Theorem resixpfo 8955

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 8952	. . . 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 7109	. . 3 ⊢ (𝐵 ⊆ 𝐴 → 𝐹:X𝑥 ∈ 𝐴 𝐶⟶X𝑥 ∈ 𝐵 𝐶)
4	3	adantr 480	. 2 ⊢ ((𝐵 ⊆ 𝐴 ∧ X𝑥 ∈ 𝐴 𝐶 ≠ ∅) → 𝐹:X𝑥 ∈ 𝐴 𝐶⟶X𝑥 ∈ 𝐵 𝐶)
5		n0 4333	. . . 4 ⊢ (X𝑥 ∈ 𝐴 𝐶 ≠ ∅ ↔ ∃𝑔 𝑔 ∈ X𝑥 ∈ 𝐴 𝐶)
6		eleq1w 2818	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
7	6	ifbid 4529	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → if(𝑧 ∈ 𝐵, ℎ, 𝑔) = if(𝑥 ∈ 𝐵, ℎ, 𝑔))
8		id 22	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → 𝑧 = 𝑥)
9	7, 8	fveq12d 6888	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧) = (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥))
10	9	cbvmptv 5230	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐴 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧)) = (𝑥 ∈ 𝐴 ↦ (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥))
11		vex 3468	. . . . . . . . . . . . . . . 16 ⊢ ℎ ∈ V
12	11	elixp 8923	. . . . . . . . . . . . . . 15 ⊢ (ℎ ∈ X𝑥 ∈ 𝐵 𝐶 ↔ (ℎ Fn 𝐵 ∧ ∀𝑥 ∈ 𝐵 (ℎ‘𝑥) ∈ 𝐶))
13	12	simprbi 496	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ X𝑥 ∈ 𝐵 𝐶 → ∀𝑥 ∈ 𝐵 (ℎ‘𝑥) ∈ 𝐶)
14		fveq1 6880	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ = if(𝑥 ∈ 𝐵, ℎ, 𝑔) → (ℎ‘𝑥) = (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥))
15	14	eleq1d 2820	. . . . . . . . . . . . . . . . 17 ⊢ (ℎ = if(𝑥 ∈ 𝐵, ℎ, 𝑔) → ((ℎ‘𝑥) ∈ 𝐶 ↔ (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥) ∈ 𝐶))
16		fveq1 6880	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 = if(𝑥 ∈ 𝐵, ℎ, 𝑔) → (𝑔‘𝑥) = (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥))
17	16	eleq1d 2820	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = if(𝑥 ∈ 𝐵, ℎ, 𝑔) → ((𝑔‘𝑥) ∈ 𝐶 ↔ (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥) ∈ 𝐶))
18		simpl 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ 𝐵 → (ℎ‘𝑥) ∈ 𝐶) ∧ (𝑥 ∈ 𝐴 ∧ (𝑔‘𝑥) ∈ 𝐶)) → (𝑥 ∈ 𝐵 → (ℎ‘𝑥) ∈ 𝐶))
19	18	imp 406	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ 𝐵 → (ℎ‘𝑥) ∈ 𝐶) ∧ (𝑥 ∈ 𝐴 ∧ (𝑔‘𝑥) ∈ 𝐶)) ∧ 𝑥 ∈ 𝐵) → (ℎ‘𝑥) ∈ 𝐶)
20		simplrr 777	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ 𝐵 → (ℎ‘𝑥) ∈ 𝐶) ∧ (𝑥 ∈ 𝐴 ∧ (𝑔‘𝑥) ∈ 𝐶)) ∧ ¬ 𝑥 ∈ 𝐵) → (𝑔‘𝑥) ∈ 𝐶)
21	15, 17, 19, 20	ifbothda 4544	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ 𝐵 → (ℎ‘𝑥) ∈ 𝐶) ∧ (𝑥 ∈ 𝐴 ∧ (𝑔‘𝑥) ∈ 𝐶)) → (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥) ∈ 𝐶)
22	21	exp32 420	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝐵 → (ℎ‘𝑥) ∈ 𝐶) → (𝑥 ∈ 𝐴 → ((𝑔‘𝑥) ∈ 𝐶 → (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥) ∈ 𝐶)))
23	22	ralimi2 3069	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ 𝐵 (ℎ‘𝑥) ∈ 𝐶 → ∀𝑥 ∈ 𝐴 ((𝑔‘𝑥) ∈ 𝐶 → (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥) ∈ 𝐶))
24	13, 23	syl 17	. . . . . . . . . . . . 13 ⊢ (ℎ ∈ X𝑥 ∈ 𝐵 𝐶 → ∀𝑥 ∈ 𝐴 ((𝑔‘𝑥) ∈ 𝐶 → (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥) ∈ 𝐶))
25	24	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝐵 ⊆ 𝐴 ∧ ℎ ∈ X𝑥 ∈ 𝐵 𝐶) → ∀𝑥 ∈ 𝐴 ((𝑔‘𝑥) ∈ 𝐶 → (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥) ∈ 𝐶))
26		ralim 3077	. . . . . . . . . . . 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 17	. . . . . . . . . . 11 ⊢ ((𝐵 ⊆ 𝐴 ∧ ℎ ∈ X𝑥 ∈ 𝐵 𝐶) → (∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐶 → ∀𝑥 ∈ 𝐴 (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥) ∈ 𝐶))
28		vex 3468	. . . . . . . . . . . . 13 ⊢ 𝑔 ∈ V
29	28	elixp 8923	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 𝐶 ↔ (𝑔 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐶))
30	29	simprbi 496	. . . . . . . . . . 11 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 𝐶 → ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐶)
31	27, 30	impel 505	. . . . . . . . . 10 ⊢ (((𝐵 ⊆ 𝐴 ∧ ℎ ∈ X𝑥 ∈ 𝐵 𝐶) ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐶) → ∀𝑥 ∈ 𝐴 (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥) ∈ 𝐶)
32		n0i 4320	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 𝐶 → ¬ X𝑥 ∈ 𝐴 𝐶 = ∅)
33		ixpprc 8938	. . . . . . . . . . . . 13 ⊢ (¬ 𝐴 ∈ V → X𝑥 ∈ 𝐴 𝐶 = ∅)
34	32, 33	nsyl2 141	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 𝐶 → 𝐴 ∈ V)
35	34	adantl 481	. . . . . . . . . . 11 ⊢ (((𝐵 ⊆ 𝐴 ∧ ℎ ∈ X𝑥 ∈ 𝐵 𝐶) ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐶) → 𝐴 ∈ V)
36		mptelixpg 8954	. . . . . . . . . . 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 17	. . . . . . . . . 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 257	. . . . . . . . 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 2839	. . . . . . . 8 ⊢ (((𝐵 ⊆ 𝐴 ∧ ℎ ∈ X𝑥 ∈ 𝐵 𝐶) ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐶) → (𝑧 ∈ 𝐴 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧)) ∈ X𝑥 ∈ 𝐴 𝐶)
40		reseq1 5965	. . . . . . . . . 10 ⊢ (𝑓 = (𝑧 ∈ 𝐴 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧)) → (𝑓 ↾ 𝐵) = ((𝑧 ∈ 𝐴 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧)) ↾ 𝐵))
41		iftrue 4511	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝐵 → if(𝑧 ∈ 𝐵, ℎ, 𝑔) = ℎ)
42	41	fveq1d 6883	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝐵 → (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧) = (ℎ‘𝑧))
43	42	mpteq2ia 5221	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝐵 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧)) = (𝑧 ∈ 𝐵 ↦ (ℎ‘𝑧))
44		resmpt 6029	. . . . . . . . . . . . 13 ⊢ (𝐵 ⊆ 𝐴 → ((𝑧 ∈ 𝐴 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧)) ↾ 𝐵) = (𝑧 ∈ 𝐵 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧)))
45	44	ad2antrr 726	. . . . . . . . . . . 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 8922	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ X𝑥 ∈ 𝐵 𝐶 → ℎ Fn 𝐵)
47	46	ad2antlr 727	. . . . . . . . . . . . 13 ⊢ (((𝐵 ⊆ 𝐴 ∧ ℎ ∈ X𝑥 ∈ 𝐵 𝐶) ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐶) → ℎ Fn 𝐵)
48		dffn5 6942	. . . . . . . . . . . . 13 ⊢ (ℎ Fn 𝐵 ↔ ℎ = (𝑧 ∈ 𝐵 ↦ (ℎ‘𝑧)))
49	47, 48	sylib 218	. . . . . . . . . . . 12 ⊢ (((𝐵 ⊆ 𝐴 ∧ ℎ ∈ X𝑥 ∈ 𝐵 𝐶) ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐶) → ℎ = (𝑧 ∈ 𝐵 ↦ (ℎ‘𝑧)))
50	43, 45, 49	3eqtr4a 2797	. . . . . . . . . . 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 2843	. . . . . . . . . 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 2772	. . . . . . . 8 ⊢ (((𝐵 ⊆ 𝐴 ∧ ℎ ∈ X𝑥 ∈ 𝐵 𝐶) ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐶) → ℎ = (𝐹‘(𝑧 ∈ 𝐴 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧))))
54		fveq2 6881	. . . . . . . . 9 ⊢ (𝑦 = (𝑧 ∈ 𝐴 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧)) → (𝐹‘𝑦) = (𝐹‘(𝑧 ∈ 𝐴 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧))))
55	54	rspceeqv 3629	. . . . . . . 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 584	. . . . . . 7 ⊢ (((𝐵 ⊆ 𝐴 ∧ ℎ ∈ X𝑥 ∈ 𝐵 𝐶) ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐶) → ∃𝑦 ∈ X 𝑥 ∈ 𝐴 𝐶ℎ = (𝐹‘𝑦))
57	56	ex 412	. . . . . 6 ⊢ ((𝐵 ⊆ 𝐴 ∧ ℎ ∈ X𝑥 ∈ 𝐵 𝐶) → (𝑔 ∈ X𝑥 ∈ 𝐴 𝐶 → ∃𝑦 ∈ X 𝑥 ∈ 𝐴 𝐶ℎ = (𝐹‘𝑦)))
58	57	ralrimdva 3141	. . . . 5 ⊢ (𝐵 ⊆ 𝐴 → (𝑔 ∈ X𝑥 ∈ 𝐴 𝐶 → ∀ℎ ∈ X 𝑥 ∈ 𝐵 𝐶∃𝑦 ∈ X 𝑥 ∈ 𝐴 𝐶ℎ = (𝐹‘𝑦)))
59	58	exlimdv 1933	. . . 4 ⊢ (𝐵 ⊆ 𝐴 → (∃𝑔 𝑔 ∈ X𝑥 ∈ 𝐴 𝐶 → ∀ℎ ∈ X 𝑥 ∈ 𝐵 𝐶∃𝑦 ∈ X 𝑥 ∈ 𝐴 𝐶ℎ = (𝐹‘𝑦)))
60	5, 59	biimtrid 242	. . 3 ⊢ (𝐵 ⊆ 𝐴 → (X𝑥 ∈ 𝐴 𝐶 ≠ ∅ → ∀ℎ ∈ X 𝑥 ∈ 𝐵 𝐶∃𝑦 ∈ X 𝑥 ∈ 𝐴 𝐶ℎ = (𝐹‘𝑦)))
61	60	imp 406	. 2 ⊢ ((𝐵 ⊆ 𝐴 ∧ X𝑥 ∈ 𝐴 𝐶 ≠ ∅) → ∀ℎ ∈ X 𝑥 ∈ 𝐵 𝐶∃𝑦 ∈ X 𝑥 ∈ 𝐴 𝐶ℎ = (𝐹‘𝑦))
62		dffo3 7097	. 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 583	1 ⊢ ((𝐵 ⊆ 𝐴 ∧ X𝑥 ∈ 𝐴 𝐶 ≠ ∅) → 𝐹:X𝑥 ∈ 𝐴 𝐶–onto→X𝑥 ∈ 𝐵 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2933  ∀wral 3052  ∃wrex 3061  Vcvv 3464   ⊆ wss 3931  ∅c0 4313  ifcif 4505   ↦ cmpt 5206   ↾ cres 5661   Fn wfn 6531  ⟶wf 6532  –onto→wfo 6534  ‘cfv 6536  Xcixp 8916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ixp 8917

This theorem is referenced by:  ptcmplem2  23996