Theorem resixpfo 8934

Description: Restriction of elements of an infinite Cartesian product creates a surjection, if the original Cartesian product is nonempty. (Contributed by Mario Carneiro, 27-Aug-2015.)

Hypothesis

Ref	Expression
resixpfo.1	⊢ 𝐹 = (𝑓 ∈ X𝑥 ∈ 𝐴 𝐶 ↦ (𝑓 ↾ 𝐵))

Assertion

Ref	Expression
resixpfo	⊢ ((𝐵 ⊆ 𝐴 ∧ X𝑥 ∈ 𝐴 𝐶 ≠ ∅) → 𝐹:X𝑥 ∈ 𝐴 𝐶–onto→X𝑥 ∈ 𝐵 𝐶)

Distinct variable groups:   𝑥,𝑓,𝐴   𝐵,𝑓,𝑥   𝐶,𝑓

Allowed substitution hints:   𝐶(𝑥)   𝐹(𝑥,𝑓)

Proof of Theorem resixpfo

Dummy variables 𝑔 ℎ 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		resixp 8931	. . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑓 ∈ X𝑥 ∈ 𝐴 𝐶) → (𝑓 ↾ 𝐵) ∈ X𝑥 ∈ 𝐵 𝐶)
2		resixpfo.1	. . . 4 ⊢ 𝐹 = (𝑓 ∈ X𝑥 ∈ 𝐴 𝐶 ↦ (𝑓 ↾ 𝐵))
3	1, 2	fmptd 7110	. . 3 ⊢ (𝐵 ⊆ 𝐴 → 𝐹:X𝑥 ∈ 𝐴 𝐶⟶X𝑥 ∈ 𝐵 𝐶)
4	3	adantr 485	. 2 ⊢ ((𝐵 ⊆ 𝐴 ∧ X𝑥 ∈ 𝐴 𝐶 ≠ ∅) → 𝐹:X𝑥 ∈ 𝐴 𝐶⟶X𝑥 ∈ 𝐵 𝐶)
5		n0 4315	. . . 4 ⊢ (X𝑥 ∈ 𝐴 𝐶 ≠ ∅ ↔ ∃𝑔 𝑔 ∈ X𝑥 ∈ 𝐴 𝐶)
6		eleq1w 2852	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
7	6	ifbid 4516	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → if(𝑧 ∈ 𝐵, ℎ, 𝑔) = if(𝑥 ∈ 𝐵, ℎ, 𝑔))
8		id 23	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → 𝑧 = 𝑥)
9	7, 8	fveq12d 6889	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧) = (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥))
10	9	cbvmptv 5219	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐴 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧)) = (𝑥 ∈ 𝐴 ↦ (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥))
11		vex 3467	. . . . . . . . . . . . . . . 16 ⊢ ℎ ∈ V
12	11	elixp 8902	. . . . . . . . . . . . . . 15 ⊢ (ℎ ∈ X𝑥 ∈ 𝐵 𝐶 ↔ (ℎ Fn 𝐵 ∧ ∀𝑥 ∈ 𝐵 (ℎ‘𝑥) ∈ 𝐶))
13	12	simprbi 502	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ X𝑥 ∈ 𝐵 𝐶 → ∀𝑥 ∈ 𝐵 (ℎ‘𝑥) ∈ 𝐶)
14		fveq1 6881	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ = if(𝑥 ∈ 𝐵, ℎ, 𝑔) → (ℎ‘𝑥) = (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥))
15	14	eleq1d 2854	. . . . . . . . . . . . . . . . 17 ⊢ (ℎ = if(𝑥 ∈ 𝐵, ℎ, 𝑔) → ((ℎ‘𝑥) ∈ 𝐶 ↔ (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥) ∈ 𝐶))
16		fveq1 6881	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 = if(𝑥 ∈ 𝐵, ℎ, 𝑔) → (𝑔‘𝑥) = (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥))
17	16	eleq1d 2854	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = if(𝑥 ∈ 𝐵, ℎ, 𝑔) → ((𝑔‘𝑥) ∈ 𝐶 ↔ (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥) ∈ 𝐶))
18		simpl 487	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ 𝐵 → (ℎ‘𝑥) ∈ 𝐶) ∧ (𝑥 ∈ 𝐴 ∧ (𝑔‘𝑥) ∈ 𝐶)) → (𝑥 ∈ 𝐵 → (ℎ‘𝑥) ∈ 𝐶))
19	18	imp 411	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ 𝐵 → (ℎ‘𝑥) ∈ 𝐶) ∧ (𝑥 ∈ 𝐴 ∧ (𝑔‘𝑥) ∈ 𝐶)) ∧ 𝑥 ∈ 𝐵) → (ℎ‘𝑥) ∈ 𝐶)
20		simplrr 789	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ 𝐵 → (ℎ‘𝑥) ∈ 𝐶) ∧ (𝑥 ∈ 𝐴 ∧ (𝑔‘𝑥) ∈ 𝐶)) ∧ ¬ 𝑥 ∈ 𝐵) → (𝑔‘𝑥) ∈ 𝐶)
21	15, 17, 19, 20	ifbothda 4531	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ 𝐵 → (ℎ‘𝑥) ∈ 𝐶) ∧ (𝑥 ∈ 𝐴 ∧ (𝑔‘𝑥) ∈ 𝐶)) → (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥) ∈ 𝐶)
22	21	exp32 425	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝐵 → (ℎ‘𝑥) ∈ 𝐶) → (𝑥 ∈ 𝐴 → ((𝑔‘𝑥) ∈ 𝐶 → (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥) ∈ 𝐶)))
23	22	ralimi2 3103	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ 𝐵 (ℎ‘𝑥) ∈ 𝐶 → ∀𝑥 ∈ 𝐴 ((𝑔‘𝑥) ∈ 𝐶 → (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥) ∈ 𝐶))
24	13, 23	syl 18	. . . . . . . . . . . . 13 ⊢ (ℎ ∈ X𝑥 ∈ 𝐵 𝐶 → ∀𝑥 ∈ 𝐴 ((𝑔‘𝑥) ∈ 𝐶 → (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥) ∈ 𝐶))
25	24	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝐵 ⊆ 𝐴 ∧ ℎ ∈ X𝑥 ∈ 𝐵 𝐶) → ∀𝑥 ∈ 𝐴 ((𝑔‘𝑥) ∈ 𝐶 → (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥) ∈ 𝐶))
26		ralim 3111	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝐴 ((𝑔‘𝑥) ∈ 𝐶 → (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥) ∈ 𝐶) → (∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐶 → ∀𝑥 ∈ 𝐴 (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥) ∈ 𝐶))
27	25, 26	syl 18	. . . . . . . . . . 11 ⊢ ((𝐵 ⊆ 𝐴 ∧ ℎ ∈ X𝑥 ∈ 𝐵 𝐶) → (∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐶 → ∀𝑥 ∈ 𝐴 (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥) ∈ 𝐶))
28		vex 3467	. . . . . . . . . . . . 13 ⊢ 𝑔 ∈ V
29	28	elixp 8902	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 𝐶 ↔ (𝑔 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐶))
30	29	simprbi 502	. . . . . . . . . . 11 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 𝐶 → ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐶)
31	27, 30	impel 514	. . . . . . . . . 10 ⊢ (((𝐵 ⊆ 𝐴 ∧ ℎ ∈ X𝑥 ∈ 𝐵 𝐶) ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐶) → ∀𝑥 ∈ 𝐴 (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥) ∈ 𝐶)
32		n0i 4301	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 𝐶 → ¬ X𝑥 ∈ 𝐴 𝐶 = ∅)
33		ixpprc 8917	. . . . . . . . . . . . 13 ⊢ (¬ 𝐴 ∈ V → X𝑥 ∈ 𝐴 𝐶 = ∅)
34	32, 33	nsyl2 142	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 𝐶 → 𝐴 ∈ V)
35	34	adantl 486	. . . . . . . . . . 11 ⊢ (((𝐵 ⊆ 𝐴 ∧ ℎ ∈ X𝑥 ∈ 𝐵 𝐶) ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐶) → 𝐴 ∈ V)
36		mptelixpg 8933	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → ((𝑥 ∈ 𝐴 ↦ (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥)) ∈ X𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥) ∈ 𝐶))
37	35, 36	syl 18	. . . . . . . . . 10 ⊢ (((𝐵 ⊆ 𝐴 ∧ ℎ ∈ X𝑥 ∈ 𝐵 𝐶) ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐶) → ((𝑥 ∈ 𝐴 ↦ (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥)) ∈ X𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥) ∈ 𝐶))
38	31, 37	mpbird 260	. . . . . . . . 9 ⊢ (((𝐵 ⊆ 𝐴 ∧ ℎ ∈ X𝑥 ∈ 𝐵 𝐶) ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐶) → (𝑥 ∈ 𝐴 ↦ (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥)) ∈ X𝑥 ∈ 𝐴 𝐶)
39	10, 38	eqeltrid 2873	. . . . . . . 8 ⊢ (((𝐵 ⊆ 𝐴 ∧ ℎ ∈ X𝑥 ∈ 𝐵 𝐶) ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐶) → (𝑧 ∈ 𝐴 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧)) ∈ X𝑥 ∈ 𝐴 𝐶)
40		reseq1 5973	. . . . . . . . . 10 ⊢ (𝑓 = (𝑧 ∈ 𝐴 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧)) → (𝑓 ↾ 𝐵) = ((𝑧 ∈ 𝐴 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧)) ↾ 𝐵))
41		iftrue 4498	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝐵 → if(𝑧 ∈ 𝐵, ℎ, 𝑔) = ℎ)
42	41	fveq1d 6884	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝐵 → (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧) = (ℎ‘𝑧))
43	42	mpteq2ia 5210	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝐵 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧)) = (𝑧 ∈ 𝐵 ↦ (ℎ‘𝑧))
44		resmpt 6040	. . . . . . . . . . . . 13 ⊢ (𝐵 ⊆ 𝐴 → ((𝑧 ∈ 𝐴 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧)) ↾ 𝐵) = (𝑧 ∈ 𝐵 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧)))
45	44	ad2antrr 738	. . . . . . . . . . . 12 ⊢ (((𝐵 ⊆ 𝐴 ∧ ℎ ∈ X𝑥 ∈ 𝐵 𝐶) ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐶) → ((𝑧 ∈ 𝐴 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧)) ↾ 𝐵) = (𝑧 ∈ 𝐵 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧)))
46		ixpfn 8901	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ X𝑥 ∈ 𝐵 𝐶 → ℎ Fn 𝐵)
47	46	ad2antlr 739	. . . . . . . . . . . . 13 ⊢ (((𝐵 ⊆ 𝐴 ∧ ℎ ∈ X𝑥 ∈ 𝐵 𝐶) ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐶) → ℎ Fn 𝐵)
48		dffn5 6940	. . . . . . . . . . . . 13 ⊢ (ℎ Fn 𝐵 ↔ ℎ = (𝑧 ∈ 𝐵 ↦ (ℎ‘𝑧)))
49	47, 48	sylib 221	. . . . . . . . . . . 12 ⊢ (((𝐵 ⊆ 𝐴 ∧ ℎ ∈ X𝑥 ∈ 𝐵 𝐶) ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐶) → ℎ = (𝑧 ∈ 𝐵 ↦ (ℎ‘𝑧)))
50	43, 45, 49	3eqtr4a 2830	. . . . . . . . . . 11 ⊢ (((𝐵 ⊆ 𝐴 ∧ ℎ ∈ X𝑥 ∈ 𝐵 𝐶) ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐶) → ((𝑧 ∈ 𝐴 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧)) ↾ 𝐵) = ℎ)
51	50, 11	eqeltrdi 2877	. . . . . . . . . 10 ⊢ (((𝐵 ⊆ 𝐴 ∧ ℎ ∈ X𝑥 ∈ 𝐵 𝐶) ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐶) → ((𝑧 ∈ 𝐴 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧)) ↾ 𝐵) ∈ V)
52	2, 40, 39, 51	fvmptd3 7014	. . . . . . . . 9 ⊢ (((𝐵 ⊆ 𝐴 ∧ ℎ ∈ X𝑥 ∈ 𝐵 𝐶) ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐶) → (𝐹‘(𝑧 ∈ 𝐴 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧))) = ((𝑧 ∈ 𝐴 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧)) ↾ 𝐵))
53	52, 50	eqtr2d 2805	. . . . . . . 8 ⊢ (((𝐵 ⊆ 𝐴 ∧ ℎ ∈ X𝑥 ∈ 𝐵 𝐶) ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐶) → ℎ = (𝐹‘(𝑧 ∈ 𝐴 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧))))
54		fveq2 6882	. . . . . . . . 9 ⊢ (𝑦 = (𝑧 ∈ 𝐴 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧)) → (𝐹‘𝑦) = (𝐹‘(𝑧 ∈ 𝐴 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧))))
55	54	rspceeqv 3613	. . . . . . . 8 ⊢ (((𝑧 ∈ 𝐴 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧)) ∈ X𝑥 ∈ 𝐴 𝐶 ∧ ℎ = (𝐹‘(𝑧 ∈ 𝐴 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧)))) → ∃𝑦 ∈ X 𝑥 ∈ 𝐴 𝐶ℎ = (𝐹‘𝑦))
56	39, 53, 55	syl2anc 595	. . . . . . 7 ⊢ (((𝐵 ⊆ 𝐴 ∧ ℎ ∈ X𝑥 ∈ 𝐵 𝐶) ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐶) → ∃𝑦 ∈ X 𝑥 ∈ 𝐴 𝐶ℎ = (𝐹‘𝑦))
57	56	ex 417	. . . . . 6 ⊢ ((𝐵 ⊆ 𝐴 ∧ ℎ ∈ X𝑥 ∈ 𝐵 𝐶) → (𝑔 ∈ X𝑥 ∈ 𝐴 𝐶 → ∃𝑦 ∈ X 𝑥 ∈ 𝐴 𝐶ℎ = (𝐹‘𝑦)))
58	57	ralrimdva 3171	. . . . 5 ⊢ (𝐵 ⊆ 𝐴 → (𝑔 ∈ X𝑥 ∈ 𝐴 𝐶 → ∀ℎ ∈ X 𝑥 ∈ 𝐵 𝐶∃𝑦 ∈ X 𝑥 ∈ 𝐴 𝐶ℎ = (𝐹‘𝑦)))
59	58	exlimdv 1960	. . . 4 ⊢ (𝐵 ⊆ 𝐴 → (∃𝑔 𝑔 ∈ X𝑥 ∈ 𝐴 𝐶 → ∀ℎ ∈ X 𝑥 ∈ 𝐵 𝐶∃𝑦 ∈ X 𝑥 ∈ 𝐴 𝐶ℎ = (𝐹‘𝑦)))
60	5, 59	biimtrid 245	. . 3 ⊢ (𝐵 ⊆ 𝐴 → (X𝑥 ∈ 𝐴 𝐶 ≠ ∅ → ∀ℎ ∈ X 𝑥 ∈ 𝐵 𝐶∃𝑦 ∈ X 𝑥 ∈ 𝐴 𝐶ℎ = (𝐹‘𝑦)))
61	60	imp 411	. 2 ⊢ ((𝐵 ⊆ 𝐴 ∧ X𝑥 ∈ 𝐴 𝐶 ≠ ∅) → ∀ℎ ∈ X 𝑥 ∈ 𝐵 𝐶∃𝑦 ∈ X 𝑥 ∈ 𝐴 𝐶ℎ = (𝐹‘𝑦))
62		dffo3 7098	. 2 ⊢ (𝐹:X𝑥 ∈ 𝐴 𝐶–onto→X𝑥 ∈ 𝐵 𝐶 ↔ (𝐹:X𝑥 ∈ 𝐴 𝐶⟶X𝑥 ∈ 𝐵 𝐶 ∧ ∀ℎ ∈ X 𝑥 ∈ 𝐵 𝐶∃𝑦 ∈ X 𝑥 ∈ 𝐴 𝐶ℎ = (𝐹‘𝑦)))
63	4, 61, 62	sylanbrc 594	1 ⊢ ((𝐵 ⊆ 𝐴 ∧ X𝑥 ∈ 𝐴 𝐶 ≠ ∅) → 𝐹:X𝑥 ∈ 𝐴 𝐶–onto→X𝑥 ∈ 𝐵 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∃wrex 3095  Vcvv 3463   ⊆ wss 3913  ∅c0 4294  ifcif 4492   ↦ cmpt 5196   ↾ cres 5664   Fn wfn 6532  ⟶wf 6533  –onto→wfo 6535  ‘cfv 6537  Xcixp 8895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ixp 8896

This theorem is referenced by:  ptcmplem2  24179