Theorem mapxpen 6972

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		fnmap 6767	. . 3 ⊢ ↑𝑚 Fn (V × V)
2		elex 2789	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
3	2	3ad2ant1 1021	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐴 ∈ V)
4		elex 2789	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
5	4	3ad2ant2 1022	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐵 ∈ V)
6		fnovex 6002	. . . 4 ⊢ (( ↑𝑚 Fn (V × V) ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ↑𝑚 𝐵) ∈ V)
7	1, 3, 5, 6	mp3an2i 1355	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 ↑𝑚 𝐵) ∈ V)
8		elex 2789	. . . 4 ⊢ (𝐶 ∈ 𝑋 → 𝐶 ∈ V)
9	8	3ad2ant3 1023	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐶 ∈ V)
10		fnovex 6002	. . 3 ⊢ (( ↑𝑚 Fn (V × V) ∧ (𝐴 ↑𝑚 𝐵) ∈ V ∧ 𝐶 ∈ V) → ((𝐴 ↑𝑚 𝐵) ↑𝑚 𝐶) ∈ V)
11	1, 7, 9, 10	mp3an2i 1355	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 ↑𝑚 𝐵) ↑𝑚 𝐶) ∈ V)
12		xpexg 4808	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐵 × 𝐶) ∈ V)
13	12	3adant1 1018	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐵 × 𝐶) ∈ V)
14		fnovex 6002	. . 3 ⊢ (( ↑𝑚 Fn (V × V) ∧ 𝐴 ∈ V ∧ (𝐵 × 𝐶) ∈ V) → (𝐴 ↑𝑚 (𝐵 × 𝐶)) ∈ V)
15	1, 3, 13, 14	mp3an2i 1355	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 ↑𝑚 (𝐵 × 𝐶)) ∈ V)
16		elmapi 6782	. . . . . . . . . 10 ⊢ (𝑓 ∈ ((𝐴 ↑𝑚 𝐵) ↑𝑚 𝐶) → 𝑓:𝐶⟶(𝐴 ↑𝑚 𝐵))
17	16	ffvelcdmda 5740	. . . . . . . . 9 ⊢ ((𝑓 ∈ ((𝐴 ↑𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ 𝐶) → (𝑓‘𝑦) ∈ (𝐴 ↑𝑚 𝐵))
18		elmapi 6782	. . . . . . . . 9 ⊢ ((𝑓‘𝑦) ∈ (𝐴 ↑𝑚 𝐵) → (𝑓‘𝑦):𝐵⟶𝐴)
19	17, 18	syl 14	. . . . . . . 8 ⊢ ((𝑓 ∈ ((𝐴 ↑𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ 𝐶) → (𝑓‘𝑦):𝐵⟶𝐴)
20	19	ffvelcdmda 5740	. . . . . . 7 ⊢ (((𝑓 ∈ ((𝐴 ↑𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ∈ 𝐵) → ((𝑓‘𝑦)‘𝑥) ∈ 𝐴)
21	20	an32s 568	. . . . . 6 ⊢ (((𝑓 ∈ ((𝐴 ↑𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐶) → ((𝑓‘𝑦)‘𝑥) ∈ 𝐴)
22	21	ralrimiva 2581	. . . . 5 ⊢ ((𝑓 ∈ ((𝐴 ↑𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑥 ∈ 𝐵) → ∀𝑦 ∈ 𝐶 ((𝑓‘𝑦)‘𝑥) ∈ 𝐴)
23	22	ralrimiva 2581	. . . 4 ⊢ (𝑓 ∈ ((𝐴 ↑𝑚 𝐵) ↑𝑚 𝐶) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 ((𝑓‘𝑦)‘𝑥) ∈ 𝐴)
24		eqid 2207	. . . . 5 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥))
25	24	fmpo 6312	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 ((𝑓‘𝑦)‘𝑥) ∈ 𝐴 ↔ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)):(𝐵 × 𝐶)⟶𝐴)
26	23, 25	sylib 122	. . 3 ⊢ (𝑓 ∈ ((𝐴 ↑𝑚 𝐵) ↑𝑚 𝐶) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)):(𝐵 × 𝐶)⟶𝐴)
27		simp1 1000	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐴 ∈ 𝑉)
28	27, 13	elmapd 6774	. . 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))
29	26, 28	imbitrrid 156	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝑓 ∈ ((𝐴 ↑𝑚 𝐵) ↑𝑚 𝐶) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) ∈ (𝐴 ↑𝑚 (𝐵 × 𝐶))))
30		elmapi 6782	. . . . . . . . 9 ⊢ (𝑔 ∈ (𝐴 ↑𝑚 (𝐵 × 𝐶)) → 𝑔:(𝐵 × 𝐶)⟶𝐴)
31	30	adantl 277	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ 𝑔 ∈ (𝐴 ↑𝑚 (𝐵 × 𝐶))) → 𝑔:(𝐵 × 𝐶)⟶𝐴)
32		fovcdm 6114	. . . . . . . . . 10 ⊢ ((𝑔:(𝐵 × 𝐶)⟶𝐴 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → (𝑥𝑔𝑦) ∈ 𝐴)
33	32	3expa 1206	. . . . . . . . 9 ⊢ (((𝑔:(𝐵 × 𝐶)⟶𝐴 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐶) → (𝑥𝑔𝑦) ∈ 𝐴)
34	33	an32s 568	. . . . . . . 8 ⊢ (((𝑔:(𝐵 × 𝐶)⟶𝐴 ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ∈ 𝐵) → (𝑥𝑔𝑦) ∈ 𝐴)
35	31, 34	sylanl1 402	. . . . . . 7 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ 𝑔 ∈ (𝐴 ↑𝑚 (𝐵 × 𝐶))) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ∈ 𝐵) → (𝑥𝑔𝑦) ∈ 𝐴)
36		eqid 2207	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)) = (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))
37	35, 36	fmptd 5759	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ 𝑔 ∈ (𝐴 ↑𝑚 (𝐵 × 𝐶))) ∧ 𝑦 ∈ 𝐶) → (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)):𝐵⟶𝐴)
38		elmapg 6773	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)) ∈ (𝐴 ↑𝑚 𝐵) ↔ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)):𝐵⟶𝐴))
39	38	3adant3 1020	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)) ∈ (𝐴 ↑𝑚 𝐵) ↔ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)):𝐵⟶𝐴))
40	39	ad2antrr 488	. . . . . 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))
41	37, 40	mpbird 167	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ 𝑔 ∈ (𝐴 ↑𝑚 (𝐵 × 𝐶))) ∧ 𝑦 ∈ 𝐶) → (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)) ∈ (𝐴 ↑𝑚 𝐵))
42		eqid 2207	. . . . 5 ⊢ (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))
43	41, 42	fmptd 5759	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ 𝑔 ∈ (𝐴 ↑𝑚 (𝐵 × 𝐶))) → (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))):𝐶⟶(𝐴 ↑𝑚 𝐵))
44	43	ex 115	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝑔 ∈ (𝐴 ↑𝑚 (𝐵 × 𝐶)) → (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))):𝐶⟶(𝐴 ↑𝑚 𝐵)))
45		simp3 1002	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐶 ∈ 𝑋)
46	7, 45	elmapd 6774	. . 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)))
47	44, 46	sylibrd 169	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝑔 ∈ (𝐴 ↑𝑚 (𝐵 × 𝐶)) → (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) ∈ ((𝐴 ↑𝑚 𝐵) ↑𝑚 𝐶)))
48		elmapfn 6783	. . . . . . . 8 ⊢ (𝑔 ∈ (𝐴 ↑𝑚 (𝐵 × 𝐶)) → 𝑔 Fn (𝐵 × 𝐶))
49	48	ad2antll 491	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴 ↑𝑚 (𝐵 × 𝐶)))) → 𝑔 Fn (𝐵 × 𝐶))
50		fnovim 6079	. . . . . . 7 ⊢ (𝑔 Fn (𝐵 × 𝐶) → 𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (𝑥𝑔𝑦)))
51	49, 50	syl 14	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴 ↑𝑚 (𝐵 × 𝐶)))) → 𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (𝑥𝑔𝑦)))
52		simp3 1002	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴 ↑𝑚 (𝐵 × 𝐶)))) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → 𝑦 ∈ 𝐶)
53	37	adantlrl 482	. . . . . . . . . . . 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)
54	53	3adant2 1019	. . . . . . . . . . 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)
55		simp1l2 1094	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴 ↑𝑚 (𝐵 × 𝐶)))) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → 𝐵 ∈ 𝑊)
56		simp1l1 1093	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴 ↑𝑚 (𝐵 × 𝐶)))) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝑉)
57		fex2 5465	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)):𝐵⟶𝐴 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)) ∈ V)
58	54, 55, 56, 57	syl3anc 1250	. . . . . . . . . 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)
59	42	fvmpt2 5688	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)) ∈ V) → ((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦) = (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))
60	52, 58, 59	syl2anc 411	. . . . . . . . 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)))
61	60	fveq1d 5602	. . . . . . . 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))
62		simp2 1001	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴 ↑𝑚 (𝐵 × 𝐶)))) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → 𝑥 ∈ 𝐵)
63		vex 2780	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
64		vex 2780	. . . . . . . . . 10 ⊢ 𝑔 ∈ V
65		vex 2780	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
66		ovexg 6003	. . . . . . . . . 10 ⊢ ((𝑥 ∈ V ∧ 𝑔 ∈ V ∧ 𝑦 ∈ V) → (𝑥𝑔𝑦) ∈ V)
67	63, 64, 65, 66	mp3an 1350	. . . . . . . . 9 ⊢ (𝑥𝑔𝑦) ∈ V
68	36	fvmpt2 5688	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐵 ∧ (𝑥𝑔𝑦) ∈ V) → ((𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))‘𝑥) = (𝑥𝑔𝑦))
69	62, 67, 68	sylancl 413	. . . . . . . 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))
70	61, 69	eqtrd 2240	. . . . . . 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))
71	70	mpoeq3dva 6034	. . . . . 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)))
72	51, 71	eqtr4d 2243	. . . . 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)))
73		eqid 2207	. . . . . . 7 ⊢ 𝐵 = 𝐵
74		nfcv 2350	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐶
75		nfmpt1 4154	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))
76	74, 75	nfmpt 4153	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))
77	76	nfeq2 2362	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))
78		nfmpt1 4154	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦(𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))
79	78	nfeq2 2362	. . . . . . . . . . 11 ⊢ Ⅎ𝑦 𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))
80		fveq1 5599	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) → (𝑓‘𝑦) = ((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦))
81	80	fveq1d 5602	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) → ((𝑓‘𝑦)‘𝑥) = (((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥))
82	81	a1d 22	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) → (𝑦 ∈ 𝐶 → ((𝑓‘𝑦)‘𝑥) = (((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)))
83	79, 82	ralrimi 2579	. . . . . . . . . 10 ⊢ (𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) → ∀𝑦 ∈ 𝐶 ((𝑓‘𝑦)‘𝑥) = (((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥))
84		eqid 2207	. . . . . . . . . 10 ⊢ 𝐶 = 𝐶
85	83, 84	jctil 312	. . . . . . . . 9 ⊢ (𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) → (𝐶 = 𝐶 ∧ ∀𝑦 ∈ 𝐶 ((𝑓‘𝑦)‘𝑥) = (((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)))
86	85	a1d 22	. . . . . . . 8 ⊢ (𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) → (𝑥 ∈ 𝐵 → (𝐶 = 𝐶 ∧ ∀𝑦 ∈ 𝐶 ((𝑓‘𝑦)‘𝑥) = (((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥))))
87	77, 86	ralrimi 2579	. . . . . . 7 ⊢ (𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) → ∀𝑥 ∈ 𝐵 (𝐶 = 𝐶 ∧ ∀𝑦 ∈ 𝐶 ((𝑓‘𝑦)‘𝑥) = (((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)))
88		mpoeq123 6029	. . . . . . 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)))
89	73, 87, 88	sylancr 414	. . . . . 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)))
90	89	eqeq2d 2219	. . . . 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))))
91	72, 90	syl5ibrcom 157	. . . 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))))
92	16	ad2antrl 490	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴 ↑𝑚 (𝐵 × 𝐶)))) → 𝑓:𝐶⟶(𝐴 ↑𝑚 𝐵))
93	92	feqmptd 5657	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴 ↑𝑚 (𝐵 × 𝐶)))) → 𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑓‘𝑦)))
94		simprl 529	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴 ↑𝑚 (𝐵 × 𝐶)))) → 𝑓 ∈ ((𝐴 ↑𝑚 𝐵) ↑𝑚 𝐶))
95	94, 19	sylan 283	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴 ↑𝑚 (𝐵 × 𝐶)))) ∧ 𝑦 ∈ 𝐶) → (𝑓‘𝑦):𝐵⟶𝐴)
96	95	feqmptd 5657	. . . . . . 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)))
97	96	mpteq2dva 4151	. . . . . 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))))
98	93, 97	eqtrd 2240	. . . . 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))))
99		nfmpo2 6038	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥))
100	99	nfeq2 2362	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥))
101		eqidd 2208	. . . . . . . . 9 ⊢ (𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) → 𝐵 = 𝐵)
102		nfmpo1 6037	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥))
103	102	nfeq2 2362	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥))
104		nfv 1552	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐶
105		vex 2780	. . . . . . . . . . . . . . 15 ⊢ 𝑓 ∈ V
106	105, 65	fvex 5620	. . . . . . . . . . . . . 14 ⊢ (𝑓‘𝑦) ∈ V
107	106, 63	fvex 5620	. . . . . . . . . . . . 13 ⊢ ((𝑓‘𝑦)‘𝑥) ∈ V
108	24	ovmpt4g 6093	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶 ∧ ((𝑓‘𝑦)‘𝑥) ∈ V) → (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥))𝑦) = ((𝑓‘𝑦)‘𝑥))
109	107, 108	mp3an3 1339	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥))𝑦) = ((𝑓‘𝑦)‘𝑥))
110		oveq 5975	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) → (𝑥𝑔𝑦) = (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥))𝑦))
111	110	eqeq1d 2216	. . . . . . . . . . . 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)))
112	109, 111	imbitrrid 156	. . . . . . . . . . 11 ⊢ (𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → (𝑥𝑔𝑦) = ((𝑓‘𝑦)‘𝑥)))
113	112	expcomd 1462	. . . . . . . . . 10 ⊢ (𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) → (𝑦 ∈ 𝐶 → (𝑥 ∈ 𝐵 → (𝑥𝑔𝑦) = ((𝑓‘𝑦)‘𝑥))))
114	103, 104, 113	ralrimd 2586	. . . . . . . . 9 ⊢ (𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) → (𝑦 ∈ 𝐶 → ∀𝑥 ∈ 𝐵 (𝑥𝑔𝑦) = ((𝑓‘𝑦)‘𝑥)))
115		mpteq12 4144	. . . . . . . . 9 ⊢ ((𝐵 = 𝐵 ∧ ∀𝑥 ∈ 𝐵 (𝑥𝑔𝑦) = ((𝑓‘𝑦)‘𝑥)) → (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)) = (𝑥 ∈ 𝐵 ↦ ((𝑓‘𝑦)‘𝑥)))
116	101, 114, 115	syl6an 1454	. . . . . . . 8 ⊢ (𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) → (𝑦 ∈ 𝐶 → (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)) = (𝑥 ∈ 𝐵 ↦ ((𝑓‘𝑦)‘𝑥))))
117	100, 116	ralrimi 2579	. . . . . . 7 ⊢ (𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) → ∀𝑦 ∈ 𝐶 (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)) = (𝑥 ∈ 𝐵 ↦ ((𝑓‘𝑦)‘𝑥)))
118		mpteq12 4144	. . . . . . 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))))
119	84, 117, 118	sylancr 414	. . . . . 6 ⊢ (𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) → (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ ((𝑓‘𝑦)‘𝑥))))
120	119	eqeq2d 2219	. . . . 5 ⊢ (𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) → (𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) ↔ 𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ ((𝑓‘𝑦)‘𝑥)))))
121	98, 120	syl5ibrcom 157	. . . 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)))))
122	91, 121	impbid 129	. . 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))))
123	122	ex 115	. 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)))))
124	11, 15, 29, 47, 123	en3d 6885	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 ↑𝑚 𝐵) ↑𝑚 𝐶) ≈ (𝐴 ↑𝑚 (𝐵 × 𝐶)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 981   = wceq 1373   ∈ wcel 2178  ∀wral 2486  Vcvv 2777   class class class wbr 4060   ↦ cmpt 4122   × cxp 4692   Fn wfn 5286  ⟶wf 5287  ‘cfv 5291  (class class class)co 5969   ∈ cmpo 5971   ↑_𝑚 cmap 6760   ≈ cen 6850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4179  ax-pow 4235  ax-pr 4270  ax-un 4499  ax-setind 4604

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2779  df-sbc 3007  df-csb 3103  df-dif 3177  df-un 3179  df-in 3181  df-ss 3188  df-pw 3629  df-sn 3650  df-pr 3651  df-op 3653  df-uni 3866  df-iun 3944  df-br 4061  df-opab 4123  df-mpt 4124  df-id 4359  df-xp 4700  df-rel 4701  df-cnv 4702  df-co 4703  df-dm 4704  df-rn 4705  df-res 4706  df-ima 4707  df-iota 5252  df-fun 5293  df-fn 5294  df-f 5295  df-f1 5296  df-fo 5297  df-f1o 5298  df-fv 5299  df-ov 5972  df-oprab 5973  df-mpo 5974  df-1st 6251  df-2nd 6252  df-map 6762  df-en 6853

This theorem is referenced by: (None)