Theorem xpdom2 6845

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 6763	. 2 ⊢ (𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵)
2		f1f 5433	. . . . . . . 8 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴⟶𝐵)
3		ffvelcdm 5662	. . . . . . . . 9 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∪ ran {𝑥} ∈ 𝐴) → (𝑓‘∪ ran {𝑥}) ∈ 𝐵)
4	3	ex 115	. . . . . . . 8 ⊢ (𝑓:𝐴⟶𝐵 → (∪ ran {𝑥} ∈ 𝐴 → (𝑓‘∪ ran {𝑥}) ∈ 𝐵))
5	2, 4	syl 14	. . . . . . 7 ⊢ (𝑓:𝐴–1-1→𝐵 → (∪ ran {𝑥} ∈ 𝐴 → (𝑓‘∪ ran {𝑥}) ∈ 𝐵))
6	5	anim2d 337	. . . . . 6 ⊢ (𝑓:𝐴–1-1→𝐵 → ((∪ dom {𝑥} ∈ 𝐶 ∧ ∪ ran {𝑥} ∈ 𝐴) → (∪ dom {𝑥} ∈ 𝐶 ∧ (𝑓‘∪ ran {𝑥}) ∈ 𝐵)))
7	6	adantld 278	. . . . 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 5128	. . . . 5 ⊢ (𝑥 ∈ (𝐶 × 𝐴) ↔ (𝑥 = ⟨∪ dom {𝑥}, ∪ ran {𝑥}⟩ ∧ (∪ dom {𝑥} ∈ 𝐶 ∧ ∪ ran {𝑥} ∈ 𝐴)))
9		opelxp 4668	. . . . 5 ⊢ (⟨∪ dom {𝑥}, (𝑓‘∪ ran {𝑥})⟩ ∈ (𝐶 × 𝐵) ↔ (∪ dom {𝑥} ∈ 𝐶 ∧ (𝑓‘∪ ran {𝑥}) ∈ 𝐵))
10	7, 8, 9	3imtr4g 205	. . . 4 ⊢ (𝑓:𝐴–1-1→𝐵 → (𝑥 ∈ (𝐶 × 𝐴) → ⟨∪ dom {𝑥}, (𝑓‘∪ ran {𝑥})⟩ ∈ (𝐶 × 𝐵)))
11	10	adantl 277	. . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → (𝑥 ∈ (𝐶 × 𝐴) → ⟨∪ dom {𝑥}, (𝑓‘∪ ran {𝑥})⟩ ∈ (𝐶 × 𝐵)))
12		elxp2 4656	. . . . . 6 ⊢ (𝑥 ∈ (𝐶 × 𝐴) ↔ ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐴 𝑥 = ⟨𝑧, 𝑤⟩)
13		elxp2 4656	. . . . . 6 ⊢ (𝑦 ∈ (𝐶 × 𝐴) ↔ ∃𝑣 ∈ 𝐶 ∃𝑢 ∈ 𝐴 𝑦 = ⟨𝑣, 𝑢⟩)
14		vex 2752	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑧 ∈ V
15		vex 2752	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑓 ∈ V
16		vex 2752	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑤 ∈ V
17	15, 16	fvex 5547	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓‘𝑤) ∈ V
18	14, 17	opth 4249	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑧, (𝑓‘𝑤)⟩ = ⟨𝑣, (𝑓‘𝑢)⟩ ↔ (𝑧 = 𝑣 ∧ (𝑓‘𝑤) = (𝑓‘𝑢)))
19		f1fveq 5786	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ (𝑤 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴)) → ((𝑓‘𝑤) = (𝑓‘𝑢) ↔ 𝑤 = 𝑢))
20	19	ancoms 268	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴) ∧ 𝑓:𝐴–1-1→𝐵) → ((𝑓‘𝑤) = (𝑓‘𝑢) ↔ 𝑤 = 𝑢))
21	20	anbi2d 464	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴) ∧ 𝑓:𝐴–1-1→𝐵) → ((𝑧 = 𝑣 ∧ (𝑓‘𝑤) = (𝑓‘𝑢)) ↔ (𝑧 = 𝑣 ∧ 𝑤 = 𝑢)))
22	18, 21	bitrid 192	. . . . . . . . . . . . . . . 16 ⊢ (((𝑤 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴) ∧ 𝑓:𝐴–1-1→𝐵) → (⟨𝑧, (𝑓‘𝑤)⟩ = ⟨𝑣, (𝑓‘𝑢)⟩ ↔ (𝑧 = 𝑣 ∧ 𝑤 = 𝑢)))
23	22	ex 115	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴) → (𝑓:𝐴–1-1→𝐵 → (⟨𝑧, (𝑓‘𝑤)⟩ = ⟨𝑣, (𝑓‘𝑢)⟩ ↔ (𝑧 = 𝑣 ∧ 𝑤 = 𝑢))))
24	23	ad2ant2l 508	. . . . . . . . . . . . . 14 ⊢ (((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐴) ∧ (𝑣 ∈ 𝐶 ∧ 𝑢 ∈ 𝐴)) → (𝑓:𝐴–1-1→𝐵 → (⟨𝑧, (𝑓‘𝑤)⟩ = ⟨𝑣, (𝑓‘𝑢)⟩ ↔ (𝑧 = 𝑣 ∧ 𝑤 = 𝑢))))
25	24	imp 124	. . . . . . . . . . . . 13 ⊢ ((((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐴) ∧ (𝑣 ∈ 𝐶 ∧ 𝑢 ∈ 𝐴)) ∧ 𝑓:𝐴–1-1→𝐵) → (⟨𝑧, (𝑓‘𝑤)⟩ = ⟨𝑣, (𝑓‘𝑢)⟩ ↔ (𝑧 = 𝑣 ∧ 𝑤 = 𝑢)))
26	25	adantlr 477	. . . . . . . . . . . 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)))
27		sneq 3615	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ⟨𝑧, 𝑤⟩ → {𝑥} = {⟨𝑧, 𝑤⟩})
28	27	dmeqd 4841	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ⟨𝑧, 𝑤⟩ → dom {𝑥} = dom {⟨𝑧, 𝑤⟩})
29	28	unieqd 3832	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ⟨𝑧, 𝑤⟩ → ∪ dom {𝑥} = ∪ dom {⟨𝑧, 𝑤⟩})
30	14, 16	op1sta 5122	. . . . . . . . . . . . . . . 16 ⊢ ∪ dom {⟨𝑧, 𝑤⟩} = 𝑧
31	29, 30	eqtrdi 2236	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ⟨𝑧, 𝑤⟩ → ∪ dom {𝑥} = 𝑧)
32	27	rneqd 4868	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ⟨𝑧, 𝑤⟩ → ran {𝑥} = ran {⟨𝑧, 𝑤⟩})
33	32	unieqd 3832	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ⟨𝑧, 𝑤⟩ → ∪ ran {𝑥} = ∪ ran {⟨𝑧, 𝑤⟩})
34	14, 16	op2nda 5125	. . . . . . . . . . . . . . . . 17 ⊢ ∪ ran {⟨𝑧, 𝑤⟩} = 𝑤
35	33, 34	eqtrdi 2236	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ⟨𝑧, 𝑤⟩ → ∪ ran {𝑥} = 𝑤)
36	35	fveq2d 5531	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ⟨𝑧, 𝑤⟩ → (𝑓‘∪ ran {𝑥}) = (𝑓‘𝑤))
37	31, 36	opeq12d 3798	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ⟨𝑧, 𝑤⟩ → ⟨∪ dom {𝑥}, (𝑓‘∪ ran {𝑥})⟩ = ⟨𝑧, (𝑓‘𝑤)⟩)
38		sneq 3615	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ⟨𝑣, 𝑢⟩ → {𝑦} = {⟨𝑣, 𝑢⟩})
39	38	dmeqd 4841	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ⟨𝑣, 𝑢⟩ → dom {𝑦} = dom {⟨𝑣, 𝑢⟩})
40	39	unieqd 3832	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ⟨𝑣, 𝑢⟩ → ∪ dom {𝑦} = ∪ dom {⟨𝑣, 𝑢⟩})
41		vex 2752	. . . . . . . . . . . . . . . . 17 ⊢ 𝑣 ∈ V
42		vex 2752	. . . . . . . . . . . . . . . . 17 ⊢ 𝑢 ∈ V
43	41, 42	op1sta 5122	. . . . . . . . . . . . . . . 16 ⊢ ∪ dom {⟨𝑣, 𝑢⟩} = 𝑣
44	40, 43	eqtrdi 2236	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ⟨𝑣, 𝑢⟩ → ∪ dom {𝑦} = 𝑣)
45	38	rneqd 4868	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ⟨𝑣, 𝑢⟩ → ran {𝑦} = ran {⟨𝑣, 𝑢⟩})
46	45	unieqd 3832	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ⟨𝑣, 𝑢⟩ → ∪ ran {𝑦} = ∪ ran {⟨𝑣, 𝑢⟩})
47	41, 42	op2nda 5125	. . . . . . . . . . . . . . . . 17 ⊢ ∪ ran {⟨𝑣, 𝑢⟩} = 𝑢
48	46, 47	eqtrdi 2236	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ⟨𝑣, 𝑢⟩ → ∪ ran {𝑦} = 𝑢)
49	48	fveq2d 5531	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ⟨𝑣, 𝑢⟩ → (𝑓‘∪ ran {𝑦}) = (𝑓‘𝑢))
50	44, 49	opeq12d 3798	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ⟨𝑣, 𝑢⟩ → ⟨∪ dom {𝑦}, (𝑓‘∪ ran {𝑦})⟩ = ⟨𝑣, (𝑓‘𝑢)⟩)
51	37, 50	eqeqan12d 2203	. . . . . . . . . . . . 13 ⊢ ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) → (⟨∪ dom {𝑥}, (𝑓‘∪ ran {𝑥})⟩ = ⟨∪ dom {𝑦}, (𝑓‘∪ ran {𝑦})⟩ ↔ ⟨𝑧, (𝑓‘𝑤)⟩ = ⟨𝑣, (𝑓‘𝑢)⟩))
52	51	ad2antlr 489	. . . . . . . . . . . 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)⟩))
53		eqeq12 2200	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) → (𝑥 = 𝑦 ↔ ⟨𝑧, 𝑤⟩ = ⟨𝑣, 𝑢⟩))
54	14, 16	opth 4249	. . . . . . . . . . . . . 14 ⊢ (⟨𝑧, 𝑤⟩ = ⟨𝑣, 𝑢⟩ ↔ (𝑧 = 𝑣 ∧ 𝑤 = 𝑢))
55	53, 54	bitrdi 196	. . . . . . . . . . . . 13 ⊢ ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) → (𝑥 = 𝑦 ↔ (𝑧 = 𝑣 ∧ 𝑤 = 𝑢)))
56	55	ad2antlr 489	. . . . . . . . . . . 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)))
57	26, 52, 56	3bitr4d 220	. . . . . . . . . . 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))
58	57	exp53 377	. . . . . . . . . 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))))))
59	58	com23 78	. . . . . . . . 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))))))
60	59	rexlimivv 2610	. . . . . . . 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)))))
61	60	rexlimdvv 2611	. . . . . . 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))))
62	61	imp 124	. . . . . 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)))
63	12, 13, 62	syl2anb 291	. . . . 5 ⊢ ((𝑥 ∈ (𝐶 × 𝐴) ∧ 𝑦 ∈ (𝐶 × 𝐴)) → (𝑓:𝐴–1-1→𝐵 → (⟨∪ dom {𝑥}, (𝑓‘∪ ran {𝑥})⟩ = ⟨∪ dom {𝑦}, (𝑓‘∪ ran {𝑦})⟩ ↔ 𝑥 = 𝑦)))
64	63	com12 30	. . . 4 ⊢ (𝑓:𝐴–1-1→𝐵 → ((𝑥 ∈ (𝐶 × 𝐴) ∧ 𝑦 ∈ (𝐶 × 𝐴)) → (⟨∪ dom {𝑥}, (𝑓‘∪ ran {𝑥})⟩ = ⟨∪ dom {𝑦}, (𝑓‘∪ ran {𝑦})⟩ ↔ 𝑥 = 𝑦)))
65	64	adantl 277	. . 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)))
66		xpdom.2	. . . . 5 ⊢ 𝐶 ∈ V
67		reldom 6759	. . . . . 6 ⊢ Rel ≼
68	67	brrelex1i 4681	. . . . 5 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ∈ V)
69		xpexg 4752	. . . . 5 ⊢ ((𝐶 ∈ V ∧ 𝐴 ∈ V) → (𝐶 × 𝐴) ∈ V)
70	66, 68, 69	sylancr 414	. . . 4 ⊢ (𝐴 ≼ 𝐵 → (𝐶 × 𝐴) ∈ V)
71	70	adantr 276	. . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → (𝐶 × 𝐴) ∈ V)
72	67	brrelex2i 4682	. . . . 5 ⊢ (𝐴 ≼ 𝐵 → 𝐵 ∈ V)
73		xpexg 4752	. . . . 5 ⊢ ((𝐶 ∈ V ∧ 𝐵 ∈ V) → (𝐶 × 𝐵) ∈ V)
74	66, 72, 73	sylancr 414	. . . 4 ⊢ (𝐴 ≼ 𝐵 → (𝐶 × 𝐵) ∈ V)
75	74	adantr 276	. . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → (𝐶 × 𝐵) ∈ V)
76	11, 65, 71, 75	dom3d 6788	. 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵))
77	1, 76	exlimddv 1908	1 ⊢ (𝐴 ≼ 𝐵 → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1363   ∈ wcel 2158  ∃wrex 2466  Vcvv 2749  {csn 3604  ⟨cop 3607  ∪ cuni 3821   class class class wbr 4015   × cxp 4636  dom cdm 4638  ran crn 4639  ⟶wf 5224  –1-1→wf1 5225  ‘cfv 5228   ≼ cdom 6753

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fv 5236  df-dom 6756

This theorem is referenced by:  xpdom2g  6846  xpct  12411