tz7.49 - Metamath Proof Explorer

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Theorem tz7.49 8081

Description: Proposition 7.49 of [TakeutiZaring] p. 51. (Contributed by NM, 10-Feb-1997.) (Revised by Mario Carneiro, 10-Jan-2013.)

**Hypotheses**
Ref	Expression
tz7.49.1	⊢ 𝐹 Fn On
tz7.49.2	⊢ (𝜑 ↔ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))

**Assertion**
Ref	Expression
tz7.49	⊢ ((𝐴 ∈ 𝐵 ∧ 𝜑) → ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ (𝐹 “ 𝑥) = 𝐴 ∧ Fun ^◡(𝐹 ↾ 𝑥)))

Distinct variable groups: 𝑥,𝑦,𝐴 𝑥,𝐹,𝑦 𝜑,𝑦 Allowed substitution hints: 𝜑(𝑥) 𝐵(𝑥,𝑦)

Proof of Theorem tz7.49 Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ne 3017	. . . . . . . . 9 ⊢ ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ ↔ ¬ (𝐴 ∖ (𝐹 “ 𝑥)) = ∅)
2	1	ralbii 3165	. . . . . . . 8 ⊢ (∀𝑥 ∈ On (𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ ↔ ∀𝑥 ∈ On ¬ (𝐴 ∖ (𝐹 “ 𝑥)) = ∅)
3		tz7.49.2	. . . . . . . . 9 ⊢ (𝜑 ↔ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
4		ralim 3162	. . . . . . . . 9 ⊢ (∀𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))) → (∀𝑥 ∈ On (𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → ∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
5	3, 4	sylbi 219	. . . . . . . 8 ⊢ (𝜑 → (∀𝑥 ∈ On (𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → ∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
6	2, 5	syl5bir 245	. . . . . . 7 ⊢ (𝜑 → (∀𝑥 ∈ On ¬ (𝐴 ∖ (𝐹 “ 𝑥)) = ∅ → ∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
7		tz7.49.1	. . . . . . . . 9 ⊢ 𝐹 Fn On
8	7	tz7.48-3 8080	. . . . . . . 8 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ¬ 𝐴 ∈ V)
9		elex 3512	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
10	8, 9	nsyl3 140	. . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → ¬ ∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))
11	6, 10	nsyli 160	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ 𝐵 → ¬ ∀𝑥 ∈ On ¬ (𝐴 ∖ (𝐹 “ 𝑥)) = ∅))
12		dfrex2 3239	. . . . . 6 ⊢ (∃𝑥 ∈ On (𝐴 ∖ (𝐹 “ 𝑥)) = ∅ ↔ ¬ ∀𝑥 ∈ On ¬ (𝐴 ∖ (𝐹 “ 𝑥)) = ∅)
13	11, 12	syl6ibr 254	. . . . 5 ⊢ (𝜑 → (𝐴 ∈ 𝐵 → ∃𝑥 ∈ On (𝐴 ∖ (𝐹 “ 𝑥)) = ∅))
14		imaeq2 5925	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐹 “ 𝑥) = (𝐹 “ 𝑦))
15	14	difeq2d 4099	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 ∖ (𝐹 “ 𝑥)) = (𝐴 ∖ (𝐹 “ 𝑦)))
16	15	eqeq1d 2823	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 ∖ (𝐹 “ 𝑥)) = ∅ ↔ (𝐴 ∖ (𝐹 “ 𝑦)) = ∅))
17	16	onminex 7522	. . . . 5 ⊢ (∃𝑥 ∈ On (𝐴 ∖ (𝐹 “ 𝑥)) = ∅ → ∃𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) = ∅ ∧ ∀𝑦 ∈ 𝑥 ¬ (𝐴 ∖ (𝐹 “ 𝑦)) = ∅))
18	13, 17	syl6 35	. . . 4 ⊢ (𝜑 → (𝐴 ∈ 𝐵 → ∃𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) = ∅ ∧ ∀𝑦 ∈ 𝑥 ¬ (𝐴 ∖ (𝐹 “ 𝑦)) = ∅)))
19		df-ne 3017	. . . . . . 7 ⊢ ((𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ↔ ¬ (𝐴 ∖ (𝐹 “ 𝑦)) = ∅)
20	19	ralbii 3165	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ↔ ∀𝑦 ∈ 𝑥 ¬ (𝐴 ∖ (𝐹 “ 𝑦)) = ∅)
21	20	anbi2i 624	. . . . 5 ⊢ (((𝐴 ∖ (𝐹 “ 𝑥)) = ∅ ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) ↔ ((𝐴 ∖ (𝐹 “ 𝑥)) = ∅ ∧ ∀𝑦 ∈ 𝑥 ¬ (𝐴 ∖ (𝐹 “ 𝑦)) = ∅))
22	21	rexbii 3247	. . . 4 ⊢ (∃𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) = ∅ ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) ↔ ∃𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) = ∅ ∧ ∀𝑦 ∈ 𝑥 ¬ (𝐴 ∖ (𝐹 “ 𝑦)) = ∅))
23	18, 22	syl6ibr 254	. . 3 ⊢ (𝜑 → (𝐴 ∈ 𝐵 → ∃𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) = ∅ ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅)))
24		nfra1 3219	. . . . 5 ⊢ Ⅎ𝑥∀𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))
25	3, 24	nfxfr 1853	. . . 4 ⊢ Ⅎ𝑥𝜑
26		simpllr 774	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹 “ 𝑥)) = ∅) → ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅)
27		fnfun 6453	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 Fn On → Fun 𝐹)
28	7, 27	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ Fun 𝐹
29		fvelima 6731	. . . . . . . . . . . . . . . 16 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ (𝐹 “ 𝑥)) → ∃𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑧)
30	28, 29	mpan 688	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ (𝐹 “ 𝑥) → ∃𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑧)
31		nfv 1915	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦𝜑
32		nfra1 3219	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅
33	31, 32	nfan 1900	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑦(𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅)
34		nfv 1915	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑦(𝑥 ∈ On → 𝑧 ∈ 𝐴)
35		rsp 3205	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ → (𝑦 ∈ 𝑥 → (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅))
36	35	adantld 493	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ → ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅))
37		onelon 6216	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
38	15	neeq1d 3075	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 = 𝑦 → ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ ↔ (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅))
39		fveq2 6670	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
40	39, 15	eleq12d 2907	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) ↔ (𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦))))
41	38, 40	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = 𝑦 → (((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))) ↔ ((𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ → (𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦)))))
42	41	rspcv 3618	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ On → (∀𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))) → ((𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ → (𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦)))))
43	3, 42	syl5bi 244	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ On → (𝜑 → ((𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ → (𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦)))))
44	43	com23 86	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ On → ((𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ → (𝜑 → (𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦)))))
45	37, 44	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → ((𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ → (𝜑 → (𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦)))))
46	36, 45	sylcom 30	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ → ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝜑 → (𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦)))))
47	46	com3r 87	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ → ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦)))))
48	47	imp 409	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) → ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦))))
49	48	expcomd 419	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) → (𝑦 ∈ 𝑥 → (𝑥 ∈ On → (𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦)))))
50		eldifi 4103	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦)) → (𝐹‘𝑦) ∈ 𝐴)
51		eleq1 2900	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑦) = 𝑧 → ((𝐹‘𝑦) ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
52	50, 51	syl5ibcom 247	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦)) → ((𝐹‘𝑦) = 𝑧 → 𝑧 ∈ 𝐴))
53	49, 52	syl8 76	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) → (𝑦 ∈ 𝑥 → (𝑥 ∈ On → ((𝐹‘𝑦) = 𝑧 → 𝑧 ∈ 𝐴))))
54	53	com34 91	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) → (𝑦 ∈ 𝑥 → ((𝐹‘𝑦) = 𝑧 → (𝑥 ∈ On → 𝑧 ∈ 𝐴))))
55	33, 34, 54	rexlimd 3317	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) → (∃𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑧 → (𝑥 ∈ On → 𝑧 ∈ 𝐴)))
56	30, 55	syl5 34	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) → (𝑧 ∈ (𝐹 “ 𝑥) → (𝑥 ∈ On → 𝑧 ∈ 𝐴)))
57	56	com23 86	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) → (𝑥 ∈ On → (𝑧 ∈ (𝐹 “ 𝑥) → 𝑧 ∈ 𝐴)))
58	57	imp 409	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) → (𝑧 ∈ (𝐹 “ 𝑥) → 𝑧 ∈ 𝐴))
59	58	ssrdv 3973	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) → (𝐹 “ 𝑥) ⊆ 𝐴)
60		ssdif0 4323	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ (𝐹 “ 𝑥) ↔ (𝐴 ∖ (𝐹 “ 𝑥)) = ∅)
61	60	biimpri 230	. . . . . . . . . . 11 ⊢ ((𝐴 ∖ (𝐹 “ 𝑥)) = ∅ → 𝐴 ⊆ (𝐹 “ 𝑥))
62	59, 61	anim12i 614	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹 “ 𝑥)) = ∅) → ((𝐹 “ 𝑥) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐹 “ 𝑥)))
63		eqss 3982	. . . . . . . . . 10 ⊢ ((𝐹 “ 𝑥) = 𝐴 ↔ ((𝐹 “ 𝑥) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐹 “ 𝑥)))
64	62, 63	sylibr 236	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹 “ 𝑥)) = ∅) → (𝐹 “ 𝑥) = 𝐴)
65		onss 7505	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On)
66	32, 31	nfan 1900	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦(∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑)
67		nfv 1915	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦 𝑥 ⊆ On
68	66, 67	nfan 1900	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑦((∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) ∧ 𝑥 ⊆ On)
69		nfv 1915	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑧(((∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) ∧ 𝑥 ⊆ On) ∧ 𝑦 ∈ 𝑥)
70		ssel 3961	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ⊆ On → (𝑦 ∈ 𝑥 → 𝑦 ∈ On))
71		onss 7505	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ On → 𝑦 ⊆ On)
72		fndm 6455	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐹 Fn On → dom 𝐹 = On)
73	7, 72	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ dom 𝐹 = On
74	71, 73	sseqtrrdi 4018	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ On → 𝑦 ⊆ dom 𝐹)
75		funfvima2 6993	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((Fun 𝐹 ∧ 𝑦 ⊆ dom 𝐹) → (𝑧 ∈ 𝑦 → (𝐹‘𝑧) ∈ (𝐹 “ 𝑦)))
76	28, 74, 75	sylancr 589	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ On → (𝑧 ∈ 𝑦 → (𝐹‘𝑧) ∈ (𝐹 “ 𝑦)))
77	70, 76	syl6 35	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ⊆ On → (𝑦 ∈ 𝑥 → (𝑧 ∈ 𝑦 → (𝐹‘𝑧) ∈ (𝐹 “ 𝑦))))
78	35	com12 32	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ 𝑥 → (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ → (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅))
79	78	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 ⊆ On → (𝑦 ∈ 𝑥 → (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ → (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅)))
80	70, 79, 44	syl10 79	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 ⊆ On → (𝑦 ∈ 𝑥 → (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ → (𝜑 → (𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦))))))
81	80	imp4a 425	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ⊆ On → (𝑦 ∈ 𝑥 → ((∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) → (𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦)))))
82		eldifn 4104	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦)) → ¬ (𝐹‘𝑦) ∈ (𝐹 “ 𝑦))
83		eleq1a 2908	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐹‘𝑧) ∈ (𝐹 “ 𝑦) → ((𝐹‘𝑦) = (𝐹‘𝑧) → (𝐹‘𝑦) ∈ (𝐹 “ 𝑦)))
84	83	con3d 155	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐹‘𝑧) ∈ (𝐹 “ 𝑦) → (¬ (𝐹‘𝑦) ∈ (𝐹 “ 𝑦) → ¬ (𝐹‘𝑦) = (𝐹‘𝑧)))
85	82, 84	syl5com 31	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐹‘𝑦) ∈ (𝐴 ∖ (𝐹 “ 𝑦)) → ((𝐹‘𝑧) ∈ (𝐹 “ 𝑦) → ¬ (𝐹‘𝑦) = (𝐹‘𝑧)))
86	81, 85	syl8 76	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ⊆ On → (𝑦 ∈ 𝑥 → ((∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) → ((𝐹‘𝑧) ∈ (𝐹 “ 𝑦) → ¬ (𝐹‘𝑦) = (𝐹‘𝑧)))))
87	86	com34 91	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ⊆ On → (𝑦 ∈ 𝑥 → ((𝐹‘𝑧) ∈ (𝐹 “ 𝑦) → ((∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) → ¬ (𝐹‘𝑦) = (𝐹‘𝑧)))))
88	77, 87	syldd 72	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ⊆ On → (𝑦 ∈ 𝑥 → (𝑧 ∈ 𝑦 → ((∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) → ¬ (𝐹‘𝑦) = (𝐹‘𝑧)))))
89	88	com4r 94	. . . . . . . . . . . . . . . . . . 19 ⊢ ((∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) → (𝑥 ⊆ On → (𝑦 ∈ 𝑥 → (𝑧 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑧)))))
90	89	imp31 420	. . . . . . . . . . . . . . . . . 18 ⊢ ((((∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) ∧ 𝑥 ⊆ On) ∧ 𝑦 ∈ 𝑥) → (𝑧 ∈ 𝑦 → ¬ (𝐹‘𝑦) = (𝐹‘𝑧)))
91	69, 90	ralrimi 3216	. . . . . . . . . . . . . . . . 17 ⊢ ((((∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) ∧ 𝑥 ⊆ On) ∧ 𝑦 ∈ 𝑥) → ∀𝑧 ∈ 𝑦 ¬ (𝐹‘𝑦) = (𝐹‘𝑧))
92	91	ex 415	. . . . . . . . . . . . . . . 16 ⊢ (((∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) ∧ 𝑥 ⊆ On) → (𝑦 ∈ 𝑥 → ∀𝑧 ∈ 𝑦 ¬ (𝐹‘𝑦) = (𝐹‘𝑧)))
93	68, 92	ralrimi 3216	. . . . . . . . . . . . . . 15 ⊢ (((∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) ∧ 𝑥 ⊆ On) → ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑦 ¬ (𝐹‘𝑦) = (𝐹‘𝑧))
94	93	ex 415	. . . . . . . . . . . . . 14 ⊢ ((∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) → (𝑥 ⊆ On → ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑦 ¬ (𝐹‘𝑦) = (𝐹‘𝑧)))
95	94	ancld 553	. . . . . . . . . . . . 13 ⊢ ((∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) → (𝑥 ⊆ On → (𝑥 ⊆ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑦 ¬ (𝐹‘𝑦) = (𝐹‘𝑧))))
96	7	tz7.48lem 8077	. . . . . . . . . . . . 13 ⊢ ((𝑥 ⊆ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑦 ¬ (𝐹‘𝑦) = (𝐹‘𝑧)) → Fun ^◡(𝐹 ↾ 𝑥))
97	65, 95, 96	syl56 36	. . . . . . . . . . . 12 ⊢ ((∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ 𝜑) → (𝑥 ∈ On → Fun ^◡(𝐹 ↾ 𝑥)))
98	97	ancoms 461	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) → (𝑥 ∈ On → Fun ^◡(𝐹 ↾ 𝑥)))
99	98	imp 409	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) → Fun ^◡(𝐹 ↾ 𝑥))
100	99	adantr 483	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹 “ 𝑥)) = ∅) → Fun ^◡(𝐹 ↾ 𝑥))
101	26, 64, 100	3jca 1124	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹 “ 𝑥)) = ∅) → (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ (𝐹 “ 𝑥) = 𝐴 ∧ Fun ^◡(𝐹 ↾ 𝑥)))
102	101	exp41 437	. . . . . . 7 ⊢ (𝜑 → (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ → (𝑥 ∈ On → ((𝐴 ∖ (𝐹 “ 𝑥)) = ∅ → (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ (𝐹 “ 𝑥) = 𝐴 ∧ Fun ^◡(𝐹 ↾ 𝑥))))))
103	102	com23 86	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ → ((𝐴 ∖ (𝐹 “ 𝑥)) = ∅ → (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ (𝐹 “ 𝑥) = 𝐴 ∧ Fun ^◡(𝐹 ↾ 𝑥))))))
104	103	com34 91	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ On → ((𝐴 ∖ (𝐹 “ 𝑥)) = ∅ → (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ → (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ (𝐹 “ 𝑥) = 𝐴 ∧ Fun ^◡(𝐹 ↾ 𝑥))))))
105	104	imp4a 425	. . . 4 ⊢ (𝜑 → (𝑥 ∈ On → (((𝐴 ∖ (𝐹 “ 𝑥)) = ∅ ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) → (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ (𝐹 “ 𝑥) = 𝐴 ∧ Fun ^◡(𝐹 ↾ 𝑥)))))
106	25, 105	reximdai 3311	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) = ∅ ∧ ∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅) → ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ (𝐹 “ 𝑥) = 𝐴 ∧ Fun ^◡(𝐹 ↾ 𝑥))))
107	23, 106	syld 47	. 2 ⊢ (𝜑 → (𝐴 ∈ 𝐵 → ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ (𝐹 “ 𝑥) = 𝐴 ∧ Fun ^◡(𝐹 ↾ 𝑥))))
108	107	impcom 410	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜑) → ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ (𝐹 “ 𝑥) = 𝐴 ∧ Fun ^◡(𝐹 ↾ 𝑥)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 Vcvv 3494 ∖ cdif 3933 ⊆ wss 3936 ∅c0 4291 ^◡ccnv 5554 dom cdm 5555 ↾ cres 5557 “ cima 5558 Oncon0 6191 Fun wfun 6349 Fn wfn 6350 ‘cfv 6355

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-un 7461

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-ord 6194 df-on 6195 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363

This theorem is referenced by: tz7.49c 8082

W3C validator