Theorem fpropnf1 6524

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 1081	. . . . . 6 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉))
3	2	adantr 481	. . . . 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 554	. . . . . . 7 ⊢ (𝑍 ∈ 𝑊 → (𝑍 ∈ 𝑊 ∧ 𝑍 ∈ 𝑊))
6	5	3ad2ant3 1084	. . . . . 6 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑍 ∈ 𝑊 ∧ 𝑍 ∈ 𝑊))
7	6	adantr 481	. . . . 5 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → (𝑍 ∈ 𝑊 ∧ 𝑍 ∈ 𝑊))
8		simpr 477	. . . . 5 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → 𝑋 ≠ 𝑌)
9	3, 7, 8	3jca 1242	. . . 4 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉) ∧ (𝑍 ∈ 𝑊 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌))
10		funprg 5940	. . . 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 5909	. . 3 ⊢ (Fun 𝐹 ↔ Fun {⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩})
14	11, 13	sylibr 224	. 2 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → Fun 𝐹)
15		neneq 2800	. . . 4 ⊢ (𝑋 ≠ 𝑌 → ¬ 𝑋 = 𝑌)
16	15	adantl 482	. . 3 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → ¬ 𝑋 = 𝑌)
17		fprg 6422	. . . . . 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 2631	. . . . . 6 ⊢ {⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩} = 𝐹
20	19	feq1i 6036	. . . . 5 ⊢ ({⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}:{𝑋, 𝑌}⟶{𝑍, 𝑍} ↔ 𝐹:{𝑋, 𝑌}⟶{𝑍, 𝑍})
21	18, 20	sylib 208	. . . 4 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → 𝐹:{𝑋, 𝑌}⟶{𝑍, 𝑍})
22		df-f1 5893	. . . . 5 ⊢ (𝐹:{𝑋, 𝑌}–1-1→{𝑍, 𝑍} ↔ (𝐹:{𝑋, 𝑌}⟶{𝑍, 𝑍} ∧ Fun ◡𝐹))
23		dff13 6512	. . . . . 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		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
25	24	eqeq1d 2624	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝐹‘𝑋) = (𝐹‘𝑦)))
26		eqeq1 2626	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑋 → (𝑥 = 𝑦 ↔ 𝑋 = 𝑦))
27	25, 26	imbi12d 334	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ ((𝐹‘𝑋) = (𝐹‘𝑦) → 𝑋 = 𝑦)))
28	27	ralbidv 2986	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (∀𝑦 ∈ {𝑋, 𝑌} ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ {𝑋, 𝑌} ((𝐹‘𝑋) = (𝐹‘𝑦) → 𝑋 = 𝑦)))
29		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑌 → (𝐹‘𝑥) = (𝐹‘𝑌))
30	29	eqeq1d 2624	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑌 → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝐹‘𝑌) = (𝐹‘𝑦)))
31		eqeq1 2626	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑌 → (𝑥 = 𝑦 ↔ 𝑌 = 𝑦))
32	30, 31	imbi12d 334	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑌 → (((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ ((𝐹‘𝑌) = (𝐹‘𝑦) → 𝑌 = 𝑦)))
33	32	ralbidv 2986	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑌 → (∀𝑦 ∈ {𝑋, 𝑌} ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ {𝑋, 𝑌} ((𝐹‘𝑌) = (𝐹‘𝑦) → 𝑌 = 𝑦)))
34	28, 33	ralprg 4234	. . . . . . . . . 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))))
35	34	3adant3 1081	. . . . . . . . 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))))
36	35	adantr 481	. . . . . . . 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))))
37		fveq2 6191	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑋 → (𝐹‘𝑦) = (𝐹‘𝑋))
38	37	eqeq2d 2632	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑋 → ((𝐹‘𝑋) = (𝐹‘𝑦) ↔ (𝐹‘𝑋) = (𝐹‘𝑋)))
39		eqeq2 2633	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑋 → (𝑋 = 𝑦 ↔ 𝑋 = 𝑋))
40	38, 39	imbi12d 334	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑋 → (((𝐹‘𝑋) = (𝐹‘𝑦) → 𝑋 = 𝑦) ↔ ((𝐹‘𝑋) = (𝐹‘𝑋) → 𝑋 = 𝑋)))
41		fveq2 6191	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑌 → (𝐹‘𝑦) = (𝐹‘𝑌))
42	41	eqeq2d 2632	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑌 → ((𝐹‘𝑋) = (𝐹‘𝑦) ↔ (𝐹‘𝑋) = (𝐹‘𝑌)))
43		eqeq2 2633	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑌 → (𝑋 = 𝑦 ↔ 𝑋 = 𝑌))
44	42, 43	imbi12d 334	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑌 → (((𝐹‘𝑋) = (𝐹‘𝑦) → 𝑋 = 𝑦) ↔ ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))
45	40, 44	ralprg 4234	. . . . . . . . . . . 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))))
46	37	eqeq2d 2632	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑋 → ((𝐹‘𝑌) = (𝐹‘𝑦) ↔ (𝐹‘𝑌) = (𝐹‘𝑋)))
47		eqeq2 2633	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑋 → (𝑌 = 𝑦 ↔ 𝑌 = 𝑋))
48	46, 47	imbi12d 334	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑋 → (((𝐹‘𝑌) = (𝐹‘𝑦) → 𝑌 = 𝑦) ↔ ((𝐹‘𝑌) = (𝐹‘𝑋) → 𝑌 = 𝑋)))
49	41	eqeq2d 2632	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑌 → ((𝐹‘𝑌) = (𝐹‘𝑦) ↔ (𝐹‘𝑌) = (𝐹‘𝑌)))
50		eqeq2 2633	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑌 → (𝑌 = 𝑦 ↔ 𝑌 = 𝑌))
51	49, 50	imbi12d 334	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑌 → (((𝐹‘𝑌) = (𝐹‘𝑦) → 𝑌 = 𝑦) ↔ ((𝐹‘𝑌) = (𝐹‘𝑌) → 𝑌 = 𝑌)))
52	48, 51	ralprg 4234	. . . . . . . . . . . 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))))
53	45, 52	anbi12d 747	. . . . . . . . . . 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)))))
54	53	3adant3 1081	. . . . . . . . . 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)))))
55	54	adantr 481	. . . . . . . . 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)))))
56	12	fveq1i 6192	. . . . . . . . . . . . . 14 ⊢ (𝐹‘𝑋) = ({⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}‘𝑋)
57		3simpb 1059	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑋 ∈ 𝑈 ∧ 𝑍 ∈ 𝑊))
58	57	anim1i 592	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → ((𝑋 ∈ 𝑈 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌))
59		df-3an 1039	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑍 ∈ 𝑊 ∧ 𝑋 ≠ 𝑌) ↔ ((𝑋 ∈ 𝑈 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌))
60	58, 59	sylibr 224	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → (𝑋 ∈ 𝑈 ∧ 𝑍 ∈ 𝑊 ∧ 𝑋 ≠ 𝑌))
61		fvpr1g 6458	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑍 ∈ 𝑊 ∧ 𝑋 ≠ 𝑌) → ({⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}‘𝑋) = 𝑍)
62	60, 61	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → ({⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}‘𝑋) = 𝑍)
63	56, 62	syl5eq 2668	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → (𝐹‘𝑋) = 𝑍)
64	12	fveq1i 6192	. . . . . . . . . . . . . 14 ⊢ (𝐹‘𝑌) = ({⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}‘𝑌)
65		3simpc 1060	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊))
66	65	anim1i 592	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → ((𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌))
67		df-3an 1039	. . . . . . . . . . . . . . . 16 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ∧ 𝑋 ≠ 𝑌) ↔ ((𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌))
68	66, 67	sylibr 224	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → (𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ∧ 𝑋 ≠ 𝑌))
69		fvpr2g 6459	. . . . . . . . . . . . . . 15 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ∧ 𝑋 ≠ 𝑌) → ({⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}‘𝑌) = 𝑍)
70	68, 69	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → ({⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}‘𝑌) = 𝑍)
71	64, 70	syl5req 2669	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → 𝑍 = (𝐹‘𝑌))
72	63, 71	eqtrd 2656	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → (𝐹‘𝑋) = (𝐹‘𝑌))
73		idd 24	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → (𝑋 = 𝑌 → 𝑋 = 𝑌))
74	72, 73	embantd 59	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → (((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌) → 𝑋 = 𝑌))
75	74	adantld 483	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → ((((𝐹‘𝑋) = (𝐹‘𝑋) → 𝑋 = 𝑋) ∧ ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)) → 𝑋 = 𝑌))
76	75	adantrd 484	. . . . . . . . 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))
77	55, 76	sylbid 230	. . . . . . . 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))
78	36, 77	sylbid 230	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → (∀𝑥 ∈ {𝑋, 𝑌}∀𝑦 ∈ {𝑋, 𝑌} ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) → 𝑋 = 𝑌))
79	78	adantld 483	. . . . . 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))
80	23, 79	syl5bi 232	. . . . 5 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → (𝐹:{𝑋, 𝑌}–1-1→{𝑍, 𝑍} → 𝑋 = 𝑌))
81	22, 80	syl5bir 233	. . . 4 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → ((𝐹:{𝑋, 𝑌}⟶{𝑍, 𝑍} ∧ Fun ◡𝐹) → 𝑋 = 𝑌))
82	21, 81	mpand 711	. . 3 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → (Fun ◡𝐹 → 𝑋 = 𝑌))
83	16, 82	mtod 189	. 2 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → ¬ Fun ◡𝐹)
84	14, 83	jca 554	1 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → (Fun 𝐹 ∧ ¬ Fun ◡𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  {cpr 4179  ⟨cop 4183  ^◡ccnv 5113  Fun wfun 5882  ⟶wf 5884  –1-1→wf1 5885  ‘cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fv 5896

This theorem is referenced by:  ntrl2v2e  27018