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Theorem mapxpen 9084

Description: Equinumerosity law for double set exponentiation. Proposition 10.45 of [TakeutiZaring] p. 96. (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 24-Jun-2015.)

**Assertion**
Ref	Expression
mapxpen	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 ↑_m 𝐵) ↑_m 𝐶) ≈ (𝐴 ↑_m (𝐵 × 𝐶)))

Proof of Theorem mapxpen Dummy variables 𝑓 𝑔 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovexd 7404	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 ↑_m 𝐵) ↑_m 𝐶) ∈ V)
2		ovexd 7404	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 ↑_m (𝐵 × 𝐶)) ∈ V)
3		elmapi 8799	. . . . . . . . . 10 ⊢ (𝑓 ∈ ((𝐴 ↑_m 𝐵) ↑_m 𝐶) → 𝑓:𝐶⟶(𝐴 ↑_m 𝐵))
4	3	ffvelcdmda 7038	. . . . . . . . 9 ⊢ ((𝑓 ∈ ((𝐴 ↑_m 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ 𝐶) → (𝑓‘𝑦) ∈ (𝐴 ↑_m 𝐵))
5		elmapi 8799	. . . . . . . . 9 ⊢ ((𝑓‘𝑦) ∈ (𝐴 ↑_m 𝐵) → (𝑓‘𝑦):𝐵⟶𝐴)
6	4, 5	syl 17	. . . . . . . 8 ⊢ ((𝑓 ∈ ((𝐴 ↑_m 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ 𝐶) → (𝑓‘𝑦):𝐵⟶𝐴)
7	6	ffvelcdmda 7038	. . . . . . 7 ⊢ (((𝑓 ∈ ((𝐴 ↑_m 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ∈ 𝐵) → ((𝑓‘𝑦)‘𝑥) ∈ 𝐴)
8	7	an32s 652	. . . . . 6 ⊢ (((𝑓 ∈ ((𝐴 ↑_m 𝐵) ↑_m 𝐶) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐶) → ((𝑓‘𝑦)‘𝑥) ∈ 𝐴)
9	8	ralrimiva 3125	. . . . 5 ⊢ ((𝑓 ∈ ((𝐴 ↑_m 𝐵) ↑_m 𝐶) ∧ 𝑥 ∈ 𝐵) → ∀𝑦 ∈ 𝐶 ((𝑓‘𝑦)‘𝑥) ∈ 𝐴)
10	9	ralrimiva 3125	. . . 4 ⊢ (𝑓 ∈ ((𝐴 ↑_m 𝐵) ↑_m 𝐶) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 ((𝑓‘𝑦)‘𝑥) ∈ 𝐴)
11		eqid 2729	. . . . 5 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥))
12	11	fmpo 8026	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 ((𝑓‘𝑦)‘𝑥) ∈ 𝐴 ↔ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)):(𝐵 × 𝐶)⟶𝐴)
13	10, 12	sylib 218	. . 3 ⊢ (𝑓 ∈ ((𝐴 ↑_m 𝐵) ↑_m 𝐶) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)):(𝐵 × 𝐶)⟶𝐴)
14		simp1 1136	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐴 ∈ 𝑉)
15		xpexg 7706	. . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐵 × 𝐶) ∈ V)
16	15	3adant1 1130	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐵 × 𝐶) ∈ V)
17		elmapg 8789	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 × 𝐶) ∈ V) → ((𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) ∈ (𝐴 ↑_m (𝐵 × 𝐶)) ↔ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)):(𝐵 × 𝐶)⟶𝐴))
18	14, 16, 17	syl2anc 584	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) ∈ (𝐴 ↑_m (𝐵 × 𝐶)) ↔ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)):(𝐵 × 𝐶)⟶𝐴))
19	13, 18	imbitrrid 246	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝑓 ∈ ((𝐴 ↑_m 𝐵) ↑_m 𝐶) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) ∈ (𝐴 ↑_m (𝐵 × 𝐶))))
20		elmapi 8799	. . . . . . . . 9 ⊢ (𝑔 ∈ (𝐴 ↑_m (𝐵 × 𝐶)) → 𝑔:(𝐵 × 𝐶)⟶𝐴)
21	20	adantl 481	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ 𝑔 ∈ (𝐴 ↑_m (𝐵 × 𝐶))) → 𝑔:(𝐵 × 𝐶)⟶𝐴)
22		fovcdm 7539	. . . . . . . . . 10 ⊢ ((𝑔:(𝐵 × 𝐶)⟶𝐴 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → (𝑥𝑔𝑦) ∈ 𝐴)
23	22	3expa 1118	. . . . . . . . 9 ⊢ (((𝑔:(𝐵 × 𝐶)⟶𝐴 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐶) → (𝑥𝑔𝑦) ∈ 𝐴)
24	23	an32s 652	. . . . . . . 8 ⊢ (((𝑔:(𝐵 × 𝐶)⟶𝐴 ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ∈ 𝐵) → (𝑥𝑔𝑦) ∈ 𝐴)
25	21, 24	sylanl1 680	. . . . . . 7 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ 𝑔 ∈ (𝐴 ↑_m (𝐵 × 𝐶))) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ∈ 𝐵) → (𝑥𝑔𝑦) ∈ 𝐴)
26	25	fmpttd 7069	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ 𝑔 ∈ (𝐴 ↑_m (𝐵 × 𝐶))) ∧ 𝑦 ∈ 𝐶) → (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)):𝐵⟶𝐴)
27		elmapg 8789	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)) ∈ (𝐴 ↑_m 𝐵) ↔ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)):𝐵⟶𝐴))
28	27	3adant3 1132	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)) ∈ (𝐴 ↑_m 𝐵) ↔ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)):𝐵⟶𝐴))
29	28	ad2antrr 726	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ 𝑔 ∈ (𝐴 ↑_m (𝐵 × 𝐶))) ∧ 𝑦 ∈ 𝐶) → ((𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)) ∈ (𝐴 ↑_m 𝐵) ↔ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)):𝐵⟶𝐴))
30	26, 29	mpbird 257	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ 𝑔 ∈ (𝐴 ↑_m (𝐵 × 𝐶))) ∧ 𝑦 ∈ 𝐶) → (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)) ∈ (𝐴 ↑_m 𝐵))
31	30	fmpttd 7069	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ 𝑔 ∈ (𝐴 ↑_m (𝐵 × 𝐶))) → (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))):𝐶⟶(𝐴 ↑_m 𝐵))
32	31	ex 412	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝑔 ∈ (𝐴 ↑_m (𝐵 × 𝐶)) → (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))):𝐶⟶(𝐴 ↑_m 𝐵)))
33		ovex 7402	. . . 4 ⊢ (𝐴 ↑_m 𝐵) ∈ V
34		simp3 1138	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐶 ∈ 𝑋)
35		elmapg 8789	. . . 4 ⊢ (((𝐴 ↑_m 𝐵) ∈ V ∧ 𝐶 ∈ 𝑋) → ((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) ∈ ((𝐴 ↑_m 𝐵) ↑_m 𝐶) ↔ (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))):𝐶⟶(𝐴 ↑_m 𝐵)))
36	33, 34, 35	sylancr 587	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) ∈ ((𝐴 ↑_m 𝐵) ↑_m 𝐶) ↔ (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))):𝐶⟶(𝐴 ↑_m 𝐵)))
37	32, 36	sylibrd 259	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝑔 ∈ (𝐴 ↑_m (𝐵 × 𝐶)) → (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) ∈ ((𝐴 ↑_m 𝐵) ↑_m 𝐶)))
38		elmapfn 8815	. . . . . . . 8 ⊢ (𝑔 ∈ (𝐴 ↑_m (𝐵 × 𝐶)) → 𝑔 Fn (𝐵 × 𝐶))
39	38	ad2antll 729	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑_m 𝐵) ↑_m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑_m (𝐵 × 𝐶)))) → 𝑔 Fn (𝐵 × 𝐶))
40		fnov 7500	. . . . . . 7 ⊢ (𝑔 Fn (𝐵 × 𝐶) ↔ 𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (𝑥𝑔𝑦)))
41	39, 40	sylib 218	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑_m 𝐵) ↑_m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑_m (𝐵 × 𝐶)))) → 𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (𝑥𝑔𝑦)))
42		simp3 1138	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑_m 𝐵) ↑_m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑_m (𝐵 × 𝐶)))) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → 𝑦 ∈ 𝐶)
43	26	adantlrl 720	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑_m 𝐵) ↑_m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑_m (𝐵 × 𝐶)))) ∧ 𝑦 ∈ 𝐶) → (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)):𝐵⟶𝐴)
44	43	3adant2 1131	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑_m 𝐵) ↑_m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑_m (𝐵 × 𝐶)))) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)):𝐵⟶𝐴)
45		simp1l2 1268	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑_m 𝐵) ↑_m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑_m (𝐵 × 𝐶)))) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → 𝐵 ∈ 𝑊)
46		simp1l1 1267	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑_m 𝐵) ↑_m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑_m (𝐵 × 𝐶)))) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝑉)
47		fex2 7892	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)):𝐵⟶𝐴 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)) ∈ V)
48	44, 45, 46, 47	syl3anc 1373	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑_m 𝐵) ↑_m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑_m (𝐵 × 𝐶)))) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)) ∈ V)
49		eqid 2729	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))
50	49	fvmpt2 6961	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)) ∈ V) → ((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦) = (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))
51	42, 48, 50	syl2anc 584	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑_m 𝐵) ↑_m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑_m (𝐵 × 𝐶)))) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → ((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦) = (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))
52	51	fveq1d 6842	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑_m 𝐵) ↑_m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑_m (𝐵 × 𝐶)))) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → (((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥) = ((𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))‘𝑥))
53		simp2 1137	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑_m 𝐵) ↑_m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑_m (𝐵 × 𝐶)))) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → 𝑥 ∈ 𝐵)
54		ovex 7402	. . . . . . . . 9 ⊢ (𝑥𝑔𝑦) ∈ V
55		eqid 2729	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)) = (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))
56	55	fvmpt2 6961	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐵 ∧ (𝑥𝑔𝑦) ∈ V) → ((𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))‘𝑥) = (𝑥𝑔𝑦))
57	53, 54, 56	sylancl 586	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑_m 𝐵) ↑_m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑_m (𝐵 × 𝐶)))) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → ((𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))‘𝑥) = (𝑥𝑔𝑦))
58	52, 57	eqtrd 2764	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑_m 𝐵) ↑_m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑_m (𝐵 × 𝐶)))) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → (((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥) = (𝑥𝑔𝑦))
59	58	mpoeq3dva 7446	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑_m 𝐵) ↑_m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑_m (𝐵 × 𝐶)))) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (𝑥𝑔𝑦)))
60	41, 59	eqtr4d 2767	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑_m 𝐵) ↑_m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑_m (𝐵 × 𝐶)))) → 𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)))
61		eqid 2729	. . . . . . 7 ⊢ 𝐵 = 𝐵
62		nfcv 2891	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐶
63		nfmpt1 5201	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))
64	62, 63	nfmpt 5200	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))
65	64	nfeq2 2909	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))
66		nfmpt1 5201	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦(𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))
67	66	nfeq2 2909	. . . . . . . . . . 11 ⊢ Ⅎ𝑦 𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))
68		fveq1 6839	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) → (𝑓‘𝑦) = ((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦))
69	68	fveq1d 6842	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) → ((𝑓‘𝑦)‘𝑥) = (((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥))
70	69	a1d 25	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) → (𝑦 ∈ 𝐶 → ((𝑓‘𝑦)‘𝑥) = (((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)))
71	67, 70	ralrimi 3233	. . . . . . . . . 10 ⊢ (𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) → ∀𝑦 ∈ 𝐶 ((𝑓‘𝑦)‘𝑥) = (((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥))
72		eqid 2729	. . . . . . . . . 10 ⊢ 𝐶 = 𝐶
73	71, 72	jctil 519	. . . . . . . . 9 ⊢ (𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) → (𝐶 = 𝐶 ∧ ∀𝑦 ∈ 𝐶 ((𝑓‘𝑦)‘𝑥) = (((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)))
74	73	a1d 25	. . . . . . . 8 ⊢ (𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) → (𝑥 ∈ 𝐵 → (𝐶 = 𝐶 ∧ ∀𝑦 ∈ 𝐶 ((𝑓‘𝑦)‘𝑥) = (((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥))))
75	65, 74	ralrimi 3233	. . . . . . 7 ⊢ (𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) → ∀𝑥 ∈ 𝐵 (𝐶 = 𝐶 ∧ ∀𝑦 ∈ 𝐶 ((𝑓‘𝑦)‘𝑥) = (((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)))
76		mpoeq123 7441	. . . . . . 7 ⊢ ((𝐵 = 𝐵 ∧ ∀𝑥 ∈ 𝐵 (𝐶 = 𝐶 ∧ ∀𝑦 ∈ 𝐶 ((𝑓‘𝑦)‘𝑥) = (((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥))) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)))
77	61, 75, 76	sylancr 587	. . . . . 6 ⊢ (𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)))
78	77	eqeq2d 2740	. . . . 5 ⊢ (𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) → (𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) ↔ 𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥))))
79	60, 78	syl5ibrcom 247	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑_m 𝐵) ↑_m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑_m (𝐵 × 𝐶)))) → (𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) → 𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥))))
80	3	ad2antrl 728	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑_m 𝐵) ↑_m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑_m (𝐵 × 𝐶)))) → 𝑓:𝐶⟶(𝐴 ↑_m 𝐵))
81	80	feqmptd 6911	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑_m 𝐵) ↑_m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑_m (𝐵 × 𝐶)))) → 𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑓‘𝑦)))
82		simprl 770	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑_m 𝐵) ↑_m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑_m (𝐵 × 𝐶)))) → 𝑓 ∈ ((𝐴 ↑_m 𝐵) ↑_m 𝐶))
83	82, 6	sylan 580	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑_m 𝐵) ↑_m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑_m (𝐵 × 𝐶)))) ∧ 𝑦 ∈ 𝐶) → (𝑓‘𝑦):𝐵⟶𝐴)
84	83	feqmptd 6911	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑_m 𝐵) ↑_m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑_m (𝐵 × 𝐶)))) ∧ 𝑦 ∈ 𝐶) → (𝑓‘𝑦) = (𝑥 ∈ 𝐵 ↦ ((𝑓‘𝑦)‘𝑥)))
85	84	mpteq2dva 5195	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑_m 𝐵) ↑_m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑_m (𝐵 × 𝐶)))) → (𝑦 ∈ 𝐶 ↦ (𝑓‘𝑦)) = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ ((𝑓‘𝑦)‘𝑥))))
86	81, 85	eqtrd 2764	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑_m 𝐵) ↑_m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑_m (𝐵 × 𝐶)))) → 𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ ((𝑓‘𝑦)‘𝑥))))
87		nfmpo2 7450	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥))
88	87	nfeq2 2909	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥))
89		eqidd 2730	. . . . . . . . 9 ⊢ (𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) → 𝐵 = 𝐵)
90		nfmpo1 7449	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥))
91	90	nfeq2 2909	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥))
92		nfv 1914	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐶
93		fvex 6853	. . . . . . . . . . . . 13 ⊢ ((𝑓‘𝑦)‘𝑥) ∈ V
94	11	ovmpt4g 7516	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶 ∧ ((𝑓‘𝑦)‘𝑥) ∈ V) → (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥))𝑦) = ((𝑓‘𝑦)‘𝑥))
95	93, 94	mp3an3 1452	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥))𝑦) = ((𝑓‘𝑦)‘𝑥))
96		oveq 7375	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) → (𝑥𝑔𝑦) = (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥))𝑦))
97	96	eqeq1d 2731	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) → ((𝑥𝑔𝑦) = ((𝑓‘𝑦)‘𝑥) ↔ (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥))𝑦) = ((𝑓‘𝑦)‘𝑥)))
98	95, 97	imbitrrid 246	. . . . . . . . . . 11 ⊢ (𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → (𝑥𝑔𝑦) = ((𝑓‘𝑦)‘𝑥)))
99	98	expcomd 416	. . . . . . . . . 10 ⊢ (𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) → (𝑦 ∈ 𝐶 → (𝑥 ∈ 𝐵 → (𝑥𝑔𝑦) = ((𝑓‘𝑦)‘𝑥))))
100	91, 92, 99	ralrimd 3240	. . . . . . . . 9 ⊢ (𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) → (𝑦 ∈ 𝐶 → ∀𝑥 ∈ 𝐵 (𝑥𝑔𝑦) = ((𝑓‘𝑦)‘𝑥)))
101		mpteq12 5190	. . . . . . . . 9 ⊢ ((𝐵 = 𝐵 ∧ ∀𝑥 ∈ 𝐵 (𝑥𝑔𝑦) = ((𝑓‘𝑦)‘𝑥)) → (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)) = (𝑥 ∈ 𝐵 ↦ ((𝑓‘𝑦)‘𝑥)))
102	89, 100, 101	syl6an 684	. . . . . . . 8 ⊢ (𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) → (𝑦 ∈ 𝐶 → (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)) = (𝑥 ∈ 𝐵 ↦ ((𝑓‘𝑦)‘𝑥))))
103	88, 102	ralrimi 3233	. . . . . . 7 ⊢ (𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) → ∀𝑦 ∈ 𝐶 (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)) = (𝑥 ∈ 𝐵 ↦ ((𝑓‘𝑦)‘𝑥)))
104		mpteq12 5190	. . . . . . 7 ⊢ ((𝐶 = 𝐶 ∧ ∀𝑦 ∈ 𝐶 (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)) = (𝑥 ∈ 𝐵 ↦ ((𝑓‘𝑦)‘𝑥))) → (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ ((𝑓‘𝑦)‘𝑥))))
105	72, 103, 104	sylancr 587	. . . . . 6 ⊢ (𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) → (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ ((𝑓‘𝑦)‘𝑥))))
106	105	eqeq2d 2740	. . . . 5 ⊢ (𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) → (𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) ↔ 𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ ((𝑓‘𝑦)‘𝑥)))))
107	86, 106	syl5ibrcom 247	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑_m 𝐵) ↑_m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑_m (𝐵 × 𝐶)))) → (𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) → 𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))))
108	79, 107	impbid 212	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑_m 𝐵) ↑_m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑_m (𝐵 × 𝐶)))) → (𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) ↔ 𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥))))
109	108	ex 412	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝑓 ∈ ((𝐴 ↑_m 𝐵) ↑_m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑_m (𝐵 × 𝐶))) → (𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) ↔ 𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)))))
110	1, 2, 19, 37, 109	en3d 8937	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 ↑_m 𝐵) ↑_m 𝐶) ≈ (𝐴 ↑_m (𝐵 × 𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3044 Vcvv 3444 class class class wbr 5102 ↦ cmpt 5183 × cxp 5629 Fn wfn 6494 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 ∈ cmpo 7371 ↑_m cmap 8776 ≈ cen 8892

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 2701 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-ov 7372 df-oprab 7373 df-mpo 7374 df-1st 7947 df-2nd 7948 df-map 8778 df-en 8896

This theorem is referenced by: mappwen 10041 cfpwsdom 10513 rpnnen 16171 rexpen 16172 enrelmap 43959

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