Theorem xpdom2 8807

Description: Dominance law for Cartesian product. Proposition 10.33(2) of [TakeutiZaring] p. 92. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)

Hypothesis

Ref	Expression
xpdom.2	⊢ 𝐶 ∈ V

Assertion

Ref	Expression
xpdom2	⊢ (𝐴 ≼ 𝐵 → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵))

Proof of Theorem xpdom2

Dummy variables 𝑢 𝑓 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brdomi 8704	. 2 ⊢ (𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵)
2		f1f 6654	. . . . . . . 8 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴⟶𝐵)
3		ffvelrn 6941	. . . . . . . . 9 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∪ ran {𝑥} ∈ 𝐴) → (𝑓‘∪ ran {𝑥}) ∈ 𝐵)
4	3	ex 412	. . . . . . . 8 ⊢ (𝑓:𝐴⟶𝐵 → (∪ ran {𝑥} ∈ 𝐴 → (𝑓‘∪ ran {𝑥}) ∈ 𝐵))
5	2, 4	syl 17	. . . . . . 7 ⊢ (𝑓:𝐴–1-1→𝐵 → (∪ ran {𝑥} ∈ 𝐴 → (𝑓‘∪ ran {𝑥}) ∈ 𝐵))
6	5	anim2d 611	. . . . . 6 ⊢ (𝑓:𝐴–1-1→𝐵 → ((∪ dom {𝑥} ∈ 𝐶 ∧ ∪ ran {𝑥} ∈ 𝐴) → (∪ dom {𝑥} ∈ 𝐶 ∧ (𝑓‘∪ ran {𝑥}) ∈ 𝐵)))
7	6	adantld 490	. . . . 5 ⊢ (𝑓:𝐴–1-1→𝐵 → ((𝑥 = ⟨∪ dom {𝑥}, ∪ ran {𝑥}⟩ ∧ (∪ dom {𝑥} ∈ 𝐶 ∧ ∪ ran {𝑥} ∈ 𝐴)) → (∪ dom {𝑥} ∈ 𝐶 ∧ (𝑓‘∪ ran {𝑥}) ∈ 𝐵)))
8		elxp4 7743	. . . . 5 ⊢ (𝑥 ∈ (𝐶 × 𝐴) ↔ (𝑥 = ⟨∪ dom {𝑥}, ∪ ran {𝑥}⟩ ∧ (∪ dom {𝑥} ∈ 𝐶 ∧ ∪ ran {𝑥} ∈ 𝐴)))
9		opelxp 5616	. . . . 5 ⊢ (⟨∪ dom {𝑥}, (𝑓‘∪ ran {𝑥})⟩ ∈ (𝐶 × 𝐵) ↔ (∪ dom {𝑥} ∈ 𝐶 ∧ (𝑓‘∪ ran {𝑥}) ∈ 𝐵))
10	7, 8, 9	3imtr4g 295	. . . 4 ⊢ (𝑓:𝐴–1-1→𝐵 → (𝑥 ∈ (𝐶 × 𝐴) → ⟨∪ dom {𝑥}, (𝑓‘∪ ran {𝑥})⟩ ∈ (𝐶 × 𝐵)))
11	10	adantl 481	. . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → (𝑥 ∈ (𝐶 × 𝐴) → ⟨∪ dom {𝑥}, (𝑓‘∪ ran {𝑥})⟩ ∈ (𝐶 × 𝐵)))
12		elxp2 5604	. . . . . 6 ⊢ (𝑥 ∈ (𝐶 × 𝐴) ↔ ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐴 𝑥 = ⟨𝑧, 𝑤⟩)
13		elxp2 5604	. . . . . 6 ⊢ (𝑦 ∈ (𝐶 × 𝐴) ↔ ∃𝑣 ∈ 𝐶 ∃𝑢 ∈ 𝐴 𝑦 = ⟨𝑣, 𝑢⟩)
14		vex 3426	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑧 ∈ V
15		fvex 6769	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓‘𝑤) ∈ V
16	14, 15	opth 5385	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑧, (𝑓‘𝑤)⟩ = ⟨𝑣, (𝑓‘𝑢)⟩ ↔ (𝑧 = 𝑣 ∧ (𝑓‘𝑤) = (𝑓‘𝑢)))
17		f1fveq 7116	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ (𝑤 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴)) → ((𝑓‘𝑤) = (𝑓‘𝑢) ↔ 𝑤 = 𝑢))
18	17	ancoms 458	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴) ∧ 𝑓:𝐴–1-1→𝐵) → ((𝑓‘𝑤) = (𝑓‘𝑢) ↔ 𝑤 = 𝑢))
19	18	anbi2d 628	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴) ∧ 𝑓:𝐴–1-1→𝐵) → ((𝑧 = 𝑣 ∧ (𝑓‘𝑤) = (𝑓‘𝑢)) ↔ (𝑧 = 𝑣 ∧ 𝑤 = 𝑢)))
20	16, 19	syl5bb 282	. . . . . . . . . . . . . . . 16 ⊢ (((𝑤 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴) ∧ 𝑓:𝐴–1-1→𝐵) → (⟨𝑧, (𝑓‘𝑤)⟩ = ⟨𝑣, (𝑓‘𝑢)⟩ ↔ (𝑧 = 𝑣 ∧ 𝑤 = 𝑢)))
21	20	ex 412	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴) → (𝑓:𝐴–1-1→𝐵 → (⟨𝑧, (𝑓‘𝑤)⟩ = ⟨𝑣, (𝑓‘𝑢)⟩ ↔ (𝑧 = 𝑣 ∧ 𝑤 = 𝑢))))
22	21	ad2ant2l 742	. . . . . . . . . . . . . 14 ⊢ (((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐴) ∧ (𝑣 ∈ 𝐶 ∧ 𝑢 ∈ 𝐴)) → (𝑓:𝐴–1-1→𝐵 → (⟨𝑧, (𝑓‘𝑤)⟩ = ⟨𝑣, (𝑓‘𝑢)⟩ ↔ (𝑧 = 𝑣 ∧ 𝑤 = 𝑢))))
23	22	imp 406	. . . . . . . . . . . . 13 ⊢ ((((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐴) ∧ (𝑣 ∈ 𝐶 ∧ 𝑢 ∈ 𝐴)) ∧ 𝑓:𝐴–1-1→𝐵) → (⟨𝑧, (𝑓‘𝑤)⟩ = ⟨𝑣, (𝑓‘𝑢)⟩ ↔ (𝑧 = 𝑣 ∧ 𝑤 = 𝑢)))
24	23	adantlr 711	. . . . . . . . . . . 12 ⊢ (((((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐴) ∧ (𝑣 ∈ 𝐶 ∧ 𝑢 ∈ 𝐴)) ∧ (𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩)) ∧ 𝑓:𝐴–1-1→𝐵) → (⟨𝑧, (𝑓‘𝑤)⟩ = ⟨𝑣, (𝑓‘𝑢)⟩ ↔ (𝑧 = 𝑣 ∧ 𝑤 = 𝑢)))
25		sneq 4568	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ⟨𝑧, 𝑤⟩ → {𝑥} = {⟨𝑧, 𝑤⟩})
26	25	dmeqd 5803	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ⟨𝑧, 𝑤⟩ → dom {𝑥} = dom {⟨𝑧, 𝑤⟩})
27	26	unieqd 4850	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ⟨𝑧, 𝑤⟩ → ∪ dom {𝑥} = ∪ dom {⟨𝑧, 𝑤⟩})
28		vex 3426	. . . . . . . . . . . . . . . . 17 ⊢ 𝑤 ∈ V
29	14, 28	op1sta 6117	. . . . . . . . . . . . . . . 16 ⊢ ∪ dom {⟨𝑧, 𝑤⟩} = 𝑧
30	27, 29	eqtrdi 2795	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ⟨𝑧, 𝑤⟩ → ∪ dom {𝑥} = 𝑧)
31	25	rneqd 5836	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ⟨𝑧, 𝑤⟩ → ran {𝑥} = ran {⟨𝑧, 𝑤⟩})
32	31	unieqd 4850	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ⟨𝑧, 𝑤⟩ → ∪ ran {𝑥} = ∪ ran {⟨𝑧, 𝑤⟩})
33	14, 28	op2nda 6120	. . . . . . . . . . . . . . . . 17 ⊢ ∪ ran {⟨𝑧, 𝑤⟩} = 𝑤
34	32, 33	eqtrdi 2795	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ⟨𝑧, 𝑤⟩ → ∪ ran {𝑥} = 𝑤)
35	34	fveq2d 6760	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ⟨𝑧, 𝑤⟩ → (𝑓‘∪ ran {𝑥}) = (𝑓‘𝑤))
36	30, 35	opeq12d 4809	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ⟨𝑧, 𝑤⟩ → ⟨∪ dom {𝑥}, (𝑓‘∪ ran {𝑥})⟩ = ⟨𝑧, (𝑓‘𝑤)⟩)
37		sneq 4568	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ⟨𝑣, 𝑢⟩ → {𝑦} = {⟨𝑣, 𝑢⟩})
38	37	dmeqd 5803	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ⟨𝑣, 𝑢⟩ → dom {𝑦} = dom {⟨𝑣, 𝑢⟩})
39	38	unieqd 4850	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ⟨𝑣, 𝑢⟩ → ∪ dom {𝑦} = ∪ dom {⟨𝑣, 𝑢⟩})
40		vex 3426	. . . . . . . . . . . . . . . . 17 ⊢ 𝑣 ∈ V
41		vex 3426	. . . . . . . . . . . . . . . . 17 ⊢ 𝑢 ∈ V
42	40, 41	op1sta 6117	. . . . . . . . . . . . . . . 16 ⊢ ∪ dom {⟨𝑣, 𝑢⟩} = 𝑣
43	39, 42	eqtrdi 2795	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ⟨𝑣, 𝑢⟩ → ∪ dom {𝑦} = 𝑣)
44	37	rneqd 5836	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ⟨𝑣, 𝑢⟩ → ran {𝑦} = ran {⟨𝑣, 𝑢⟩})
45	44	unieqd 4850	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ⟨𝑣, 𝑢⟩ → ∪ ran {𝑦} = ∪ ran {⟨𝑣, 𝑢⟩})
46	40, 41	op2nda 6120	. . . . . . . . . . . . . . . . 17 ⊢ ∪ ran {⟨𝑣, 𝑢⟩} = 𝑢
47	45, 46	eqtrdi 2795	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ⟨𝑣, 𝑢⟩ → ∪ ran {𝑦} = 𝑢)
48	47	fveq2d 6760	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ⟨𝑣, 𝑢⟩ → (𝑓‘∪ ran {𝑦}) = (𝑓‘𝑢))
49	43, 48	opeq12d 4809	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ⟨𝑣, 𝑢⟩ → ⟨∪ dom {𝑦}, (𝑓‘∪ ran {𝑦})⟩ = ⟨𝑣, (𝑓‘𝑢)⟩)
50	36, 49	eqeqan12d 2752	. . . . . . . . . . . . 13 ⊢ ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) → (⟨∪ dom {𝑥}, (𝑓‘∪ ran {𝑥})⟩ = ⟨∪ dom {𝑦}, (𝑓‘∪ ran {𝑦})⟩ ↔ ⟨𝑧, (𝑓‘𝑤)⟩ = ⟨𝑣, (𝑓‘𝑢)⟩))
51	50	ad2antlr 723	. . . . . . . . . . . 12 ⊢ (((((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐴) ∧ (𝑣 ∈ 𝐶 ∧ 𝑢 ∈ 𝐴)) ∧ (𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩)) ∧ 𝑓:𝐴–1-1→𝐵) → (⟨∪ dom {𝑥}, (𝑓‘∪ ran {𝑥})⟩ = ⟨∪ dom {𝑦}, (𝑓‘∪ ran {𝑦})⟩ ↔ ⟨𝑧, (𝑓‘𝑤)⟩ = ⟨𝑣, (𝑓‘𝑢)⟩))
52		eqeq12 2755	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) → (𝑥 = 𝑦 ↔ ⟨𝑧, 𝑤⟩ = ⟨𝑣, 𝑢⟩))
53	14, 28	opth 5385	. . . . . . . . . . . . . 14 ⊢ (⟨𝑧, 𝑤⟩ = ⟨𝑣, 𝑢⟩ ↔ (𝑧 = 𝑣 ∧ 𝑤 = 𝑢))
54	52, 53	bitrdi 286	. . . . . . . . . . . . 13 ⊢ ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) → (𝑥 = 𝑦 ↔ (𝑧 = 𝑣 ∧ 𝑤 = 𝑢)))
55	54	ad2antlr 723	. . . . . . . . . . . 12 ⊢ (((((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐴) ∧ (𝑣 ∈ 𝐶 ∧ 𝑢 ∈ 𝐴)) ∧ (𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩)) ∧ 𝑓:𝐴–1-1→𝐵) → (𝑥 = 𝑦 ↔ (𝑧 = 𝑣 ∧ 𝑤 = 𝑢)))
56	24, 51, 55	3bitr4d 310	. . . . . . . . . . 11 ⊢ (((((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐴) ∧ (𝑣 ∈ 𝐶 ∧ 𝑢 ∈ 𝐴)) ∧ (𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩)) ∧ 𝑓:𝐴–1-1→𝐵) → (⟨∪ dom {𝑥}, (𝑓‘∪ ran {𝑥})⟩ = ⟨∪ dom {𝑦}, (𝑓‘∪ ran {𝑦})⟩ ↔ 𝑥 = 𝑦))
57	56	exp53 447	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐴) → ((𝑣 ∈ 𝐶 ∧ 𝑢 ∈ 𝐴) → (𝑥 = ⟨𝑧, 𝑤⟩ → (𝑦 = ⟨𝑣, 𝑢⟩ → (𝑓:𝐴–1-1→𝐵 → (⟨∪ dom {𝑥}, (𝑓‘∪ ran {𝑥})⟩ = ⟨∪ dom {𝑦}, (𝑓‘∪ ran {𝑦})⟩ ↔ 𝑥 = 𝑦))))))
58	57	com23 86	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐴) → (𝑥 = ⟨𝑧, 𝑤⟩ → ((𝑣 ∈ 𝐶 ∧ 𝑢 ∈ 𝐴) → (𝑦 = ⟨𝑣, 𝑢⟩ → (𝑓:𝐴–1-1→𝐵 → (⟨∪ dom {𝑥}, (𝑓‘∪ ran {𝑥})⟩ = ⟨∪ dom {𝑦}, (𝑓‘∪ ran {𝑦})⟩ ↔ 𝑥 = 𝑦))))))
59	58	rexlimivv 3220	. . . . . . . 8 ⊢ (∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐴 𝑥 = ⟨𝑧, 𝑤⟩ → ((𝑣 ∈ 𝐶 ∧ 𝑢 ∈ 𝐴) → (𝑦 = ⟨𝑣, 𝑢⟩ → (𝑓:𝐴–1-1→𝐵 → (⟨∪ dom {𝑥}, (𝑓‘∪ ran {𝑥})⟩ = ⟨∪ dom {𝑦}, (𝑓‘∪ ran {𝑦})⟩ ↔ 𝑥 = 𝑦)))))
60	59	rexlimdvv 3221	. . . . . . 7 ⊢ (∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐴 𝑥 = ⟨𝑧, 𝑤⟩ → (∃𝑣 ∈ 𝐶 ∃𝑢 ∈ 𝐴 𝑦 = ⟨𝑣, 𝑢⟩ → (𝑓:𝐴–1-1→𝐵 → (⟨∪ dom {𝑥}, (𝑓‘∪ ran {𝑥})⟩ = ⟨∪ dom {𝑦}, (𝑓‘∪ ran {𝑦})⟩ ↔ 𝑥 = 𝑦))))
61	60	imp 406	. . . . . 6 ⊢ ((∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐴 𝑥 = ⟨𝑧, 𝑤⟩ ∧ ∃𝑣 ∈ 𝐶 ∃𝑢 ∈ 𝐴 𝑦 = ⟨𝑣, 𝑢⟩) → (𝑓:𝐴–1-1→𝐵 → (⟨∪ dom {𝑥}, (𝑓‘∪ ran {𝑥})⟩ = ⟨∪ dom {𝑦}, (𝑓‘∪ ran {𝑦})⟩ ↔ 𝑥 = 𝑦)))
62	12, 13, 61	syl2anb 597	. . . . 5 ⊢ ((𝑥 ∈ (𝐶 × 𝐴) ∧ 𝑦 ∈ (𝐶 × 𝐴)) → (𝑓:𝐴–1-1→𝐵 → (⟨∪ dom {𝑥}, (𝑓‘∪ ran {𝑥})⟩ = ⟨∪ dom {𝑦}, (𝑓‘∪ ran {𝑦})⟩ ↔ 𝑥 = 𝑦)))
63	62	com12 32	. . . 4 ⊢ (𝑓:𝐴–1-1→𝐵 → ((𝑥 ∈ (𝐶 × 𝐴) ∧ 𝑦 ∈ (𝐶 × 𝐴)) → (⟨∪ dom {𝑥}, (𝑓‘∪ ran {𝑥})⟩ = ⟨∪ dom {𝑦}, (𝑓‘∪ ran {𝑦})⟩ ↔ 𝑥 = 𝑦)))
64	63	adantl 481	. . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ((𝑥 ∈ (𝐶 × 𝐴) ∧ 𝑦 ∈ (𝐶 × 𝐴)) → (⟨∪ dom {𝑥}, (𝑓‘∪ ran {𝑥})⟩ = ⟨∪ dom {𝑦}, (𝑓‘∪ ran {𝑦})⟩ ↔ 𝑥 = 𝑦)))
65		xpdom.2	. . . . 5 ⊢ 𝐶 ∈ V
66		reldom 8697	. . . . . 6 ⊢ Rel ≼
67	66	brrelex1i 5634	. . . . 5 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ∈ V)
68		xpexg 7578	. . . . 5 ⊢ ((𝐶 ∈ V ∧ 𝐴 ∈ V) → (𝐶 × 𝐴) ∈ V)
69	65, 67, 68	sylancr 586	. . . 4 ⊢ (𝐴 ≼ 𝐵 → (𝐶 × 𝐴) ∈ V)
70	69	adantr 480	. . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → (𝐶 × 𝐴) ∈ V)
71	66	brrelex2i 5635	. . . . 5 ⊢ (𝐴 ≼ 𝐵 → 𝐵 ∈ V)
72		xpexg 7578	. . . . 5 ⊢ ((𝐶 ∈ V ∧ 𝐵 ∈ V) → (𝐶 × 𝐵) ∈ V)
73	65, 71, 72	sylancr 586	. . . 4 ⊢ (𝐴 ≼ 𝐵 → (𝐶 × 𝐵) ∈ V)
74	73	adantr 480	. . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → (𝐶 × 𝐵) ∈ V)
75	11, 64, 70, 74	dom3d 8737	. 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵))
76	1, 75	exlimddv 1939	1 ⊢ (𝐴 ≼ 𝐵 → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∃wrex 3064  Vcvv 3422  {csn 4558  ⟨cop 4564  ∪ cuni 4836   class class class wbr 5070   × cxp 5578  dom cdm 5580  ran crn 5581  ⟶wf 6414  –1-1→wf1 6415  ‘cfv 6418   ≼ cdom 8689

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-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

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-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-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  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-fv 6426  df-dom 8693

This theorem is referenced by:  xpdom2g  8808  infxpenlem  9700  xpct  9703  djudom1  9869  cfpwsdom  10271  inar1  10462  rexpen  15865  2ndcctbss  22514  tx2ndc  22710  met2ndci  23584  mbfimaopnlem  24724