Theorem elovmpt3rab1 7507

Description: Implications for the value of an operation defined by the maps-to notation with a function into a class abstraction as a result having an element. The domain of the function and the base set of the class abstraction may depend on the operands, using implicit substitution. (Contributed by AV, 16-Jul-2018.) (Revised by AV, 16-May-2019.)

Hypotheses

Ref	Expression
ovmpt3rab1.o	⊢ 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑧 ∈ 𝑀 ↦ {𝑎 ∈ 𝑁 ∣ 𝜑}))
ovmpt3rab1.m	⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑀 = 𝐾)
ovmpt3rab1.n	⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑁 = 𝐿)

Assertion

Ref	Expression
elovmpt3rab1	⊢ ((𝐾 ∈ 𝑈 ∧ 𝐿 ∈ 𝑇) → (𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) → ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑍 ∈ 𝐾 ∧ 𝐴 ∈ 𝐿))))

Distinct variable groups:   𝑥,𝐾,𝑦,𝑧   𝐿,𝑎,𝑥,𝑦   𝑁,𝑎   𝑥,𝑈,𝑦   𝑋,𝑎,𝑥,𝑦,𝑧   𝑌,𝑎,𝑥,𝑦,𝑧   𝑧,𝐿   𝑧,𝑇   𝑧,𝑈   𝐴,𝑎   𝑍,𝑎,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑎)   𝐴(𝑥,𝑦,𝑧)   𝑇(𝑥,𝑦,𝑎)   𝑈(𝑎)   𝐾(𝑎)   𝑀(𝑥,𝑦,𝑧,𝑎)   𝑁(𝑥,𝑦,𝑧)   𝑂(𝑥,𝑦,𝑧,𝑎)   𝑍(𝑥,𝑦)

Proof of Theorem elovmpt3rab1

Step	Hyp	Ref	Expression
1		ovmpt3rab1.o	. . . 4 ⊢ 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑧 ∈ 𝑀 ↦ {𝑎 ∈ 𝑁 ∣ 𝜑}))
2	1	elovmpt3imp 7504	. . 3 ⊢ (𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) → (𝑋 ∈ V ∧ 𝑌 ∈ V))
3		simprl 767	. . . . 5 ⊢ ((𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾 ∈ 𝑈 ∧ 𝐿 ∈ 𝑇))) → (𝑋 ∈ V ∧ 𝑌 ∈ V))
4		elfvdm 6788	. . . . . . 7 ⊢ (𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) → 𝑍 ∈ dom (𝑋𝑂𝑌))
5		simpl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑋 ∈ V)
6	5	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾 ∈ 𝑈 ∧ 𝐿 ∈ 𝑇)) → 𝑋 ∈ V)
7		simplr 765	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾 ∈ 𝑈 ∧ 𝐿 ∈ 𝑇)) → 𝑌 ∈ V)
8		simprl 767	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾 ∈ 𝑈 ∧ 𝐿 ∈ 𝑇)) → 𝐾 ∈ 𝑈)
9		simprr 769	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾 ∈ 𝑈 ∧ 𝐿 ∈ 𝑇)) → 𝐿 ∈ 𝑇)
10		ovmpt3rab1.m	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑀 = 𝐾)
11		ovmpt3rab1.n	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑁 = 𝐿)
12	1, 10, 11	ovmpt3rabdm 7506	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝐾 ∈ 𝑈) ∧ 𝐿 ∈ 𝑇) → dom (𝑋𝑂𝑌) = 𝐾)
13	6, 7, 8, 9, 12	syl31anc 1371	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾 ∈ 𝑈 ∧ 𝐿 ∈ 𝑇)) → dom (𝑋𝑂𝑌) = 𝐾)
14	13	eleq2d 2824	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾 ∈ 𝑈 ∧ 𝐿 ∈ 𝑇)) → (𝑍 ∈ dom (𝑋𝑂𝑌) ↔ 𝑍 ∈ 𝐾))
15	14	biimpcd 248	. . . . . . . . . . 11 ⊢ (𝑍 ∈ dom (𝑋𝑂𝑌) → (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾 ∈ 𝑈 ∧ 𝐿 ∈ 𝑇)) → 𝑍 ∈ 𝐾))
16	15	adantr 480	. . . . . . . . . 10 ⊢ ((𝑍 ∈ dom (𝑋𝑂𝑌) ∧ 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍)) → (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾 ∈ 𝑈 ∧ 𝐿 ∈ 𝑇)) → 𝑍 ∈ 𝐾))
17	16	imp 406	. . . . . . . . 9 ⊢ (((𝑍 ∈ dom (𝑋𝑂𝑌) ∧ 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍)) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾 ∈ 𝑈 ∧ 𝐿 ∈ 𝑇))) → 𝑍 ∈ 𝐾)
18		simpl 482	. . . . . . . . . 10 ⊢ ((𝑍 ∈ 𝐾 ∧ ((𝑍 ∈ dom (𝑋𝑂𝑌) ∧ 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍)) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾 ∈ 𝑈 ∧ 𝐿 ∈ 𝑇)))) → 𝑍 ∈ 𝐾)
19		simplr 765	. . . . . . . . . . . . 13 ⊢ (((𝑍 ∈ dom (𝑋𝑂𝑌) ∧ 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍)) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾 ∈ 𝑈 ∧ 𝐿 ∈ 𝑇))) → 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍))
20	19	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑍 ∈ 𝐾 ∧ ((𝑍 ∈ dom (𝑋𝑂𝑌) ∧ 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍)) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾 ∈ 𝑈 ∧ 𝐿 ∈ 𝑇)))) → 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍))
21		simpl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ 𝑈 ∧ 𝐿 ∈ 𝑇) → 𝐾 ∈ 𝑈)
22	21	anim2i 616	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾 ∈ 𝑈 ∧ 𝐿 ∈ 𝑇)) → ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝐾 ∈ 𝑈))
23		df-3an 1087	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝐾 ∈ 𝑈) ↔ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝐾 ∈ 𝑈))
24	22, 23	sylibr 233	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾 ∈ 𝑈 ∧ 𝐿 ∈ 𝑇)) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝐾 ∈ 𝑈))
25	24	ad2antll 725	. . . . . . . . . . . . . 14 ⊢ ((𝑍 ∈ 𝐾 ∧ ((𝑍 ∈ dom (𝑋𝑂𝑌) ∧ 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍)) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾 ∈ 𝑈 ∧ 𝐿 ∈ 𝑇)))) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝐾 ∈ 𝑈))
26		sbceq1a 3722	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑌 → (𝜑 ↔ [𝑌 / 𝑦]𝜑))
27		sbceq1a 3722	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑋 → ([𝑌 / 𝑦]𝜑 ↔ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
28	26, 27	sylan9bbr 510	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝜑 ↔ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
29		nfsbc1v 3731	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
30		nfcv 2906	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦𝑋
31		nfsbc1v 3731	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦[𝑌 / 𝑦]𝜑
32	30, 31	nfsbcw 3733	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑦[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
33	1, 10, 11, 28, 29, 32	ovmpt3rab1 7505	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝐾 ∈ 𝑈) → (𝑋𝑂𝑌) = (𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑}))
34	33	fveq1d 6758	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝐾 ∈ 𝑈) → ((𝑋𝑂𝑌)‘𝑍) = ((𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑})‘𝑍))
35	25, 34	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑍 ∈ 𝐾 ∧ ((𝑍 ∈ dom (𝑋𝑂𝑌) ∧ 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍)) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾 ∈ 𝑈 ∧ 𝐿 ∈ 𝑇)))) → ((𝑋𝑂𝑌)‘𝑍) = ((𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑})‘𝑍))
36		rabexg 5250	. . . . . . . . . . . . . . . 16 ⊢ (𝐿 ∈ 𝑇 → {𝑎 ∈ 𝐿 ∣ [𝑍 / 𝑧][𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
37	36	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ 𝑈 ∧ 𝐿 ∈ 𝑇) → {𝑎 ∈ 𝐿 ∣ [𝑍 / 𝑧][𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
38	37	ad2antll 725	. . . . . . . . . . . . . 14 ⊢ (((𝑍 ∈ dom (𝑋𝑂𝑌) ∧ 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍)) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾 ∈ 𝑈 ∧ 𝐿 ∈ 𝑇))) → {𝑎 ∈ 𝐿 ∣ [𝑍 / 𝑧][𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
39		nfcv 2906	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑧𝑍
40		nfsbc1v 3731	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑧[𝑍 / 𝑧][𝑋 / 𝑥][𝑌 / 𝑦]𝜑
41		nfcv 2906	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑧𝐿
42	40, 41	nfrabw 3311	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑧{𝑎 ∈ 𝐿 ∣ [𝑍 / 𝑧][𝑋 / 𝑥][𝑌 / 𝑦]𝜑}
43		sbceq1a 3722	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑍 → ([𝑋 / 𝑥][𝑌 / 𝑦]𝜑 ↔ [𝑍 / 𝑧][𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
44	43	rabbidv 3404	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑍 → {𝑎 ∈ 𝐿 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑} = {𝑎 ∈ 𝐿 ∣ [𝑍 / 𝑧][𝑋 / 𝑥][𝑌 / 𝑦]𝜑})
45		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑}) = (𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑})
46	39, 42, 44, 45	fvmptf 6878	. . . . . . . . . . . . . 14 ⊢ ((𝑍 ∈ 𝐾 ∧ {𝑎 ∈ 𝐿 ∣ [𝑍 / 𝑧][𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V) → ((𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑})‘𝑍) = {𝑎 ∈ 𝐿 ∣ [𝑍 / 𝑧][𝑋 / 𝑥][𝑌 / 𝑦]𝜑})
47	38, 46	sylan2 592	. . . . . . . . . . . . 13 ⊢ ((𝑍 ∈ 𝐾 ∧ ((𝑍 ∈ dom (𝑋𝑂𝑌) ∧ 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍)) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾 ∈ 𝑈 ∧ 𝐿 ∈ 𝑇)))) → ((𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑})‘𝑍) = {𝑎 ∈ 𝐿 ∣ [𝑍 / 𝑧][𝑋 / 𝑥][𝑌 / 𝑦]𝜑})
48	35, 47	eqtr2d 2779	. . . . . . . . . . . 12 ⊢ ((𝑍 ∈ 𝐾 ∧ ((𝑍 ∈ dom (𝑋𝑂𝑌) ∧ 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍)) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾 ∈ 𝑈 ∧ 𝐿 ∈ 𝑇)))) → {𝑎 ∈ 𝐿 ∣ [𝑍 / 𝑧][𝑋 / 𝑥][𝑌 / 𝑦]𝜑} = ((𝑋𝑂𝑌)‘𝑍))
49	20, 48	eleqtrrd 2842	. . . . . . . . . . 11 ⊢ ((𝑍 ∈ 𝐾 ∧ ((𝑍 ∈ dom (𝑋𝑂𝑌) ∧ 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍)) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾 ∈ 𝑈 ∧ 𝐿 ∈ 𝑇)))) → 𝐴 ∈ {𝑎 ∈ 𝐿 ∣ [𝑍 / 𝑧][𝑋 / 𝑥][𝑌 / 𝑦]𝜑})
50		elrabi 3611	. . . . . . . . . . 11 ⊢ (𝐴 ∈ {𝑎 ∈ 𝐿 ∣ [𝑍 / 𝑧][𝑋 / 𝑥][𝑌 / 𝑦]𝜑} → 𝐴 ∈ 𝐿)
51	49, 50	syl 17	. . . . . . . . . 10 ⊢ ((𝑍 ∈ 𝐾 ∧ ((𝑍 ∈ dom (𝑋𝑂𝑌) ∧ 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍)) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾 ∈ 𝑈 ∧ 𝐿 ∈ 𝑇)))) → 𝐴 ∈ 𝐿)
52	18, 51	jca 511	. . . . . . . . 9 ⊢ ((𝑍 ∈ 𝐾 ∧ ((𝑍 ∈ dom (𝑋𝑂𝑌) ∧ 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍)) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾 ∈ 𝑈 ∧ 𝐿 ∈ 𝑇)))) → (𝑍 ∈ 𝐾 ∧ 𝐴 ∈ 𝐿))
53	17, 52	mpancom 684	. . . . . . . 8 ⊢ (((𝑍 ∈ dom (𝑋𝑂𝑌) ∧ 𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍)) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾 ∈ 𝑈 ∧ 𝐿 ∈ 𝑇))) → (𝑍 ∈ 𝐾 ∧ 𝐴 ∈ 𝐿))
54	53	exp31 419	. . . . . . 7 ⊢ (𝑍 ∈ dom (𝑋𝑂𝑌) → (𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) → (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾 ∈ 𝑈 ∧ 𝐿 ∈ 𝑇)) → (𝑍 ∈ 𝐾 ∧ 𝐴 ∈ 𝐿))))
55	4, 54	mpcom 38	. . . . . 6 ⊢ (𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) → (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾 ∈ 𝑈 ∧ 𝐿 ∈ 𝑇)) → (𝑍 ∈ 𝐾 ∧ 𝐴 ∈ 𝐿)))
56	55	imp 406	. . . . 5 ⊢ ((𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾 ∈ 𝑈 ∧ 𝐿 ∈ 𝑇))) → (𝑍 ∈ 𝐾 ∧ 𝐴 ∈ 𝐿))
57	3, 56	jca 511	. . . 4 ⊢ ((𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) ∧ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝐾 ∈ 𝑈 ∧ 𝐿 ∈ 𝑇))) → ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑍 ∈ 𝐾 ∧ 𝐴 ∈ 𝐿)))
58	57	exp32 420	. . 3 ⊢ (𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) → ((𝑋 ∈ V ∧ 𝑌 ∈ V) → ((𝐾 ∈ 𝑈 ∧ 𝐿 ∈ 𝑇) → ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑍 ∈ 𝐾 ∧ 𝐴 ∈ 𝐿)))))
59	2, 58	mpd 15	. 2 ⊢ (𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) → ((𝐾 ∈ 𝑈 ∧ 𝐿 ∈ 𝑇) → ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑍 ∈ 𝐾 ∧ 𝐴 ∈ 𝐿))))
60	59	com12 32	1 ⊢ ((𝐾 ∈ 𝑈 ∧ 𝐿 ∈ 𝑇) → (𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) → ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑍 ∈ 𝐾 ∧ 𝐴 ∈ 𝐿))))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  {crab 3067  Vcvv 3422  [wsbc 3711   ↦ cmpt 5153  dom cdm 5580  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260

This theorem is referenced by:  elovmpt3rab  7508  elovmptnn0wrd  14190