑋 ≠ 𝑌 → ¬ 𝑋 = 𝑌)
16	15	adantl 484	. . 3 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → ¬ 𝑋 = 𝑌)
17		fprg 6912	. . . . . 6 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉) ∧ (𝑍 ∈ 𝑊 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → {⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}:{𝑋, 𝑌}⟶{𝑍, 𝑍})
18	9, 17	syl 17	. . . . 5 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → {⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}:{𝑋, 𝑌}⟶{𝑍, 𝑍})
19	12	eqcomi 2830	. . . . . 6 ⊢ {⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩} = 𝐹
20	19	feq1i 6500	. . . . 5 ⊢ ({⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}:{𝑋, 𝑌}⟶{𝑍, 𝑍} ↔ 𝐹:{𝑋, 𝑌}⟶{𝑍, 𝑍})
21	18, 20	sylib 220	. . . 4 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → 𝐹:{𝑋, 𝑌}⟶{𝑍, 𝑍})
22		df-f1 6355	. . . . 5 ⊢ (𝐹:{𝑋, 𝑌}–1-1→{𝑍, 𝑍} ↔ (𝐹:{𝑋, 𝑌}⟶{𝑍, 𝑍} ∧ Fun ◡𝐹))
23		dff13 7007	. . . . . 6 ⊢ (𝐹:{𝑋, 𝑌}–1-1→{𝑍, 𝑍} ↔ (𝐹:{𝑋, 𝑌}⟶{𝑍, 𝑍} ∧ ∀𝑥 ∈ {𝑋, 𝑌}∀𝑦 ∈ {𝑋, 𝑌} ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
24		fveqeq2 6674	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝐹‘𝑋) = (𝐹‘𝑦)))
25		eqeq1 2825	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑋 → (𝑥 = 𝑦 ↔ 𝑋 = 𝑦))
26	24, 25	imbi12d 347	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ ((𝐹‘𝑋) = (𝐹‘𝑦) → 𝑋 = 𝑦)))
27	26	ralbidv 3197	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (∀𝑦 ∈ {𝑋, 𝑌} ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ {𝑋, 𝑌} ((𝐹‘𝑋) = (𝐹‘𝑦) → 𝑋 = 𝑦)))
28		fveqeq2 6674	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑌 → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝐹‘𝑌) = (𝐹‘𝑦)))
29		eqeq1 2825	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑌 → (𝑥 = 𝑦 ↔ 𝑌 = 𝑦))
30	28, 29	imbi12d 347	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑌 → (((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ ((𝐹‘𝑌) = (𝐹‘𝑦) → 𝑌 = 𝑦)))
31	30	ralbidv 3197	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑌 → (∀𝑦 ∈ {𝑋, 𝑌} ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ {𝑋, 𝑌} ((𝐹‘𝑌) = (𝐹‘𝑦) → 𝑌 = 𝑦)))
32	27, 31	ralprg 4626	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉) → (∀𝑥 ∈ {𝑋, 𝑌}∀𝑦 ∈ {𝑋, 𝑌} ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ (∀𝑦 ∈ {𝑋, 𝑌} ((𝐹‘𝑋) = (𝐹‘𝑦) → 𝑋 = 𝑦) ∧ ∀𝑦 ∈ {𝑋, 𝑌} ((𝐹‘𝑌) = (𝐹‘𝑦) → 𝑌 = 𝑦))))
33	32	3adant3 1128	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (∀𝑥 ∈ {𝑋, 𝑌}∀𝑦 ∈ {𝑋, 𝑌} ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ (∀𝑦 ∈ {𝑋, 𝑌} ((𝐹‘𝑋) = (𝐹‘𝑦) → 𝑋 = 𝑦) ∧ ∀𝑦 ∈ {𝑋, 𝑌} ((𝐹‘𝑌) = (𝐹‘𝑦) → 𝑌 = 𝑦))))
34	33	adantr 483	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → (∀𝑥 ∈ {𝑋, 𝑌}∀𝑦 ∈ {𝑋, 𝑌} ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ (∀𝑦 ∈ {𝑋, 𝑌} ((𝐹‘𝑋) = (𝐹‘𝑦) → 𝑋 = 𝑦) ∧ ∀𝑦 ∈ {𝑋, 𝑌} ((𝐹‘𝑌) = (𝐹‘𝑦) → 𝑌 = 𝑦))))
35		fveq2 6665	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑋 → (𝐹‘𝑦) = (𝐹‘𝑋))
36	35	eqeq2d 2832	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑋 → ((𝐹‘𝑋) = (𝐹‘𝑦) ↔ (𝐹‘𝑋) = (𝐹‘𝑋)))
37		eqeq2 2833	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑋 → (𝑋 = 𝑦 ↔ 𝑋 = 𝑋))
38	36, 37	imbi12d 347	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑋 → (((𝐹‘𝑋) = (𝐹‘𝑦) → 𝑋 = 𝑦) ↔ ((𝐹‘𝑋) = (𝐹‘𝑋) → 𝑋 = 𝑋)))
39		fveq2 6665	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑌 → (𝐹‘𝑦) = (𝐹‘𝑌))
40	39	eqeq2d 2832	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑌 → ((𝐹‘𝑋) = (𝐹‘𝑦) ↔ (𝐹‘𝑋) = (𝐹‘𝑌)))
41		eqeq2 2833	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑌 → (𝑋 = 𝑦 ↔ 𝑋 = 𝑌))
42	40, 41	imbi12d 347	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑌 → (((𝐹‘𝑋) = (𝐹‘𝑦) → 𝑋 = 𝑦) ↔ ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))
43	38, 42	ralprg 4626	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉) → (∀𝑦 ∈ {𝑋, 𝑌} ((𝐹‘𝑋) = (𝐹‘𝑦) → 𝑋 = 𝑦) ↔ (((𝐹‘𝑋) = (𝐹‘𝑋) → 𝑋 = 𝑋) ∧ ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌))))
44	35	eqeq2d 2832	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑋 → ((𝐹‘𝑌) = (𝐹‘𝑦) ↔ (𝐹‘𝑌) = (𝐹‘𝑋)))
45		eqeq2 2833	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑋 → (𝑌 = 𝑦 ↔ 𝑌 = 𝑋))
46	44, 45	imbi12d 347	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑋 → (((𝐹‘𝑌) = (𝐹‘𝑦) → 𝑌 = 𝑦) ↔ ((𝐹‘𝑌) = (𝐹‘𝑋) → 𝑌 = 𝑋)))
47	39	eqeq2d 2832	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑌 → ((𝐹‘𝑌) = (𝐹‘𝑦) ↔ (𝐹‘𝑌) = (𝐹‘𝑌)))
48		eqeq2 2833	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑌 → (𝑌 = 𝑦 ↔ 𝑌 = 𝑌))
49	47, 48	imbi12d 347	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑌 → (((𝐹‘𝑌) = (𝐹‘𝑦) → 𝑌 = 𝑦) ↔ ((𝐹‘𝑌) = (𝐹‘𝑌) → 𝑌 = 𝑌)))
50	46, 49	ralprg 4626	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉) → (∀𝑦 ∈ {𝑋, 𝑌} ((𝐹‘𝑌) = (𝐹‘𝑦) → 𝑌 = 𝑦) ↔ (((𝐹‘𝑌) = (𝐹‘𝑋) → 𝑌 = 𝑋) ∧ ((𝐹‘𝑌) = (𝐹‘𝑌) → 𝑌 = 𝑌))))
51	43, 50	anbi12d 632	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉) → ((∀𝑦 ∈ {𝑋, 𝑌} ((𝐹‘𝑋) = (𝐹‘𝑦) → 𝑋 = 𝑦) ∧ ∀𝑦 ∈ {𝑋, 𝑌} ((𝐹‘𝑌) = (𝐹‘𝑦) → 𝑌 = 𝑦)) ↔ ((((𝐹‘𝑋) = (𝐹‘𝑋) → 𝑋 = 𝑋) ∧ ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)) ∧ (((𝐹‘𝑌) = (𝐹‘𝑋) → 𝑌 = 𝑋) ∧ ((𝐹‘𝑌) = (𝐹‘𝑌) → 𝑌 = 𝑌)))))
52	51	3adant3 1128	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ((∀𝑦 ∈ {𝑋, 𝑌} ((𝐹‘𝑋) = (𝐹‘𝑦) → 𝑋 = 𝑦) ∧ ∀𝑦 ∈ {𝑋, 𝑌} ((𝐹‘𝑌) = (𝐹‘𝑦) → 𝑌 = 𝑦)) ↔ ((((𝐹‘𝑋) = (𝐹‘𝑋) → 𝑋 = 𝑋) ∧ ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)) ∧ (((𝐹‘𝑌) = (𝐹‘𝑋) → 𝑌 = 𝑋) ∧ ((𝐹‘𝑌) = (𝐹‘𝑌) → 𝑌 = 𝑌)))))
53	52	adantr 483	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → ((∀𝑦 ∈ {𝑋, 𝑌} ((𝐹‘𝑋) = (𝐹‘𝑦) → 𝑋 = 𝑦) ∧ ∀𝑦 ∈ {𝑋, 𝑌} ((𝐹‘𝑌) = (𝐹‘𝑦) → 𝑌 = 𝑦)) ↔ ((((𝐹‘𝑋) = (𝐹‘𝑋) → 𝑋 = 𝑋) ∧ ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)) ∧ (((𝐹‘𝑌) = (𝐹‘𝑋) → 𝑌 = 𝑋) ∧ ((𝐹‘𝑌) = (𝐹‘𝑌) → 𝑌 = 𝑌)))))
54	12	fveq1i 6666	. . . . . . . . . . . . . 14 ⊢ (𝐹‘𝑋) = ({⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}‘𝑋)
55		3simpb 1145	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑋 ∈ 𝑈 ∧ 𝑍 ∈ 𝑊))
56	55	anim1i 616	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → ((𝑋 ∈ 𝑈 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌))
57		df-3an 1085	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑍 ∈ 𝑊 ∧ 𝑋 ≠ 𝑌) ↔ ((𝑋 ∈ 𝑈 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌))
58	56, 57	sylibr 236	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → (𝑋 ∈ 𝑈 ∧ 𝑍 ∈ 𝑊 ∧ 𝑋 ≠ 𝑌))
59		fvpr1g 6949	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑍 ∈ 𝑊 ∧ 𝑋 ≠ 𝑌) → ({⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}‘𝑋) = 𝑍)
60	58, 59	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → ({⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}‘𝑋) = 𝑍)
61	54, 60	syl5eq 2868	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → (𝐹‘𝑋) = 𝑍)
62	12	fveq1i 6666	. . . . . . . . . . . . . 14 ⊢ (𝐹‘𝑌) = ({⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}‘𝑌)
63		3simpc 1146	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊))
64	63	anim1i 616	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → ((𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌))
65		df-3an 1085	. . . . . . . . . . . . . . . 16 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ∧ 𝑋 ≠ 𝑌) ↔ ((𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌))
66	64, 65	sylibr 236	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → (𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ∧ 𝑋 ≠ 𝑌))
67		fvpr2g 6950	. . . . . . . . . . . . . . 15 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ∧ 𝑋 ≠ 𝑌) → ({⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}‘𝑌) = 𝑍)
68	66, 67	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → ({⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}‘𝑌) = 𝑍)
69	62, 68	syl5req 2869	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → 𝑍 = (𝐹‘𝑌))
70	61, 69	eqtrd 2856	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → (𝐹‘𝑋) = (𝐹‘𝑌))
71		idd 24	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → (𝑋 = 𝑌 → 𝑋 = 𝑌))
72	70, 71	embantd 59	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → (((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌) → 𝑋 = 𝑌))
73	72	adantld 493	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → ((((𝐹‘𝑋) = (𝐹‘𝑋) → 𝑋 = 𝑋) ∧ ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)) → 𝑋 = 𝑌))
74	73	adantrd 494	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → (((((𝐹‘𝑋) = (𝐹‘𝑋) → 𝑋 = 𝑋) ∧ ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)) ∧ (((𝐹‘𝑌) = (𝐹‘𝑋) → 𝑌 = 𝑋) ∧ ((𝐹‘𝑌) = (𝐹‘𝑌) → 𝑌 = 𝑌))) → 𝑋 = 𝑌))
75	53, 74	sylbid 242	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → ((∀𝑦 ∈ {𝑋, 𝑌} ((𝐹‘𝑋) = (𝐹‘𝑦) → 𝑋 = 𝑦) ∧ ∀𝑦 ∈ {𝑋, 𝑌} ((𝐹‘𝑌) = (𝐹‘𝑦) → 𝑌 = 𝑦)) → 𝑋 = 𝑌))
76	34, 75	sylbid 242	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → (∀𝑥 ∈ {𝑋, 𝑌}∀𝑦 ∈ {𝑋, 𝑌} ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) → 𝑋 = 𝑌))
77	76	adantld 493	. . . . . 6 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → ((𝐹:{𝑋, 𝑌}⟶{𝑍, 𝑍} ∧ ∀𝑥 ∈ {𝑋, 𝑌}∀𝑦 ∈ {𝑋, 𝑌} ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)) → 𝑋 = 𝑌))
78	23, 77	syl5bi 244	. . . . 5 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → (𝐹:{𝑋, 𝑌}–1-1→{𝑍, 𝑍} → 𝑋 = 𝑌))
79	22, 78	syl5bir 245	. . . 4 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → ((𝐹:{𝑋, 𝑌}⟶{𝑍, 𝑍} ∧ Fun ◡𝐹) → 𝑋 = 𝑌))
80	21, 79	mpand 693	. . 3 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → (Fun ◡𝐹 → 𝑋 = 𝑌))
81	16, 80	mtod 200	. 2 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → ¬ Fun ◡𝐹)
82	14, 81	jca 514	1 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → (Fun 𝐹 ∧ ¬ Fun ◡𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ∀wral 3138  {cpr 4563  ⟨cop 4567  ^◡ccnv 5549  Fun wfun 6344  ⟶wf 6346  –1-1→wf1 6347  ‘cfv 6350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fv 6358

This theorem is referenced by:  ntrl2v2e  27931