tz7.48lem - Metamath Proof Explorer

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Theorem tz7.48lem 7423

Description: A way of showing an ordinal function is one-to-one. (Contributed by NM, 9-Feb-1997.)

**Hypothesis**
Ref	Expression
tz7.48.1	⊢ 𝐹 Fn On

**Assertion**
Ref	Expression
tz7.48lem	⊢ ((𝐴 ⊆ On ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) → Fun ^◡(𝐹 ↾ 𝐴))

Distinct variable groups: 𝑦,𝐴,𝑥 𝑥,𝐹,𝑦 𝑥,𝐴

Proof of Theorem tz7.48lem Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		r2al 2923	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
2		simpl 472	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴)
3	2	anim1i 590	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ 𝑦 ∈ 𝑥) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥))
4	3	imim1i 61	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ 𝑦 ∈ 𝑥) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
5	4	expd 451	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))))
6	5	2alimi 1731	. . . . . . 7 ⊢ (∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) → ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))))
7	1, 6	sylbi 206	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦) → ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))))
8		r2al 2923	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))))
9	7, 8	sylibr 223	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
10		elequ1 1984	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝑥 ↔ 𝑤 ∈ 𝑥))
11		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑤 → (𝐹‘𝑦) = (𝐹‘𝑤))
12	11	eqeq2d 2620	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝐹‘𝑥) = (𝐹‘𝑤)))
13	12	notbid 307	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (¬ (𝐹‘𝑥) = (𝐹‘𝑦) ↔ ¬ (𝐹‘𝑥) = (𝐹‘𝑤)))
14	10, 13	imbi12d 333	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑤 → ((𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) ↔ (𝑤 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑤))))
15	14	cbvralv 3147	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) ↔ ∀𝑤 ∈ 𝐴 (𝑤 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑤)))
16	15	ralbii 2963	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑤 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑤)))
17		elequ2 1991	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑧))
18		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
19	18	eqeq1d 2612	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥) = (𝐹‘𝑤) ↔ (𝐹‘𝑧) = (𝐹‘𝑤)))
20	19	notbid 307	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (¬ (𝐹‘𝑥) = (𝐹‘𝑤) ↔ ¬ (𝐹‘𝑧) = (𝐹‘𝑤)))
21	17, 20	imbi12d 333	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → ((𝑤 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑤)) ↔ (𝑤 ∈ 𝑧 → ¬ (𝐹‘𝑧) = (𝐹‘𝑤))))
22	21	ralbidv 2969	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (∀𝑤 ∈ 𝐴 (𝑤 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑤)) ↔ ∀𝑤 ∈ 𝐴 (𝑤 ∈ 𝑧 → ¬ (𝐹‘𝑧) = (𝐹‘𝑤))))
23	22	cbvralv 3147	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑤 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑤)) ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑤 ∈ 𝑧 → ¬ (𝐹‘𝑧) = (𝐹‘𝑤)))
24		elequ1 1984	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ 𝑧 ↔ 𝑥 ∈ 𝑧))
25		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑥 → (𝐹‘𝑤) = (𝐹‘𝑥))
26	25	eqeq2d 2620	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑥 → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ (𝐹‘𝑧) = (𝐹‘𝑥)))
27	26	notbid 307	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑥 → (¬ (𝐹‘𝑧) = (𝐹‘𝑤) ↔ ¬ (𝐹‘𝑧) = (𝐹‘𝑥)))
28	24, 27	imbi12d 333	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑥 → ((𝑤 ∈ 𝑧 → ¬ (𝐹‘𝑧) = (𝐹‘𝑤)) ↔ (𝑥 ∈ 𝑧 → ¬ (𝐹‘𝑧) = (𝐹‘𝑥))))
29	28	cbvralv 3147	. . . . . . . . . . 11 ⊢ (∀𝑤 ∈ 𝐴 (𝑤 ∈ 𝑧 → ¬ (𝐹‘𝑧) = (𝐹‘𝑤)) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝑧 → ¬ (𝐹‘𝑧) = (𝐹‘𝑥)))
30	29	ralbii 2963	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑤 ∈ 𝑧 → ¬ (𝐹‘𝑧) = (𝐹‘𝑤)) ↔ ∀𝑧 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝑧 → ¬ (𝐹‘𝑧) = (𝐹‘𝑥)))
31		elequ2 1991	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ 𝑦))
32		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑦 → (𝐹‘𝑧) = (𝐹‘𝑦))
33	32	eqeq1d 2612	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑦 → ((𝐹‘𝑧) = (𝐹‘𝑥) ↔ (𝐹‘𝑦) = (𝐹‘𝑥)))
34	33	notbid 307	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑦 → (¬ (𝐹‘𝑧) = (𝐹‘𝑥) ↔ ¬ (𝐹‘𝑦) = (𝐹‘𝑥)))
35	31, 34	imbi12d 333	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → ((𝑥 ∈ 𝑧 → ¬ (𝐹‘𝑧) = (𝐹‘𝑥)) ↔ (𝑥 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑥))))
36	35	ralbidv 2969	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝑧 → ¬ (𝐹‘𝑧) = (𝐹‘𝑥)) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑥))))
37	36	cbvralv 3147	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝑧 → ¬ (𝐹‘𝑧) = (𝐹‘𝑥)) ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑥)))
38	30, 37	bitri 263	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑤 ∈ 𝑧 → ¬ (𝐹‘𝑧) = (𝐹‘𝑤)) ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑥)))
39	16, 23, 38	3bitri 285	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑥)))
40		ralcom2 3083	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑥)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑥)))
41	39, 40	sylbi 206	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑥)))
42	41	ancri 573	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))))
43		r19.26-2 3047	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑥)) ∧ (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))))
44	42, 43	sylibr 223	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑥)) ∧ (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))))
45	9, 44	syl 17	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑥)) ∧ (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))))
46		fvres 6117	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥))
47		fvres 6117	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑦) = (𝐹‘𝑦))
48	46, 47	eqeqan12d 2626	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (((𝐹 ↾ 𝐴)‘𝑥) = ((𝐹 ↾ 𝐴)‘𝑦) ↔ (𝐹‘𝑥) = (𝐹‘𝑦)))
49	48	ad2antrl 760	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝑥 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑥)) ∧ (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))))) → (((𝐹 ↾ 𝐴)‘𝑥) = ((𝐹 ↾ 𝐴)‘𝑦) ↔ (𝐹‘𝑥) = (𝐹‘𝑦)))
50		ssel 3562	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ On → (𝑥 ∈ 𝐴 → 𝑥 ∈ On))
51		ssel 3562	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ On → (𝑦 ∈ 𝐴 → 𝑦 ∈ On))
52	50, 51	anim12d 584	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ On → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ On ∧ 𝑦 ∈ On)))
53		pm3.48 874	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑥)) ∧ (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))) → ((𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥) → (¬ (𝐹‘𝑦) = (𝐹‘𝑥) ∨ ¬ (𝐹‘𝑥) = (𝐹‘𝑦))))
54		oridm 535	. . . . . . . . . . . . . . 15 ⊢ ((¬ (𝐹‘𝑥) = (𝐹‘𝑦) ∨ ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) ↔ ¬ (𝐹‘𝑥) = (𝐹‘𝑦))
55		eqcom 2617	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝐹‘𝑦) = (𝐹‘𝑥))
56	55	notbii 309	. . . . . . . . . . . . . . . 16 ⊢ (¬ (𝐹‘𝑥) = (𝐹‘𝑦) ↔ ¬ (𝐹‘𝑦) = (𝐹‘𝑥))
57	56	orbi1i 541	. . . . . . . . . . . . . . 15 ⊢ ((¬ (𝐹‘𝑥) = (𝐹‘𝑦) ∨ ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) ↔ (¬ (𝐹‘𝑦) = (𝐹‘𝑥) ∨ ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
58	54, 57	bitr3i 265	. . . . . . . . . . . . . 14 ⊢ (¬ (𝐹‘𝑥) = (𝐹‘𝑦) ↔ (¬ (𝐹‘𝑦) = (𝐹‘𝑥) ∨ ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
59	53, 58	syl6ibr 241	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑥)) ∧ (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))) → ((𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
60	59	con2d 128	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑥)) ∧ (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))) → ((𝐹‘𝑥) = (𝐹‘𝑦) → ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥)))
61		eloni 5650	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ On → Ord 𝑥)
62		eloni 5650	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ On → Ord 𝑦)
63		ordtri3 5676	. . . . . . . . . . . . . 14 ⊢ ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 = 𝑦 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥)))
64	63	biimprd 237	. . . . . . . . . . . . 13 ⊢ ((Ord 𝑥 ∧ Ord 𝑦) → (¬ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥) → 𝑥 = 𝑦))
65	61, 62, 64	syl2an 493	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (¬ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥) → 𝑥 = 𝑦))
66	60, 65	syl9r 76	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (((𝑥 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑥)) ∧ (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
67	52, 66	syl6 34	. . . . . . . . . 10 ⊢ (𝐴 ⊆ On → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (((𝑥 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑥)) ∧ (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))))
68	67	imp32 448	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝑥 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑥)) ∧ (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))))) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
69	49, 68	sylbid 229	. . . . . . . 8 ⊢ ((𝐴 ⊆ On ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝑥 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑥)) ∧ (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))))) → (((𝐹 ↾ 𝐴)‘𝑥) = ((𝐹 ↾ 𝐴)‘𝑦) → 𝑥 = 𝑦))
70	69	exp32 629	. . . . . . 7 ⊢ (𝐴 ⊆ On → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (((𝑥 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑥)) ∧ (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))) → (((𝐹 ↾ 𝐴)‘𝑥) = ((𝐹 ↾ 𝐴)‘𝑦) → 𝑥 = 𝑦))))
71	70	a2d 29	. . . . . 6 ⊢ (𝐴 ⊆ On → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑥)) ∧ (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (((𝐹 ↾ 𝐴)‘𝑥) = ((𝐹 ↾ 𝐴)‘𝑦) → 𝑥 = 𝑦))))
72	71	2alimdv 1834	. . . . 5 ⊢ (𝐴 ⊆ On → (∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑥)) ∧ (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))) → ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (((𝐹 ↾ 𝐴)‘𝑥) = ((𝐹 ↾ 𝐴)‘𝑦) → 𝑥 = 𝑦))))
73		r2al 2923	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑥)) ∧ (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑥)) ∧ (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))))
74		r2al 2923	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (((𝐹 ↾ 𝐴)‘𝑥) = ((𝐹 ↾ 𝐴)‘𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (((𝐹 ↾ 𝐴)‘𝑥) = ((𝐹 ↾ 𝐴)‘𝑦) → 𝑥 = 𝑦)))
75	72, 73, 74	3imtr4g 284	. . . 4 ⊢ (𝐴 ⊆ On → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑥)) ∧ (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (((𝐹 ↾ 𝐴)‘𝑥) = ((𝐹 ↾ 𝐴)‘𝑦) → 𝑥 = 𝑦)))
76	45, 75	syl5 33	. . 3 ⊢ (𝐴 ⊆ On → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (((𝐹 ↾ 𝐴)‘𝑥) = ((𝐹 ↾ 𝐴)‘𝑦) → 𝑥 = 𝑦)))
77	76	imdistani 722	. 2 ⊢ ((𝐴 ⊆ On ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) → (𝐴 ⊆ On ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (((𝐹 ↾ 𝐴)‘𝑥) = ((𝐹 ↾ 𝐴)‘𝑦) → 𝑥 = 𝑦)))
78		tz7.48.1	. . . 4 ⊢ 𝐹 Fn On
79		fnssres 5918	. . . 4 ⊢ ((𝐹 Fn On ∧ 𝐴 ⊆ On) → (𝐹 ↾ 𝐴) Fn 𝐴)
80	78, 79	mpan 702	. . 3 ⊢ (𝐴 ⊆ On → (𝐹 ↾ 𝐴) Fn 𝐴)
81		dffn2 5960	. . . 4 ⊢ ((𝐹 ↾ 𝐴) Fn 𝐴 ↔ (𝐹 ↾ 𝐴):𝐴⟶V)
82		dff13 6416	. . . . . 6 ⊢ ((𝐹 ↾ 𝐴):𝐴–1-1→V ↔ ((𝐹 ↾ 𝐴):𝐴⟶V ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (((𝐹 ↾ 𝐴)‘𝑥) = ((𝐹 ↾ 𝐴)‘𝑦) → 𝑥 = 𝑦)))
83		df-f1 5809	. . . . . 6 ⊢ ((𝐹 ↾ 𝐴):𝐴–1-1→V ↔ ((𝐹 ↾ 𝐴):𝐴⟶V ∧ Fun ^◡(𝐹 ↾ 𝐴)))
84	82, 83	bitr3i 265	. . . . 5 ⊢ (((𝐹 ↾ 𝐴):𝐴⟶V ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (((𝐹 ↾ 𝐴)‘𝑥) = ((𝐹 ↾ 𝐴)‘𝑦) → 𝑥 = 𝑦)) ↔ ((𝐹 ↾ 𝐴):𝐴⟶V ∧ Fun ^◡(𝐹 ↾ 𝐴)))
85	84	simprbi 479	. . . 4 ⊢ (((𝐹 ↾ 𝐴):𝐴⟶V ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (((𝐹 ↾ 𝐴)‘𝑥) = ((𝐹 ↾ 𝐴)‘𝑦) → 𝑥 = 𝑦)) → Fun ^◡(𝐹 ↾ 𝐴))
86	81, 85	sylanb 488	. . 3 ⊢ (((𝐹 ↾ 𝐴) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (((𝐹 ↾ 𝐴)‘𝑥) = ((𝐹 ↾ 𝐴)‘𝑦) → 𝑥 = 𝑦)) → Fun ^◡(𝐹 ↾ 𝐴))
87	80, 86	sylan 487	. 2 ⊢ ((𝐴 ⊆ On ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (((𝐹 ↾ 𝐴)‘𝑥) = ((𝐹 ↾ 𝐴)‘𝑦) → 𝑥 = 𝑦)) → Fun ^◡(𝐹 ↾ 𝐴))
88	77, 87	syl 17	1 ⊢ ((𝐴 ⊆ On ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) → Fun ^◡(𝐹 ↾ 𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 ∀wal 1473 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ⊆ wss 3540 ^◡ccnv 5037 ↾ cres 5040 Ord word 5639 Oncon0 5640 Fun wfun 5798 Fn wfn 5799 ⟶wf 5800 –1-1→wf1 5801 ‘cfv 5804

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-res 5050 df-ord 5643 df-on 5644 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fv 5812

This theorem is referenced by: tz7.48-2 7424 tz7.49 7427 zorn2lem4 9204

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