Theorem resixpfo 8935

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 8932	. . . 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 7111	. . 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 4308	. . . 4 ⊢ (X𝑥 ∈ 𝐴 𝐶 ≠ ∅ ↔ ∃𝑔 𝑔 ∈ X𝑥 ∈ 𝐴 𝐶)
6		eleq1w 2846	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
7	6	ifbid 4512	. . . . . . . . . . 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 6890	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧) = (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥))
10	9	cbvmptv 5216	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐴 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧)) = (𝑥 ∈ 𝐴 ↦ (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥))
11		vex 3459	. . . . . . . . . . . . . . . 16 ⊢ ℎ ∈ V
12	11	elixp 8903	. . . . . . . . . . . . . . 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 6882	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ = if(𝑥 ∈ 𝐵, ℎ, 𝑔) → (ℎ‘𝑥) = (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥))
15	14	eleq1d 2848	. . . . . . . . . . . . . . . . 17 ⊢ (ℎ = if(𝑥 ∈ 𝐵, ℎ, 𝑔) → ((ℎ‘𝑥) ∈ 𝐶 ↔ (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥) ∈ 𝐶))
16		fveq1 6882	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 = if(𝑥 ∈ 𝐵, ℎ, 𝑔) → (𝑔‘𝑥) = (if(𝑥 ∈ 𝐵, ℎ, 𝑔)‘𝑥))
17	16	eleq1d 2848	. . . . . . . . . . . . . . . . 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 4527	. . . . . . . . . . . . . . . 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 3097	. . . . . . . . . . . . . 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 3105	. . . . . . . . . . . 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 3459	. . . . . . . . . . . . 13 ⊢ 𝑔 ∈ V
29	28	elixp 8903	. . . . . . . . . . . 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 4294	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 𝐶 → ¬ X𝑥 ∈ 𝐴 𝐶 = ∅)
33		ixpprc 8918	. . . . . . . . . . . . 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 8934	. . . . . . . . . . 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 2867	. . . . . . . 8 ⊢ (((𝐵 ⊆ 𝐴 ∧ ℎ ∈ X𝑥 ∈ 𝐵 𝐶) ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐶) → (𝑧 ∈ 𝐴 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧)) ∈ X𝑥 ∈ 𝐴 𝐶)
40		reseq1 5974	. . . . . . . . . 10 ⊢ (𝑓 = (𝑧 ∈ 𝐴 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧)) → (𝑓 ↾ 𝐵) = ((𝑧 ∈ 𝐴 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧)) ↾ 𝐵))
41		iftrue 4494	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝐵 → if(𝑧 ∈ 𝐵, ℎ, 𝑔) = ℎ)
42	41	fveq1d 6885	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝐵 → (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧) = (ℎ‘𝑧))
43	42	mpteq2ia 5207	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝐵 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧)) = (𝑧 ∈ 𝐵 ↦ (ℎ‘𝑧))
44		resmpt 6041	. . . . . . . . . . . . 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 8902	. . . . . . . . . . . . . 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 6941	. . . . . . . . . . . . 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 2824	. . . . . . . . . . 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 2871	. . . . . . . . . 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 7015	. . . . . . . . 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 2799	. . . . . . . 8 ⊢ (((𝐵 ⊆ 𝐴 ∧ ℎ ∈ X𝑥 ∈ 𝐵 𝐶) ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐶) → ℎ = (𝐹‘(𝑧 ∈ 𝐴 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧))))
54		fveq2 6883	. . . . . . . . 9 ⊢ (𝑦 = (𝑧 ∈ 𝐴 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧)) → (𝐹‘𝑦) = (𝐹‘(𝑧 ∈ 𝐴 ↦ (if(𝑧 ∈ 𝐵, ℎ, 𝑔)‘𝑧))))
55	54	rspceeqv 3605	. . . . . . . 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 3165	. . . . 5 ⊢ (𝐵 ⊆ 𝐴 → (𝑔 ∈ X𝑥 ∈ 𝐴 𝐶 → ∀ℎ ∈ X 𝑥 ∈ 𝐵 𝐶∃𝑦 ∈ X 𝑥 ∈ 𝐴 𝐶ℎ = (𝐹‘𝑦)))
59	58	exlimdv 1963	. . . 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 7099	. 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 1570  ∃wex 1809   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  ∃wrex 3089  Vcvv 3455   ⊆ wss 3906  ∅c0 4287  ifcif 4488   ↦ cmpt 5193   ↾ cres 5665   Fn wfn 6533  ⟶wf 6534  –onto→wfo 6536  ‘cfv 6538  Xcixp 8896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ixp 8897

This theorem is referenced by:  ptcmplem2  24191