Theorem resixpfo 8914

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 8911	. . . 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 7091	. . 3 ⊢ (𝐵 ⊆ 𝐴 → 𝐹:X𝑥 ∈ 𝐴 𝐶⟶X𝑥 ∈ 𝐵 𝐶)
4	3	adantr 484	. 2 ⊢ ((𝐵 ⊆ 𝐴 ∧ X𝑥 ∈ 𝐴 𝐶 ≠ ∅) → 𝐹:X𝑥 ∈ 𝐴 𝐶⟶X𝑥 ∈ 𝐵 𝐶)
5		n0 4305	. . . 4 ⊢ (X𝑥 ∈ 𝐴 𝐶 ≠ ∅ ↔ ∃𝑔 𝑔 ∈ X𝑥 ∈ 𝐴 𝐶)
6		eleq1w 2844	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
7	6	ifbid 4503	. . . . . . . . . . 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 6870	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧) = (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥))
10	9	cbvmptv 5203	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐴 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧)) = (𝑥 ∈ 𝐴 ↦ (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥))
11		vex 3457	. . . . . . . . . . . . . . . 16 ⊢ ℎ ∈ V
12	11	elixp 8882	. . . . . . . . . . . . . . 15 ⊢ (ℎ ∈ X𝑥 ∈ 𝐵 𝐶 ↔ (ℎ Fn 𝐵 ∧ ∀𝑥 ∈ 𝐵 (ℎ‘𝑥) ∈ 𝐶))
13	12	simprbi 501	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ X𝑥 ∈ 𝐵 𝐶 → ∀𝑥 ∈ 𝐵 (ℎ‘𝑥) ∈ 𝐶)
14		fveq1 6862	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ = if(𝑥 ∈ 𝐵, ℎ, 𝑔) → (ℎ‘𝑥) = (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥))
15	14	eleq1d 2846	. . . . . . . . . . . . . . . . 17 ⊢ (ℎ = if(𝑥 ∈ 𝐵, ℎ, 𝑔) → ((ℎ‘𝑥) ∈ 𝐶 ↔ (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥) ∈ 𝐶))
16		fveq1 6862	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 = if(𝑥 ∈ 𝐵, ℎ, 𝑔) → (𝑔‘𝑥) = (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥))
17	16	eleq1d 2846	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = if(𝑥 ∈ 𝐵, ℎ, 𝑔) → ((𝑔‘𝑥) ∈ 𝐶 ↔ (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥) ∈ 𝐶))
18		simpl 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ 𝐵 → (ℎ‘𝑥) ∈ 𝐶) ∧ (𝑥 ∈ 𝐴 ∧ (𝑔‘𝑥) ∈ 𝐶)) → (𝑥 ∈ 𝐵 → (ℎ‘𝑥) ∈ 𝐶))
19	18	imp 410	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ 𝐵 → (ℎ‘𝑥) ∈ 𝐶) ∧ (𝑥 ∈ 𝐴 ∧ (𝑔‘𝑥) ∈ 𝐶)) ∧ 𝑥 ∈ 𝐵) → (ℎ‘𝑥) ∈ 𝐶)
20		simplrr 787	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ 𝐵 → (ℎ‘𝑥) ∈ 𝐶) ∧ (𝑥 ∈ 𝐴 ∧ (𝑔‘𝑥) ∈ 𝐶)) ∧ ¬ 𝑥 ∈ 𝐵) → (𝑔‘𝑥) ∈ 𝐶)
21	15, 17, 19, 20	ifbothda 4518	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ 𝐵 → (ℎ‘𝑥) ∈ 𝐶) ∧ (𝑥 ∈ 𝐴 ∧ (𝑔‘𝑥) ∈ 𝐶)) → (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥) ∈ 𝐶)
22	21	exp32 424	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝐵 → (ℎ‘𝑥) ∈ 𝐶) → (𝑥 ∈ 𝐴 → ((𝑔‘𝑥) ∈ 𝐶 → (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥) ∈ 𝐶)))
23	22	ralimi2 3093	. . . . . . . . . . . . . 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 485	. . . . . . . . . . . 12 ⊢ ((𝐵 ⊆ 𝐴 ∧ ℎ ∈ X𝑥 ∈ 𝐵 𝐶) → ∀𝑥 ∈ 𝐴 ((𝑔‘𝑥) ∈ 𝐶 → (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥) ∈ 𝐶))
26		ralim 3101	. . . . . . . . . . . 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 3457	. . . . . . . . . . . . 13 ⊢ 𝑔 ∈ V
29	28	elixp 8882	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 𝐶 ↔ (𝑔 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐶))
30	29	simprbi 501	. . . . . . . . . . 11 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 𝐶 → ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐶)
31	27, 30	impel 513	. . . . . . . . . 10 ⊢ (((𝐵 ⊆ 𝐴 ∧ ℎ ∈ X𝑥 ∈ 𝐵 𝐶) ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐶) → ∀𝑥 ∈ 𝐴 (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥) ∈ 𝐶)
32		n0i 4292	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 𝐶 → ¬ X𝑥 ∈ 𝐴 𝐶 = ∅)
33		ixpprc 8897	. . . . . . . . . . . . 13 ⊢ (¬ 𝐴 ∈ V → X𝑥 ∈ 𝐴 𝐶 = ∅)
34	32, 33	nsyl2 141	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 𝐶 → 𝐴 ∈ V)
35	34	adantl 485	. . . . . . . . . . 11 ⊢ (((𝐵 ⊆ 𝐴 ∧ ℎ ∈ X𝑥 ∈ 𝐵 𝐶) ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐶) → 𝐴 ∈ V)
36		mptelixpg 8913	. . . . . . . . . . 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 259	. . . . . . . . 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 2865	. . . . . . . 8 ⊢ (((𝐵 ⊆ 𝐴 ∧ ℎ ∈ X𝑥 ∈ 𝐵 𝐶) ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐶) → (𝑧 ∈ 𝐴 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧)) ∈ X𝑥 ∈ 𝐴 𝐶)
40		reseq1 5957	. . . . . . . . . 10 ⊢ (𝑓 = (𝑧 ∈ 𝐴 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧)) → (𝑓 ↾ 𝐵) = ((𝑧 ∈ 𝐴 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧)) ↾ 𝐵))
41		iftrue 4485	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝐵 → if(𝑧 ∈ 𝐵, ℎ, 𝑔) = ℎ)
42	41	fveq1d 6865	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝐵 → (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧) = (ℎ‘𝑧))
43	42	mpteq2ia 5194	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝐵 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧)) = (𝑧 ∈ 𝐵 ↦ (ℎ‘𝑧))
44		resmpt 6023	. . . . . . . . . . . . 13 ⊢ (𝐵 ⊆ 𝐴 → ((𝑧 ∈ 𝐴 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧)) ↾ 𝐵) = (𝑧 ∈ 𝐵 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧)))
45	44	ad2antrr 736	. . . . . . . . . . . 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 8881	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ X𝑥 ∈ 𝐵 𝐶 → ℎ Fn 𝐵)
47	46	ad2antlr 737	. . . . . . . . . . . . 13 ⊢ (((𝐵 ⊆ 𝐴 ∧ ℎ ∈ X𝑥 ∈ 𝐵 𝐶) ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐶) → ℎ Fn 𝐵)
48		dffn5 6921	. . . . . . . . . . . . 13 ⊢ (ℎ Fn 𝐵 ↔ ℎ = (𝑧 ∈ 𝐵 ↦ (ℎ‘𝑧)))
49	47, 48	sylib 220	. . . . . . . . . . . 12 ⊢ (((𝐵 ⊆ 𝐴 ∧ ℎ ∈ X𝑥 ∈ 𝐵 𝐶) ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐶) → ℎ = (𝑧 ∈ 𝐵 ↦ (ℎ‘𝑧)))
50	43, 45, 49	3eqtr4a 2822	. . . . . . . . . . 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 2869	. . . . . . . . . 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 6995	. . . . . . . . 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 2797	. . . . . . . 8 ⊢ (((𝐵 ⊆ 𝐴 ∧ ℎ ∈ X𝑥 ∈ 𝐵 𝐶) ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐶) → ℎ = (𝐹‘(𝑧 ∈ 𝐴 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧))))
54		fveq2 6863	. . . . . . . . 9 ⊢ (𝑦 = (𝑧 ∈ 𝐴 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧)) → (𝐹‘𝑦) = (𝐹‘(𝑧 ∈ 𝐴 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧))))
55	54	rspceeqv 3604	. . . . . . . 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 593	. . . . . . 7 ⊢ (((𝐵 ⊆ 𝐴 ∧ ℎ ∈ X𝑥 ∈ 𝐵 𝐶) ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐶) → ∃𝑦 ∈ X 𝑥 ∈ 𝐴 𝐶ℎ = (𝐹‘𝑦))
57	56	ex 416	. . . . . 6 ⊢ ((𝐵 ⊆ 𝐴 ∧ ℎ ∈ X𝑥 ∈ 𝐵 𝐶) → (𝑔 ∈ X𝑥 ∈ 𝐴 𝐶 → ∃𝑦 ∈ X 𝑥 ∈ 𝐴 𝐶ℎ = (𝐹‘𝑦)))
58	57	ralrimdva 3161	. . . . 5 ⊢ (𝐵 ⊆ 𝐴 → (𝑔 ∈ X𝑥 ∈ 𝐴 𝐶 → ∀ℎ ∈ X 𝑥 ∈ 𝐵 𝐶∃𝑦 ∈ X 𝑥 ∈ 𝐴 𝐶ℎ = (𝐹‘𝑦)))
59	58	exlimdv 1952	. . . 4 ⊢ (𝐵 ⊆ 𝐴 → (∃𝑔 𝑔 ∈ X𝑥 ∈ 𝐴 𝐶 → ∀ℎ ∈ X 𝑥 ∈ 𝐵 𝐶∃𝑦 ∈ X 𝑥 ∈ 𝐴 𝐶ℎ = (𝐹‘𝑦)))
60	5, 59	biimtrid 244	. . 3 ⊢ (𝐵 ⊆ 𝐴 → (X𝑥 ∈ 𝐴 𝐶 ≠ ∅ → ∀ℎ ∈ X 𝑥 ∈ 𝐵 𝐶∃𝑦 ∈ X 𝑥 ∈ 𝐴 𝐶ℎ = (𝐹‘𝑦)))
61	60	imp 410	. 2 ⊢ ((𝐵 ⊆ 𝐴 ∧ X𝑥 ∈ 𝐴 𝐶 ≠ ∅) → ∀ℎ ∈ X 𝑥 ∈ 𝐵 𝐶∃𝑦 ∈ X 𝑥 ∈ 𝐴 𝐶ℎ = (𝐹‘𝑦))
62		dffo3 7079	. 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 592	1 ⊢ ((𝐵 ⊆ 𝐴 ∧ X𝑥 ∈ 𝐴 𝐶 ≠ ∅) → 𝐹:X𝑥 ∈ 𝐴 𝐶–onto→X𝑥 ∈ 𝐵 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559  ∃wex 1798   ∈ wcel 2141   ≠ wne 2956  ∀wral 3075  ∃wrex 3085  Vcvv 3453   ⊆ wss 3904  ∅c0 4285  ifcif 4479   ↦ cmpt 5180   ↾ cres 5647   Fn wfn 6512  ⟶wf 6513  –onto→wfo 6515  ‘cfv 6517  Xcixp 8875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ixp 8876

This theorem is referenced by:  ptcmplem2  24093