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Theorem resf1extb 7939

Description: Extension of an injection which is a restriction of a function. (Contributed by AV, 3-Oct-2025.)

**Assertion**
Ref	Expression
resf1extb	⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → (((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 ∧ (𝐹‘𝑋) ∉ (𝐹 “ 𝐶)) ↔ (𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})–1-1→𝐵))

Proof of Theorem resf1extb Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1136	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → 𝐹:𝐴⟶𝐵)
2		simp3 1138	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → 𝐶 ⊆ 𝐴)
3		eldifi 4113	. . . . . . . 8 ⊢ (𝑋 ∈ (𝐴 ∖ 𝐶) → 𝑋 ∈ 𝐴)
4	3	snssd 4791	. . . . . . 7 ⊢ (𝑋 ∈ (𝐴 ∖ 𝐶) → {𝑋} ⊆ 𝐴)
5	4	3ad2ant2 1134	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → {𝑋} ⊆ 𝐴)
6	2, 5	unssd 4174	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → (𝐶 ∪ {𝑋}) ⊆ 𝐴)
7	1, 6	fssresd 6756	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})⟶𝐵)
8	7	adantr 480	. . 3 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ ((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 ∧ (𝐹‘𝑋) ∉ (𝐹 “ 𝐶))) → (𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})⟶𝐵)
9		elun 4135	. . . . . 6 ⊢ (𝑦 ∈ (𝐶 ∪ {𝑋}) ↔ (𝑦 ∈ 𝐶 ∨ 𝑦 ∈ {𝑋}))
10		elun 4135	. . . . . 6 ⊢ (𝑧 ∈ (𝐶 ∪ {𝑋}) ↔ (𝑧 ∈ 𝐶 ∨ 𝑧 ∈ {𝑋}))
11	9, 10	anbi12i 628	. . . . 5 ⊢ ((𝑦 ∈ (𝐶 ∪ {𝑋}) ∧ 𝑧 ∈ (𝐶 ∪ {𝑋})) ↔ ((𝑦 ∈ 𝐶 ∨ 𝑦 ∈ {𝑋}) ∧ (𝑧 ∈ 𝐶 ∨ 𝑧 ∈ {𝑋})))
12		dff14a 7273	. . . . . . . . 9 ⊢ ((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 ↔ ((𝐹 ↾ 𝐶):𝐶⟶𝐵 ∧ ∀𝑤 ∈ 𝐶 ∀𝑥 ∈ 𝐶 (𝑤 ≠ 𝑥 → ((𝐹 ↾ 𝐶)‘𝑤) ≠ ((𝐹 ↾ 𝐶)‘𝑥))))
13		neeq1 2993	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑦 → (𝑤 ≠ 𝑥 ↔ 𝑦 ≠ 𝑥))
14		fveq2 6887	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑦 → ((𝐹 ↾ 𝐶)‘𝑤) = ((𝐹 ↾ 𝐶)‘𝑦))
15	14	neeq1d 2990	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑦 → (((𝐹 ↾ 𝐶)‘𝑤) ≠ ((𝐹 ↾ 𝐶)‘𝑥) ↔ ((𝐹 ↾ 𝐶)‘𝑦) ≠ ((𝐹 ↾ 𝐶)‘𝑥)))
16	13, 15	imbi12d 344	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑦 → ((𝑤 ≠ 𝑥 → ((𝐹 ↾ 𝐶)‘𝑤) ≠ ((𝐹 ↾ 𝐶)‘𝑥)) ↔ (𝑦 ≠ 𝑥 → ((𝐹 ↾ 𝐶)‘𝑦) ≠ ((𝐹 ↾ 𝐶)‘𝑥))))
17		neeq2 2994	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝑦 ≠ 𝑥 ↔ 𝑦 ≠ 𝑧))
18		fveq2 6887	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → ((𝐹 ↾ 𝐶)‘𝑥) = ((𝐹 ↾ 𝐶)‘𝑧))
19	18	neeq2d 2991	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (((𝐹 ↾ 𝐶)‘𝑦) ≠ ((𝐹 ↾ 𝐶)‘𝑥) ↔ ((𝐹 ↾ 𝐶)‘𝑦) ≠ ((𝐹 ↾ 𝐶)‘𝑧)))
20	17, 19	imbi12d 344	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → ((𝑦 ≠ 𝑥 → ((𝐹 ↾ 𝐶)‘𝑦) ≠ ((𝐹 ↾ 𝐶)‘𝑥)) ↔ (𝑦 ≠ 𝑧 → ((𝐹 ↾ 𝐶)‘𝑦) ≠ ((𝐹 ↾ 𝐶)‘𝑧))))
21	16, 20	rspc2v 3617	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶) → (∀𝑤 ∈ 𝐶 ∀𝑥 ∈ 𝐶 (𝑤 ≠ 𝑥 → ((𝐹 ↾ 𝐶)‘𝑤) ≠ ((𝐹 ↾ 𝐶)‘𝑥)) → (𝑦 ≠ 𝑧 → ((𝐹 ↾ 𝐶)‘𝑦) ≠ ((𝐹 ↾ 𝐶)‘𝑧))))
22		simpl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶) → 𝑦 ∈ 𝐶)
23	22	fvresd 6907	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶) → ((𝐹 ↾ 𝐶)‘𝑦) = (𝐹‘𝑦))
24		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ 𝐶)
25	24	fvresd 6907	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶) → ((𝐹 ↾ 𝐶)‘𝑧) = (𝐹‘𝑧))
26	23, 25	neeq12d 2992	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶) → (((𝐹 ↾ 𝐶)‘𝑦) ≠ ((𝐹 ↾ 𝐶)‘𝑧) ↔ (𝐹‘𝑦) ≠ (𝐹‘𝑧)))
27	26	imbi2d 340	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶) → ((𝑦 ≠ 𝑧 → ((𝐹 ↾ 𝐶)‘𝑦) ≠ ((𝐹 ↾ 𝐶)‘𝑧)) ↔ (𝑦 ≠ 𝑧 → (𝐹‘𝑦) ≠ (𝐹‘𝑧))))
28	27	bi23imp13 1115	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶) ∧ (𝑦 ≠ 𝑧 → ((𝐹 ↾ 𝐶)‘𝑦) ≠ ((𝐹 ↾ 𝐶)‘𝑧)) ∧ 𝑦 ≠ 𝑧) → (𝐹‘𝑦) ≠ (𝐹‘𝑧))
29		elun1 4164	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ 𝐶 → 𝑦 ∈ (𝐶 ∪ {𝑋}))
30	29	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶) → 𝑦 ∈ (𝐶 ∪ {𝑋}))
31	30	fvresd 6907	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶) → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) = (𝐹‘𝑦))
32		elun1 4164	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ 𝐶 → 𝑧 ∈ (𝐶 ∪ {𝑋}))
33	32	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ (𝐶 ∪ {𝑋}))
34	33	fvresd 6907	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶) → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧) = (𝐹‘𝑧))
35	31, 34	neeq12d 2992	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶) → (((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧) ↔ (𝐹‘𝑦) ≠ (𝐹‘𝑧)))
36	35	3ad2ant1 1133	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶) ∧ (𝑦 ≠ 𝑧 → ((𝐹 ↾ 𝐶)‘𝑦) ≠ ((𝐹 ↾ 𝐶)‘𝑧)) ∧ 𝑦 ≠ 𝑧) → (((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧) ↔ (𝐹‘𝑦) ≠ (𝐹‘𝑧)))
37	28, 36	mpbird 257	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶) ∧ (𝑦 ≠ 𝑧 → ((𝐹 ↾ 𝐶)‘𝑦) ≠ ((𝐹 ↾ 𝐶)‘𝑧)) ∧ 𝑦 ≠ 𝑧) → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧))
38	37	3exp 1119	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶) → ((𝑦 ≠ 𝑧 → ((𝐹 ↾ 𝐶)‘𝑦) ≠ ((𝐹 ↾ 𝐶)‘𝑧)) → (𝑦 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧))))
39	21, 38	syldc 48	. . . . . . . . . . 11 ⊢ (∀𝑤 ∈ 𝐶 ∀𝑥 ∈ 𝐶 (𝑤 ≠ 𝑥 → ((𝐹 ↾ 𝐶)‘𝑤) ≠ ((𝐹 ↾ 𝐶)‘𝑥)) → ((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶) → (𝑦 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧))))
40	39	adantl 481	. . . . . . . . . 10 ⊢ (((𝐹 ↾ 𝐶):𝐶⟶𝐵 ∧ ∀𝑤 ∈ 𝐶 ∀𝑥 ∈ 𝐶 (𝑤 ≠ 𝑥 → ((𝐹 ↾ 𝐶)‘𝑤) ≠ ((𝐹 ↾ 𝐶)‘𝑥))) → ((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶) → (𝑦 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧))))
41	40	a1i 11	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → (((𝐹 ↾ 𝐶):𝐶⟶𝐵 ∧ ∀𝑤 ∈ 𝐶 ∀𝑥 ∈ 𝐶 (𝑤 ≠ 𝑥 → ((𝐹 ↾ 𝐶)‘𝑤) ≠ ((𝐹 ↾ 𝐶)‘𝑥))) → ((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶) → (𝑦 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧)))))
42	12, 41	biimtrid 242	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → ((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 → ((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶) → (𝑦 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧)))))
43	42	a1dd 50	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → ((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 → ((𝐹‘𝑋) ∉ (𝐹 “ 𝐶) → ((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶) → (𝑦 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧))))))
44	43	imp32 418	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ ((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 ∧ (𝐹‘𝑋) ∉ (𝐹 “ 𝐶))) → ((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶) → (𝑦 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧))))
45		ffn 6717	. . . . . . . . . . . . 13 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
46	45	3ad2ant1 1133	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → 𝐹 Fn 𝐴)
47	46, 2	fvelimabd 6963	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → ((𝐹‘𝑋) ∈ (𝐹 “ 𝐶) ↔ ∃𝑥 ∈ 𝐶 (𝐹‘𝑥) = (𝐹‘𝑋)))
48	47	notbid 318	. . . . . . . . . 10 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → (¬ (𝐹‘𝑋) ∈ (𝐹 “ 𝐶) ↔ ¬ ∃𝑥 ∈ 𝐶 (𝐹‘𝑥) = (𝐹‘𝑋)))
49		df-nel 3036	. . . . . . . . . 10 ⊢ ((𝐹‘𝑋) ∉ (𝐹 “ 𝐶) ↔ ¬ (𝐹‘𝑋) ∈ (𝐹 “ 𝐶))
50		ralnex 3061	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐶 ¬ (𝐹‘𝑥) = (𝐹‘𝑋) ↔ ¬ ∃𝑥 ∈ 𝐶 (𝐹‘𝑥) = (𝐹‘𝑋))
51	48, 49, 50	3bitr4g 314	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → ((𝐹‘𝑋) ∉ (𝐹 “ 𝐶) ↔ ∀𝑥 ∈ 𝐶 ¬ (𝐹‘𝑥) = (𝐹‘𝑋)))
52		df-ne 2932	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑥) ≠ (𝐹‘𝑋) ↔ ¬ (𝐹‘𝑥) = (𝐹‘𝑋))
53		fveq2 6887	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
54	53	neeq1d 2990	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥) ≠ (𝐹‘𝑋) ↔ (𝐹‘𝑧) ≠ (𝐹‘𝑋)))
55	52, 54	bitr3id 285	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → (¬ (𝐹‘𝑥) = (𝐹‘𝑋) ↔ (𝐹‘𝑧) ≠ (𝐹‘𝑋)))
56	55	rspcv 3602	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝐶 → (∀𝑥 ∈ 𝐶 ¬ (𝐹‘𝑥) = (𝐹‘𝑋) → (𝐹‘𝑧) ≠ (𝐹‘𝑋)))
57	56	ad2antll 729	. . . . . . . . . . . . 13 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝑦 ∈ {𝑋} ∧ 𝑧 ∈ 𝐶)) → (∀𝑥 ∈ 𝐶 ¬ (𝐹‘𝑥) = (𝐹‘𝑋) → (𝐹‘𝑧) ≠ (𝐹‘𝑋)))
58	32	ad2antll 729	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝑦 ∈ {𝑋} ∧ 𝑧 ∈ 𝐶)) → 𝑧 ∈ (𝐶 ∪ {𝑋}))
59	58	fvresd 6907	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝑦 ∈ {𝑋} ∧ 𝑧 ∈ 𝐶)) → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧) = (𝐹‘𝑧))
60	59	eqcomd 2740	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝑦 ∈ {𝑋} ∧ 𝑧 ∈ 𝐶)) → (𝐹‘𝑧) = ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧))
61		elsni 4625	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ {𝑋} → 𝑦 = 𝑋)
62	61	eqcomd 2740	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ {𝑋} → 𝑋 = 𝑦)
63	62	ad2antrl 728	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝑦 ∈ {𝑋} ∧ 𝑧 ∈ 𝐶)) → 𝑋 = 𝑦)
64	63	fveq2d 6891	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝑦 ∈ {𝑋} ∧ 𝑧 ∈ 𝐶)) → (𝐹‘𝑋) = (𝐹‘𝑦))
65		elun2 4165	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ {𝑋} → 𝑦 ∈ (𝐶 ∪ {𝑋}))
66	65	ad2antrl 728	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝑦 ∈ {𝑋} ∧ 𝑧 ∈ 𝐶)) → 𝑦 ∈ (𝐶 ∪ {𝑋}))
67	66	fvresd 6907	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝑦 ∈ {𝑋} ∧ 𝑧 ∈ 𝐶)) → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) = (𝐹‘𝑦))
68	64, 67	eqtr4d 2772	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝑦 ∈ {𝑋} ∧ 𝑧 ∈ 𝐶)) → (𝐹‘𝑋) = ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦))
69	60, 68	neeq12d 2992	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝑦 ∈ {𝑋} ∧ 𝑧 ∈ 𝐶)) → ((𝐹‘𝑧) ≠ (𝐹‘𝑋) ↔ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦)))
70	69	biimpa 476	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝑦 ∈ {𝑋} ∧ 𝑧 ∈ 𝐶)) ∧ (𝐹‘𝑧) ≠ (𝐹‘𝑋)) → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦))
71	70	necomd 2986	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝑦 ∈ {𝑋} ∧ 𝑧 ∈ 𝐶)) ∧ (𝐹‘𝑧) ≠ (𝐹‘𝑋)) → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧))
72	71	a1d 25	. . . . . . . . . . . . . 14 ⊢ ((((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝑦 ∈ {𝑋} ∧ 𝑧 ∈ 𝐶)) ∧ (𝐹‘𝑧) ≠ (𝐹‘𝑋)) → (𝑦 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧)))
73	72	ex 412	. . . . . . . . . . . . 13 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝑦 ∈ {𝑋} ∧ 𝑧 ∈ 𝐶)) → ((𝐹‘𝑧) ≠ (𝐹‘𝑋) → (𝑦 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧))))
74	57, 73	syld 47	. . . . . . . . . . . 12 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝑦 ∈ {𝑋} ∧ 𝑧 ∈ 𝐶)) → (∀𝑥 ∈ 𝐶 ¬ (𝐹‘𝑥) = (𝐹‘𝑋) → (𝑦 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧))))
75	74	a1d 25	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝑦 ∈ {𝑋} ∧ 𝑧 ∈ 𝐶)) → ((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 → (∀𝑥 ∈ 𝐶 ¬ (𝐹‘𝑥) = (𝐹‘𝑋) → (𝑦 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧)))))
76	75	ex 412	. . . . . . . . . 10 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → ((𝑦 ∈ {𝑋} ∧ 𝑧 ∈ 𝐶) → ((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 → (∀𝑥 ∈ 𝐶 ¬ (𝐹‘𝑥) = (𝐹‘𝑋) → (𝑦 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧))))))
77	76	com24 95	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → (∀𝑥 ∈ 𝐶 ¬ (𝐹‘𝑥) = (𝐹‘𝑋) → ((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 → ((𝑦 ∈ {𝑋} ∧ 𝑧 ∈ 𝐶) → (𝑦 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧))))))
78	51, 77	sylbid 240	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → ((𝐹‘𝑋) ∉ (𝐹 “ 𝐶) → ((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 → ((𝑦 ∈ {𝑋} ∧ 𝑧 ∈ 𝐶) → (𝑦 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧))))))
79	78	impcomd 411	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → (((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 ∧ (𝐹‘𝑋) ∉ (𝐹 “ 𝐶)) → ((𝑦 ∈ {𝑋} ∧ 𝑧 ∈ 𝐶) → (𝑦 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧)))))
80	79	imp 406	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ ((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 ∧ (𝐹‘𝑋) ∉ (𝐹 “ 𝐶))) → ((𝑦 ∈ {𝑋} ∧ 𝑧 ∈ 𝐶) → (𝑦 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧))))
81		fveq2 6887	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
82	81	neeq1d 2990	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) ≠ (𝐹‘𝑋) ↔ (𝐹‘𝑦) ≠ (𝐹‘𝑋)))
83	52, 82	bitr3id 285	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (¬ (𝐹‘𝑥) = (𝐹‘𝑋) ↔ (𝐹‘𝑦) ≠ (𝐹‘𝑋)))
84	83	rspcv 3602	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝐶 → (∀𝑥 ∈ 𝐶 ¬ (𝐹‘𝑥) = (𝐹‘𝑋) → (𝐹‘𝑦) ≠ (𝐹‘𝑋)))
85	84	ad2antrl 728	. . . . . . . . . . . . 13 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ {𝑋})) → (∀𝑥 ∈ 𝐶 ¬ (𝐹‘𝑥) = (𝐹‘𝑋) → (𝐹‘𝑦) ≠ (𝐹‘𝑋)))
86	29	ad2antrl 728	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ {𝑋})) → 𝑦 ∈ (𝐶 ∪ {𝑋}))
87	86	fvresd 6907	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ {𝑋})) → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) = (𝐹‘𝑦))
88	87	eqcomd 2740	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ {𝑋})) → (𝐹‘𝑦) = ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦))
89		elsni 4625	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ {𝑋} → 𝑧 = 𝑋)
90	89	eqcomd 2740	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ {𝑋} → 𝑋 = 𝑧)
91	90	ad2antll 729	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ {𝑋})) → 𝑋 = 𝑧)
92	91	fveq2d 6891	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ {𝑋})) → (𝐹‘𝑋) = (𝐹‘𝑧))
93		elun2 4165	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ {𝑋} → 𝑧 ∈ (𝐶 ∪ {𝑋}))
94	93	ad2antll 729	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ {𝑋})) → 𝑧 ∈ (𝐶 ∪ {𝑋}))
95	94	fvresd 6907	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ {𝑋})) → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧) = (𝐹‘𝑧))
96	92, 95	eqtr4d 2772	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ {𝑋})) → (𝐹‘𝑋) = ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧))
97	88, 96	neeq12d 2992	. . . . . . . . . . . . . . 15 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ {𝑋})) → ((𝐹‘𝑦) ≠ (𝐹‘𝑋) ↔ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧)))
98	97	biimpd 229	. . . . . . . . . . . . . 14 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ {𝑋})) → ((𝐹‘𝑦) ≠ (𝐹‘𝑋) → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧)))
99	98	a1dd 50	. . . . . . . . . . . . 13 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ {𝑋})) → ((𝐹‘𝑦) ≠ (𝐹‘𝑋) → (𝑦 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧))))
100	85, 99	syld 47	. . . . . . . . . . . 12 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ {𝑋})) → (∀𝑥 ∈ 𝐶 ¬ (𝐹‘𝑥) = (𝐹‘𝑋) → (𝑦 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧))))
101	100	a1d 25	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ {𝑋})) → ((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 → (∀𝑥 ∈ 𝐶 ¬ (𝐹‘𝑥) = (𝐹‘𝑋) → (𝑦 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧)))))
102	101	ex 412	. . . . . . . . . 10 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → ((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ {𝑋}) → ((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 → (∀𝑥 ∈ 𝐶 ¬ (𝐹‘𝑥) = (𝐹‘𝑋) → (𝑦 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧))))))
103	102	com24 95	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → (∀𝑥 ∈ 𝐶 ¬ (𝐹‘𝑥) = (𝐹‘𝑋) → ((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 → ((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ {𝑋}) → (𝑦 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧))))))
104	51, 103	sylbid 240	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → ((𝐹‘𝑋) ∉ (𝐹 “ 𝐶) → ((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 → ((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ {𝑋}) → (𝑦 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧))))))
105	104	impcomd 411	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → (((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 ∧ (𝐹‘𝑋) ∉ (𝐹 “ 𝐶)) → ((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ {𝑋}) → (𝑦 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧)))))
106	105	imp 406	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ ((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 ∧ (𝐹‘𝑋) ∉ (𝐹 “ 𝐶))) → ((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ {𝑋}) → (𝑦 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧))))
107		velsn 4624	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑋} ↔ 𝑦 = 𝑋)
108		velsn 4624	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑋} ↔ 𝑧 = 𝑋)
109		eqtr3 2756	. . . . . . . . 9 ⊢ ((𝑦 = 𝑋 ∧ 𝑧 = 𝑋) → 𝑦 = 𝑧)
110		eqneqall 2942	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝑦 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧)))
111	109, 110	syl 17	. . . . . . . 8 ⊢ ((𝑦 = 𝑋 ∧ 𝑧 = 𝑋) → (𝑦 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧)))
112	107, 108, 111	syl2anb 598	. . . . . . 7 ⊢ ((𝑦 ∈ {𝑋} ∧ 𝑧 ∈ {𝑋}) → (𝑦 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧)))
113	112	a1i 11	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ ((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 ∧ (𝐹‘𝑋) ∉ (𝐹 “ 𝐶))) → ((𝑦 ∈ {𝑋} ∧ 𝑧 ∈ {𝑋}) → (𝑦 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧))))
114	44, 80, 106, 113	ccased 1038	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ ((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 ∧ (𝐹‘𝑋) ∉ (𝐹 “ 𝐶))) → (((𝑦 ∈ 𝐶 ∨ 𝑦 ∈ {𝑋}) ∧ (𝑧 ∈ 𝐶 ∨ 𝑧 ∈ {𝑋})) → (𝑦 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧))))
115	11, 114	biimtrid 242	. . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ ((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 ∧ (𝐹‘𝑋) ∉ (𝐹 “ 𝐶))) → ((𝑦 ∈ (𝐶 ∪ {𝑋}) ∧ 𝑧 ∈ (𝐶 ∪ {𝑋})) → (𝑦 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧))))
116	115	ralrimivv 3187	. . 3 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ ((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 ∧ (𝐹‘𝑋) ∉ (𝐹 “ 𝐶))) → ∀𝑦 ∈ (𝐶 ∪ {𝑋})∀𝑧 ∈ (𝐶 ∪ {𝑋})(𝑦 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧)))
117		dff14a 7273	. . 3 ⊢ ((𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})–1-1→𝐵 ↔ ((𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})⟶𝐵 ∧ ∀𝑦 ∈ (𝐶 ∪ {𝑋})∀𝑧 ∈ (𝐶 ∪ {𝑋})(𝑦 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧))))
118	8, 116, 117	sylanbrc 583	. 2 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ ((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 ∧ (𝐹‘𝑋) ∉ (𝐹 “ 𝐶))) → (𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})–1-1→𝐵)
119		fssres 6755	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
120	119	3adant2 1131	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
121	120	adantr 480	. . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})–1-1→𝐵) → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
122		df-f1 6547	. . . . . . 7 ⊢ ((𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})–1-1→𝐵 ↔ ((𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})⟶𝐵 ∧ Fun ^◡(𝐹 ↾ (𝐶 ∪ {𝑋}))))
123		funres11 6624	. . . . . . 7 ⊢ (Fun ^◡(𝐹 ↾ (𝐶 ∪ {𝑋})) → Fun ^◡((𝐹 ↾ (𝐶 ∪ {𝑋})) ↾ 𝐶))
124	122, 123	simplbiim 504	. . . . . 6 ⊢ ((𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})–1-1→𝐵 → Fun ^◡((𝐹 ↾ (𝐶 ∪ {𝑋})) ↾ 𝐶))
125	124	adantl 481	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})–1-1→𝐵) → Fun ^◡((𝐹 ↾ (𝐶 ∪ {𝑋})) ↾ 𝐶))
126		ssun1 4160	. . . . . . . . 9 ⊢ 𝐶 ⊆ (𝐶 ∪ {𝑋})
127	126	resabs1i 6007	. . . . . . . 8 ⊢ ((𝐹 ↾ (𝐶 ∪ {𝑋})) ↾ 𝐶) = (𝐹 ↾ 𝐶)
128	127	eqcomi 2743	. . . . . . 7 ⊢ (𝐹 ↾ 𝐶) = ((𝐹 ↾ (𝐶 ∪ {𝑋})) ↾ 𝐶)
129	128	cnveqi 5867	. . . . . 6 ⊢ ^◡(𝐹 ↾ 𝐶) = ^◡((𝐹 ↾ (𝐶 ∪ {𝑋})) ↾ 𝐶)
130	129	funeqi 6568	. . . . 5 ⊢ (Fun ^◡(𝐹 ↾ 𝐶) ↔ Fun ^◡((𝐹 ↾ (𝐶 ∪ {𝑋})) ↾ 𝐶))
131	125, 130	sylibr 234	. . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})–1-1→𝐵) → Fun ^◡(𝐹 ↾ 𝐶))
132		df-f1 6547	. . . 4 ⊢ ((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 ↔ ((𝐹 ↾ 𝐶):𝐶⟶𝐵 ∧ Fun ^◡(𝐹 ↾ 𝐶)))
133	121, 131, 132	sylanbrc 583	. . 3 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})–1-1→𝐵) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐵)
134		elun1 4164	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐶 → 𝑥 ∈ (𝐶 ∪ {𝑋}))
135		snidg 4642	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ (𝐴 ∖ 𝐶) → 𝑋 ∈ {𝑋})
136	135	3ad2ant2 1134	. . . . . . . . . . . . . 14 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → 𝑋 ∈ {𝑋})
137		elun2 4165	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ {𝑋} → 𝑋 ∈ (𝐶 ∪ {𝑋}))
138	136, 137	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → 𝑋 ∈ (𝐶 ∪ {𝑋}))
139		neeq1 2993	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → (𝑦 ≠ 𝑧 ↔ 𝑥 ≠ 𝑧))
140		fveq2 6887	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑥 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) = ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑥))
141	140	neeq1d 2990	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → (((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧) ↔ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑥) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧)))
142	139, 141	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → ((𝑦 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧)) ↔ (𝑥 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑥) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧))))
143		neeq2 2994	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑋 → (𝑥 ≠ 𝑧 ↔ 𝑥 ≠ 𝑋))
144		fveq2 6887	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑋 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧) = ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑋))
145	144	neeq2d 2991	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑋 → (((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑥) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧) ↔ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑥) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑋)))
146	143, 145	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑋 → ((𝑥 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑥) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧)) ↔ (𝑥 ≠ 𝑋 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑥) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑋))))
147	142, 146	rspc2v 3617	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝐶 ∪ {𝑋}) ∧ 𝑋 ∈ (𝐶 ∪ {𝑋})) → (∀𝑦 ∈ (𝐶 ∪ {𝑋})∀𝑧 ∈ (𝐶 ∪ {𝑋})(𝑦 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧)) → (𝑥 ≠ 𝑋 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑥) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑋))))
148	134, 138, 147	syl2anr 597	. . . . . . . . . . . 12 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ 𝑥 ∈ 𝐶) → (∀𝑦 ∈ (𝐶 ∪ {𝑋})∀𝑧 ∈ (𝐶 ∪ {𝑋})(𝑦 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧)) → (𝑥 ≠ 𝑋 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑥) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑋))))
149	148	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ 𝑥 ∈ 𝐶) ∧ (𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})⟶𝐵) → (∀𝑦 ∈ (𝐶 ∪ {𝑋})∀𝑧 ∈ (𝐶 ∪ {𝑋})(𝑦 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧)) → (𝑥 ≠ 𝑋 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑥) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑋))))
150		eldifn 4114	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ (𝐴 ∖ 𝐶) → ¬ 𝑋 ∈ 𝐶)
151		nelelne 3030	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑋 ∈ 𝐶 → (𝑥 ∈ 𝐶 → 𝑥 ≠ 𝑋))
152	150, 151	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 ∈ (𝐴 ∖ 𝐶) → (𝑥 ∈ 𝐶 → 𝑥 ≠ 𝑋))
153	152	3ad2ant2 1134	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → (𝑥 ∈ 𝐶 → 𝑥 ≠ 𝑋))
154	153	imp 406	. . . . . . . . . . . . . 14 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ 𝑥 ∈ 𝐶) → 𝑥 ≠ 𝑋)
155	154	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ 𝑥 ∈ 𝐶) ∧ (𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})⟶𝐵) → 𝑥 ≠ 𝑋)
156		pm2.27 42	. . . . . . . . . . . . 13 ⊢ (𝑥 ≠ 𝑋 → ((𝑥 ≠ 𝑋 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑥) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑋)) → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑥) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑋)))
157	155, 156	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ 𝑥 ∈ 𝐶) ∧ (𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})⟶𝐵) → ((𝑥 ≠ 𝑋 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑥) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑋)) → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑥) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑋)))
158	134	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ (𝐶 ∪ {𝑋}))
159	158	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ 𝑥 ∈ 𝐶) ∧ (𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})⟶𝐵) → 𝑥 ∈ (𝐶 ∪ {𝑋}))
160	159	fvresd 6907	. . . . . . . . . . . . 13 ⊢ ((((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ 𝑥 ∈ 𝐶) ∧ (𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})⟶𝐵) → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑥) = (𝐹‘𝑥))
161	135, 137	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ (𝐴 ∖ 𝐶) → 𝑋 ∈ (𝐶 ∪ {𝑋}))
162	161	3ad2ant2 1134	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → 𝑋 ∈ (𝐶 ∪ {𝑋}))
163	162	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ 𝑥 ∈ 𝐶) → 𝑋 ∈ (𝐶 ∪ {𝑋}))
164	163	fvresd 6907	. . . . . . . . . . . . . 14 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ 𝑥 ∈ 𝐶) → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑋) = (𝐹‘𝑋))
165	164	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ 𝑥 ∈ 𝐶) ∧ (𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})⟶𝐵) → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑋) = (𝐹‘𝑋))
166	160, 165	neeq12d 2992	. . . . . . . . . . . 12 ⊢ ((((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ 𝑥 ∈ 𝐶) ∧ (𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})⟶𝐵) → (((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑥) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑋) ↔ (𝐹‘𝑥) ≠ (𝐹‘𝑋)))
167	157, 166	sylibd 239	. . . . . . . . . . 11 ⊢ ((((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ 𝑥 ∈ 𝐶) ∧ (𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})⟶𝐵) → ((𝑥 ≠ 𝑋 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑥) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑋)) → (𝐹‘𝑥) ≠ (𝐹‘𝑋)))
168	149, 167	syld 47	. . . . . . . . . 10 ⊢ ((((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ 𝑥 ∈ 𝐶) ∧ (𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})⟶𝐵) → (∀𝑦 ∈ (𝐶 ∪ {𝑋})∀𝑧 ∈ (𝐶 ∪ {𝑋})(𝑦 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧)) → (𝐹‘𝑥) ≠ (𝐹‘𝑋)))
169	168	expimpd 453	. . . . . . . . 9 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ 𝑥 ∈ 𝐶) → (((𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})⟶𝐵 ∧ ∀𝑦 ∈ (𝐶 ∪ {𝑋})∀𝑧 ∈ (𝐶 ∪ {𝑋})(𝑦 ≠ 𝑧 → ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑦) ≠ ((𝐹 ↾ (𝐶 ∪ {𝑋}))‘𝑧))) → (𝐹‘𝑥) ≠ (𝐹‘𝑋)))
170	117, 169	biimtrid 242	. . . . . . . 8 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ 𝑥 ∈ 𝐶) → ((𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})–1-1→𝐵 → (𝐹‘𝑥) ≠ (𝐹‘𝑋)))
171	170	impancom 451	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})–1-1→𝐵) → (𝑥 ∈ 𝐶 → (𝐹‘𝑥) ≠ (𝐹‘𝑋)))
172	171	imp 406	. . . . . 6 ⊢ ((((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})–1-1→𝐵) ∧ 𝑥 ∈ 𝐶) → (𝐹‘𝑥) ≠ (𝐹‘𝑋))
173	172	neneqd 2936	. . . . 5 ⊢ ((((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})–1-1→𝐵) ∧ 𝑥 ∈ 𝐶) → ¬ (𝐹‘𝑥) = (𝐹‘𝑋))
174	173	ralrimiva 3133	. . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})–1-1→𝐵) → ∀𝑥 ∈ 𝐶 ¬ (𝐹‘𝑥) = (𝐹‘𝑋))
175	51	adantr 480	. . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})–1-1→𝐵) → ((𝐹‘𝑋) ∉ (𝐹 “ 𝐶) ↔ ∀𝑥 ∈ 𝐶 ¬ (𝐹‘𝑥) = (𝐹‘𝑋)))
176	174, 175	mpbird 257	. . 3 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})–1-1→𝐵) → (𝐹‘𝑋) ∉ (𝐹 “ 𝐶))
177	133, 176	jca 511	. 2 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) ∧ (𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})–1-1→𝐵) → ((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 ∧ (𝐹‘𝑋) ∉ (𝐹 “ 𝐶)))
178	118, 177	impbida 800	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → (((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 ∧ (𝐹‘𝑋) ∉ (𝐹 “ 𝐶)) ↔ (𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})–1-1→𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ≠ wne 2931 ∉ wnel 3035 ∀wral 3050 ∃wrex 3059 ∖ cdif 3930 ∪ cun 3931 ⊆ wss 3933 {csn 4608 ^◡ccnv 5666 ↾ cres 5669 “ cima 5670 Fun wfun 6536 Fn wfn 6537 ⟶wf 6538 –1-1→wf1 6539 ‘cfv 6542

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-sep 5278 ax-nul 5288 ax-pr 5414

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rab 3421 df-v 3466 df-dif 3936 df-un 3938 df-in 3940 df-ss 3950 df-nul 4316 df-if 4508 df-sn 4609 df-pr 4611 df-op 4615 df-uni 4890 df-br 5126 df-opab 5188 df-id 5560 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6495 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fv 6550

This theorem is referenced by: resf1ext2b 7940

W3C validator