tz7.48lem - Metamath Proof Explorer

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Theorem tz7.48lem 7819

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 3120	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
2		simpl 476	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴)
3	2	anim1i 608	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ 𝑦 ∈ 𝑥) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥))
4	3	imim1i 63	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ 𝑦 ∈ 𝑥) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
5	4	expd 406	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))))
6	5	2alimi 1856	. . . . . . 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 209	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦) → ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))))
8		r2al 3120	. . . . . 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 226	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
10		elequ1 2113	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝑥 ↔ 𝑤 ∈ 𝑥))
11		fveq2 6446	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑤 → (𝐹‘𝑦) = (𝐹‘𝑤))
12	11	eqeq2d 2787	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝐹‘𝑥) = (𝐹‘𝑤)))
13	12	notbid 310	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (¬ (𝐹‘𝑥) = (𝐹‘𝑦) ↔ ¬ (𝐹‘𝑥) = (𝐹‘𝑤)))
14	10, 13	imbi12d 336	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑤 → ((𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) ↔ (𝑤 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑤))))
15	14	cbvralv 3366	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) ↔ ∀𝑤 ∈ 𝐴 (𝑤 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑤)))
16	15	ralbii 3161	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑤 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑤)))
17		elequ2 2120	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑧))
18		fveqeq2 6455	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥) = (𝐹‘𝑤) ↔ (𝐹‘𝑧) = (𝐹‘𝑤)))
19	18	notbid 310	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (¬ (𝐹‘𝑥) = (𝐹‘𝑤) ↔ ¬ (𝐹‘𝑧) = (𝐹‘𝑤)))
20	17, 19	imbi12d 336	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → ((𝑤 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑤)) ↔ (𝑤 ∈ 𝑧 → ¬ (𝐹‘𝑧) = (𝐹‘𝑤))))
21	20	ralbidv 3167	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (∀𝑤 ∈ 𝐴 (𝑤 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑤)) ↔ ∀𝑤 ∈ 𝐴 (𝑤 ∈ 𝑧 → ¬ (𝐹‘𝑧) = (𝐹‘𝑤))))
22	21	cbvralv 3366	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑤 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑤)) ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑤 ∈ 𝑧 → ¬ (𝐹‘𝑧) = (𝐹‘𝑤)))
23		elequ1 2113	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ 𝑧 ↔ 𝑥 ∈ 𝑧))
24		fveq2 6446	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑥 → (𝐹‘𝑤) = (𝐹‘𝑥))
25	24	eqeq2d 2787	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑥 → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ (𝐹‘𝑧) = (𝐹‘𝑥)))
26	25	notbid 310	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑥 → (¬ (𝐹‘𝑧) = (𝐹‘𝑤) ↔ ¬ (𝐹‘𝑧) = (𝐹‘𝑥)))
27	23, 26	imbi12d 336	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑥 → ((𝑤 ∈ 𝑧 → ¬ (𝐹‘𝑧) = (𝐹‘𝑤)) ↔ (𝑥 ∈ 𝑧 → ¬ (𝐹‘𝑧) = (𝐹‘𝑥))))
28	27	cbvralv 3366	. . . . . . . . . . 11 ⊢ (∀𝑤 ∈ 𝐴 (𝑤 ∈ 𝑧 → ¬ (𝐹‘𝑧) = (𝐹‘𝑤)) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝑧 → ¬ (𝐹‘𝑧) = (𝐹‘𝑥)))
29	28	ralbii 3161	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑤 ∈ 𝑧 → ¬ (𝐹‘𝑧) = (𝐹‘𝑤)) ↔ ∀𝑧 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝑧 → ¬ (𝐹‘𝑧) = (𝐹‘𝑥)))
30		elequ2 2120	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ 𝑦))
31		fveqeq2 6455	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑦 → ((𝐹‘𝑧) = (𝐹‘𝑥) ↔ (𝐹‘𝑦) = (𝐹‘𝑥)))
32	31	notbid 310	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑦 → (¬ (𝐹‘𝑧) = (𝐹‘𝑥) ↔ ¬ (𝐹‘𝑦) = (𝐹‘𝑥)))
33	30, 32	imbi12d 336	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → ((𝑥 ∈ 𝑧 → ¬ (𝐹‘𝑧) = (𝐹‘𝑥)) ↔ (𝑥 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑥))))
34	33	ralbidv 3167	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝑧 → ¬ (𝐹‘𝑧) = (𝐹‘𝑥)) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑥))))
35	34	cbvralv 3366	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝑧 → ¬ (𝐹‘𝑧) = (𝐹‘𝑥)) ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑥)))
36	29, 35	bitri 267	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑤 ∈ 𝑧 → ¬ (𝐹‘𝑧) = (𝐹‘𝑤)) ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑥)))
37	16, 22, 36	3bitri 289	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑥)))
38		ralcom2 3289	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑥)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑥)))
39	37, 38	sylbi 209	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑥)))
40	39	ancri 545	. . . . . 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))))
41		r19.26-2 3250	. . . . . 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))))
42	40, 41	sylibr 226	. . . . 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))))
43	9, 42	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))))
44		fvres 6465	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥))
45		fvres 6465	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑦) = (𝐹‘𝑦))
46	44, 45	eqeqan12d 2793	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (((𝐹 ↾ 𝐴)‘𝑥) = ((𝐹 ↾ 𝐴)‘𝑦) ↔ (𝐹‘𝑥) = (𝐹‘𝑦)))
47	46	ad2antrl 718	. . . . . . . . 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)))
48		ssel 3814	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ On → (𝑥 ∈ 𝐴 → 𝑥 ∈ On))
49		ssel 3814	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ On → (𝑦 ∈ 𝐴 → 𝑦 ∈ On))
50	48, 49	anim12d 602	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ On → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ On ∧ 𝑦 ∈ On)))
51		pm3.48 949	. . . . . . . . . . . . . 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))))
52		oridm 891	. . . . . . . . . . . . . . 15 ⊢ ((¬ (𝐹‘𝑥) = (𝐹‘𝑦) ∨ ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) ↔ ¬ (𝐹‘𝑥) = (𝐹‘𝑦))
53		eqcom 2784	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝐹‘𝑦) = (𝐹‘𝑥))
54	53	notbii 312	. . . . . . . . . . . . . . . 16 ⊢ (¬ (𝐹‘𝑥) = (𝐹‘𝑦) ↔ ¬ (𝐹‘𝑦) = (𝐹‘𝑥))
55	54	orbi1i 900	. . . . . . . . . . . . . . 15 ⊢ ((¬ (𝐹‘𝑥) = (𝐹‘𝑦) ∨ ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) ↔ (¬ (𝐹‘𝑦) = (𝐹‘𝑥) ∨ ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
56	52, 55	bitr3i 269	. . . . . . . . . . . . . 14 ⊢ (¬ (𝐹‘𝑥) = (𝐹‘𝑦) ↔ (¬ (𝐹‘𝑦) = (𝐹‘𝑥) ∨ ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
57	51, 56	syl6ibr 244	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑥)) ∧ (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))) → ((𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
58	57	con2d 132	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑥)) ∧ (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))) → ((𝐹‘𝑥) = (𝐹‘𝑦) → ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥)))
59		eloni 5986	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ On → Ord 𝑥)
60		eloni 5986	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ On → Ord 𝑦)
61		ordtri3 6012	. . . . . . . . . . . . . 14 ⊢ ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 = 𝑦 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥)))
62	61	biimprd 240	. . . . . . . . . . . . 13 ⊢ ((Ord 𝑥 ∧ Ord 𝑦) → (¬ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥) → 𝑥 = 𝑦))
63	59, 60, 62	syl2an 589	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (¬ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥) → 𝑥 = 𝑦))
64	58, 63	syl9r 78	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (((𝑥 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑥)) ∧ (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
65	50, 64	syl6 35	. . . . . . . . . 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))))
66	65	imp32 411	. . . . . . . . 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))
67	47, 66	sylbid 232	. . . . . . . 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))
68	67	exp32 413	. . . . . . 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))))
69	68	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))))
70	69	2alimdv 1961	. . . . 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))))
71		r2al 3120	. . . . 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)))))
72		r2al 3120	. . . . 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)))
73	70, 71, 72	3imtr4g 288	. . . 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)))
74	43, 73	syl5 34	. . 3 ⊢ (𝐴 ⊆ On → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (((𝐹 ↾ 𝐴)‘𝑥) = ((𝐹 ↾ 𝐴)‘𝑦) → 𝑥 = 𝑦)))
75	74	imdistani 564	. 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)))
76		tz7.48.1	. . . 4 ⊢ 𝐹 Fn On
77		fnssres 6250	. . . 4 ⊢ ((𝐹 Fn On ∧ 𝐴 ⊆ On) → (𝐹 ↾ 𝐴) Fn 𝐴)
78	76, 77	mpan 680	. . 3 ⊢ (𝐴 ⊆ On → (𝐹 ↾ 𝐴) Fn 𝐴)
79		dffn2 6293	. . . 4 ⊢ ((𝐹 ↾ 𝐴) Fn 𝐴 ↔ (𝐹 ↾ 𝐴):𝐴⟶V)
80		dff13 6784	. . . . . 6 ⊢ ((𝐹 ↾ 𝐴):𝐴–1-1→V ↔ ((𝐹 ↾ 𝐴):𝐴⟶V ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (((𝐹 ↾ 𝐴)‘𝑥) = ((𝐹 ↾ 𝐴)‘𝑦) → 𝑥 = 𝑦)))
81		df-f1 6140	. . . . . 6 ⊢ ((𝐹 ↾ 𝐴):𝐴–1-1→V ↔ ((𝐹 ↾ 𝐴):𝐴⟶V ∧ Fun ^◡(𝐹 ↾ 𝐴)))
82	80, 81	bitr3i 269	. . . . 5 ⊢ (((𝐹 ↾ 𝐴):𝐴⟶V ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (((𝐹 ↾ 𝐴)‘𝑥) = ((𝐹 ↾ 𝐴)‘𝑦) → 𝑥 = 𝑦)) ↔ ((𝐹 ↾ 𝐴):𝐴⟶V ∧ Fun ^◡(𝐹 ↾ 𝐴)))
83	82	simprbi 492	. . . 4 ⊢ (((𝐹 ↾ 𝐴):𝐴⟶V ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (((𝐹 ↾ 𝐴)‘𝑥) = ((𝐹 ↾ 𝐴)‘𝑦) → 𝑥 = 𝑦)) → Fun ^◡(𝐹 ↾ 𝐴))
84	79, 83	sylanb 576	. . 3 ⊢ (((𝐹 ↾ 𝐴) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (((𝐹 ↾ 𝐴)‘𝑥) = ((𝐹 ↾ 𝐴)‘𝑦) → 𝑥 = 𝑦)) → Fun ^◡(𝐹 ↾ 𝐴))
85	78, 84	sylan 575	. 2 ⊢ ((𝐴 ⊆ On ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (((𝐹 ↾ 𝐴)‘𝑥) = ((𝐹 ↾ 𝐴)‘𝑦) → 𝑥 = 𝑦)) → Fun ^◡(𝐹 ↾ 𝐴))
86	75, 85	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 198 ∧ wa 386 ∨ wo 836 ∀wal 1599 = wceq 1601 ∈ wcel 2106 ∀wral 3089 Vcvv 3397 ⊆ wss 3791 ^◡ccnv 5354 ↾ cres 5357 Ord word 5975 Oncon0 5976 Fun wfun 6129 Fn wfn 6130 ⟶wf 6131 –1-1→wf1 6132 ‘cfv 6135

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pr 5138

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-res 5367 df-ord 5979 df-on 5980 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fv 6143

This theorem is referenced by: tz7.48-2 7820 tz7.49 7823 zorn2lem4 9656

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