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Theorem mapxpen 8746
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 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴m 𝐵) ↑m 𝐶) ≈ (𝐴m (𝐵 × 𝐶)))

Proof of Theorem mapxpen
Dummy variables 𝑓 𝑔 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovexd 7218 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴m 𝐵) ↑m 𝐶) ∈ V)
2 ovexd 7218 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴m (𝐵 × 𝐶)) ∈ V)
3 elmapi 8472 . . . . . . . . . 10 (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) → 𝑓:𝐶⟶(𝐴m 𝐵))
43ffvelrnda 6874 . . . . . . . . 9 ((𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑦𝐶) → (𝑓𝑦) ∈ (𝐴m 𝐵))
5 elmapi 8472 . . . . . . . . 9 ((𝑓𝑦) ∈ (𝐴m 𝐵) → (𝑓𝑦):𝐵𝐴)
64, 5syl 17 . . . . . . . 8 ((𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑦𝐶) → (𝑓𝑦):𝐵𝐴)
76ffvelrnda 6874 . . . . . . 7 (((𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑦𝐶) ∧ 𝑥𝐵) → ((𝑓𝑦)‘𝑥) ∈ 𝐴)
87an32s 652 . . . . . 6 (((𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑥𝐵) ∧ 𝑦𝐶) → ((𝑓𝑦)‘𝑥) ∈ 𝐴)
98ralrimiva 3097 . . . . 5 ((𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑥𝐵) → ∀𝑦𝐶 ((𝑓𝑦)‘𝑥) ∈ 𝐴)
109ralrimiva 3097 . . . 4 (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) → ∀𝑥𝐵𝑦𝐶 ((𝑓𝑦)‘𝑥) ∈ 𝐴)
11 eqid 2739 . . . . 5 (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))
1211fmpo 7804 . . . 4 (∀𝑥𝐵𝑦𝐶 ((𝑓𝑦)‘𝑥) ∈ 𝐴 ↔ (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)):(𝐵 × 𝐶)⟶𝐴)
1310, 12sylib 221 . . 3 (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) → (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)):(𝐵 × 𝐶)⟶𝐴)
14 simp1 1137 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐴𝑉)
15 xpexg 7504 . . . . 5 ((𝐵𝑊𝐶𝑋) → (𝐵 × 𝐶) ∈ V)
16153adant1 1131 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐵 × 𝐶) ∈ V)
17 elmapg 8463 . . . 4 ((𝐴𝑉 ∧ (𝐵 × 𝐶) ∈ V) → ((𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) ∈ (𝐴m (𝐵 × 𝐶)) ↔ (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)):(𝐵 × 𝐶)⟶𝐴))
1814, 16, 17syl2anc 587 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) ∈ (𝐴m (𝐵 × 𝐶)) ↔ (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)):(𝐵 × 𝐶)⟶𝐴))
1913, 18syl5ibr 249 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) → (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) ∈ (𝐴m (𝐵 × 𝐶))))
20 elmapi 8472 . . . . . . . . 9 (𝑔 ∈ (𝐴m (𝐵 × 𝐶)) → 𝑔:(𝐵 × 𝐶)⟶𝐴)
2120adantl 485 . . . . . . . 8 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶))) → 𝑔:(𝐵 × 𝐶)⟶𝐴)
22 fovrn 7347 . . . . . . . . . 10 ((𝑔:(𝐵 × 𝐶)⟶𝐴𝑥𝐵𝑦𝐶) → (𝑥𝑔𝑦) ∈ 𝐴)
23223expa 1119 . . . . . . . . 9 (((𝑔:(𝐵 × 𝐶)⟶𝐴𝑥𝐵) ∧ 𝑦𝐶) → (𝑥𝑔𝑦) ∈ 𝐴)
2423an32s 652 . . . . . . . 8 (((𝑔:(𝐵 × 𝐶)⟶𝐴𝑦𝐶) ∧ 𝑥𝐵) → (𝑥𝑔𝑦) ∈ 𝐴)
2521, 24sylanl1 680 . . . . . . 7 (((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶))) ∧ 𝑦𝐶) ∧ 𝑥𝐵) → (𝑥𝑔𝑦) ∈ 𝐴)
2625fmpttd 6902 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶))) ∧ 𝑦𝐶) → (𝑥𝐵 ↦ (𝑥𝑔𝑦)):𝐵𝐴)
27 elmapg 8463 . . . . . . . 8 ((𝐴𝑉𝐵𝑊) → ((𝑥𝐵 ↦ (𝑥𝑔𝑦)) ∈ (𝐴m 𝐵) ↔ (𝑥𝐵 ↦ (𝑥𝑔𝑦)):𝐵𝐴))
28273adant3 1133 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝑥𝐵 ↦ (𝑥𝑔𝑦)) ∈ (𝐴m 𝐵) ↔ (𝑥𝐵 ↦ (𝑥𝑔𝑦)):𝐵𝐴))
2928ad2antrr 726 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶))) ∧ 𝑦𝐶) → ((𝑥𝐵 ↦ (𝑥𝑔𝑦)) ∈ (𝐴m 𝐵) ↔ (𝑥𝐵 ↦ (𝑥𝑔𝑦)):𝐵𝐴))
3026, 29mpbird 260 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶))) ∧ 𝑦𝐶) → (𝑥𝐵 ↦ (𝑥𝑔𝑦)) ∈ (𝐴m 𝐵))
3130fmpttd 6902 . . . 4 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶))) → (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))):𝐶⟶(𝐴m 𝐵))
3231ex 416 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝑔 ∈ (𝐴m (𝐵 × 𝐶)) → (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))):𝐶⟶(𝐴m 𝐵)))
33 ovex 7216 . . . 4 (𝐴m 𝐵) ∈ V
34 simp3 1139 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐶𝑋)
35 elmapg 8463 . . . 4 (((𝐴m 𝐵) ∈ V ∧ 𝐶𝑋) → ((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) ∈ ((𝐴m 𝐵) ↑m 𝐶) ↔ (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))):𝐶⟶(𝐴m 𝐵)))
3633, 34, 35sylancr 590 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) ∈ ((𝐴m 𝐵) ↑m 𝐶) ↔ (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))):𝐶⟶(𝐴m 𝐵)))
3732, 36sylibrd 262 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝑔 ∈ (𝐴m (𝐵 × 𝐶)) → (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) ∈ ((𝐴m 𝐵) ↑m 𝐶)))
38 elmapfn 8488 . . . . . . . 8 (𝑔 ∈ (𝐴m (𝐵 × 𝐶)) → 𝑔 Fn (𝐵 × 𝐶))
3938ad2antll 729 . . . . . . 7 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) → 𝑔 Fn (𝐵 × 𝐶))
40 fnov 7310 . . . . . . 7 (𝑔 Fn (𝐵 × 𝐶) ↔ 𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑔𝑦)))
4139, 40sylib 221 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) → 𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑔𝑦)))
42 simp3 1139 . . . . . . . . . 10 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → 𝑦𝐶)
4326adantlrl 720 . . . . . . . . . . . 12 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) ∧ 𝑦𝐶) → (𝑥𝐵 ↦ (𝑥𝑔𝑦)):𝐵𝐴)
44433adant2 1132 . . . . . . . . . . 11 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → (𝑥𝐵 ↦ (𝑥𝑔𝑦)):𝐵𝐴)
45 simp1l2 1268 . . . . . . . . . . 11 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → 𝐵𝑊)
46 simp1l1 1267 . . . . . . . . . . 11 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → 𝐴𝑉)
47 fex2 7677 . . . . . . . . . . 11 (((𝑥𝐵 ↦ (𝑥𝑔𝑦)):𝐵𝐴𝐵𝑊𝐴𝑉) → (𝑥𝐵 ↦ (𝑥𝑔𝑦)) ∈ V)
4844, 45, 46, 47syl3anc 1372 . . . . . . . . . 10 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → (𝑥𝐵 ↦ (𝑥𝑔𝑦)) ∈ V)
49 eqid 2739 . . . . . . . . . . 11 (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))
5049fvmpt2 6799 . . . . . . . . . 10 ((𝑦𝐶 ∧ (𝑥𝐵 ↦ (𝑥𝑔𝑦)) ∈ V) → ((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦) = (𝑥𝐵 ↦ (𝑥𝑔𝑦)))
5142, 48, 50syl2anc 587 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → ((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦) = (𝑥𝐵 ↦ (𝑥𝑔𝑦)))
5251fveq1d 6689 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥) = ((𝑥𝐵 ↦ (𝑥𝑔𝑦))‘𝑥))
53 simp2 1138 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → 𝑥𝐵)
54 ovex 7216 . . . . . . . . 9 (𝑥𝑔𝑦) ∈ V
55 eqid 2739 . . . . . . . . . 10 (𝑥𝐵 ↦ (𝑥𝑔𝑦)) = (𝑥𝐵 ↦ (𝑥𝑔𝑦))
5655fvmpt2 6799 . . . . . . . . 9 ((𝑥𝐵 ∧ (𝑥𝑔𝑦) ∈ V) → ((𝑥𝐵 ↦ (𝑥𝑔𝑦))‘𝑥) = (𝑥𝑔𝑦))
5753, 54, 56sylancl 589 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → ((𝑥𝐵 ↦ (𝑥𝑔𝑦))‘𝑥) = (𝑥𝑔𝑦))
5852, 57eqtrd 2774 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥) = (𝑥𝑔𝑦))
5958mpoeq3dva 7258 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) → (𝑥𝐵, 𝑦𝐶 ↦ (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)) = (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑔𝑦)))
6041, 59eqtr4d 2777 . . . . 5 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) → 𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)))
61 eqid 2739 . . . . . . 7 𝐵 = 𝐵
62 nfcv 2900 . . . . . . . . . 10 𝑥𝐶
63 nfmpt1 5138 . . . . . . . . . 10 𝑥(𝑥𝐵 ↦ (𝑥𝑔𝑦))
6462, 63nfmpt 5137 . . . . . . . . 9 𝑥(𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))
6564nfeq2 2917 . . . . . . . 8 𝑥 𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))
66 nfmpt1 5138 . . . . . . . . . . . 12 𝑦(𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))
6766nfeq2 2917 . . . . . . . . . . 11 𝑦 𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))
68 fveq1 6686 . . . . . . . . . . . . 13 (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) → (𝑓𝑦) = ((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦))
6968fveq1d 6689 . . . . . . . . . . . 12 (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) → ((𝑓𝑦)‘𝑥) = (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥))
7069a1d 25 . . . . . . . . . . 11 (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) → (𝑦𝐶 → ((𝑓𝑦)‘𝑥) = (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)))
7167, 70ralrimi 3129 . . . . . . . . . 10 (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) → ∀𝑦𝐶 ((𝑓𝑦)‘𝑥) = (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥))
72 eqid 2739 . . . . . . . . . 10 𝐶 = 𝐶
7371, 72jctil 523 . . . . . . . . 9 (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) → (𝐶 = 𝐶 ∧ ∀𝑦𝐶 ((𝑓𝑦)‘𝑥) = (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)))
7473a1d 25 . . . . . . . 8 (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) → (𝑥𝐵 → (𝐶 = 𝐶 ∧ ∀𝑦𝐶 ((𝑓𝑦)‘𝑥) = (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥))))
7565, 74ralrimi 3129 . . . . . . 7 (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) → ∀𝑥𝐵 (𝐶 = 𝐶 ∧ ∀𝑦𝐶 ((𝑓𝑦)‘𝑥) = (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)))
76 mpoeq123 7253 . . . . . . 7 ((𝐵 = 𝐵 ∧ ∀𝑥𝐵 (𝐶 = 𝐶 ∧ ∀𝑦𝐶 ((𝑓𝑦)‘𝑥) = (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥))) → (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) = (𝑥𝐵, 𝑦𝐶 ↦ (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)))
7761, 75, 76sylancr 590 . . . . . 6 (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) → (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) = (𝑥𝐵, 𝑦𝐶 ↦ (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)))
7877eqeq2d 2750 . . . . 5 (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) → (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) ↔ 𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥))))
7960, 78syl5ibrcom 250 . . . 4 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) → (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) → 𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))))
803ad2antrl 728 . . . . . . 7 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) → 𝑓:𝐶⟶(𝐴m 𝐵))
8180feqmptd 6750 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) → 𝑓 = (𝑦𝐶 ↦ (𝑓𝑦)))
82 simprl 771 . . . . . . . . 9 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) → 𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶))
8382, 6sylan 583 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) ∧ 𝑦𝐶) → (𝑓𝑦):𝐵𝐴)
8483feqmptd 6750 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) ∧ 𝑦𝐶) → (𝑓𝑦) = (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥)))
8584mpteq2dva 5135 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) → (𝑦𝐶 ↦ (𝑓𝑦)) = (𝑦𝐶 ↦ (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥))))
8681, 85eqtrd 2774 . . . . 5 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) → 𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥))))
87 nfmpo2 7262 . . . . . . . . 9 𝑦(𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))
8887nfeq2 2917 . . . . . . . 8 𝑦 𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))
89 eqidd 2740 . . . . . . . . 9 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → 𝐵 = 𝐵)
90 nfmpo1 7261 . . . . . . . . . . 11 𝑥(𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))
9190nfeq2 2917 . . . . . . . . . 10 𝑥 𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))
92 nfv 1921 . . . . . . . . . 10 𝑥 𝑦𝐶
93 fvex 6700 . . . . . . . . . . . . 13 ((𝑓𝑦)‘𝑥) ∈ V
9411ovmpt4g 7325 . . . . . . . . . . . . 13 ((𝑥𝐵𝑦𝐶 ∧ ((𝑓𝑦)‘𝑥) ∈ V) → (𝑥(𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))𝑦) = ((𝑓𝑦)‘𝑥))
9593, 94mp3an3 1451 . . . . . . . . . . . 12 ((𝑥𝐵𝑦𝐶) → (𝑥(𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))𝑦) = ((𝑓𝑦)‘𝑥))
96 oveq 7189 . . . . . . . . . . . . 13 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → (𝑥𝑔𝑦) = (𝑥(𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))𝑦))
9796eqeq1d 2741 . . . . . . . . . . . 12 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → ((𝑥𝑔𝑦) = ((𝑓𝑦)‘𝑥) ↔ (𝑥(𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))𝑦) = ((𝑓𝑦)‘𝑥)))
9895, 97syl5ibr 249 . . . . . . . . . . 11 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → ((𝑥𝐵𝑦𝐶) → (𝑥𝑔𝑦) = ((𝑓𝑦)‘𝑥)))
9998expcomd 420 . . . . . . . . . 10 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → (𝑦𝐶 → (𝑥𝐵 → (𝑥𝑔𝑦) = ((𝑓𝑦)‘𝑥))))
10091, 92, 99ralrimd 3131 . . . . . . . . 9 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → (𝑦𝐶 → ∀𝑥𝐵 (𝑥𝑔𝑦) = ((𝑓𝑦)‘𝑥)))
101 mpteq12 5127 . . . . . . . . 9 ((𝐵 = 𝐵 ∧ ∀𝑥𝐵 (𝑥𝑔𝑦) = ((𝑓𝑦)‘𝑥)) → (𝑥𝐵 ↦ (𝑥𝑔𝑦)) = (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥)))
10289, 100, 101syl6an 684 . . . . . . . 8 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → (𝑦𝐶 → (𝑥𝐵 ↦ (𝑥𝑔𝑦)) = (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥))))
10388, 102ralrimi 3129 . . . . . . 7 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → ∀𝑦𝐶 (𝑥𝐵 ↦ (𝑥𝑔𝑦)) = (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥)))
104 mpteq12 5127 . . . . . . 7 ((𝐶 = 𝐶 ∧ ∀𝑦𝐶 (𝑥𝐵 ↦ (𝑥𝑔𝑦)) = (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥))) → (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) = (𝑦𝐶 ↦ (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥))))
10572, 103, 104sylancr 590 . . . . . 6 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) = (𝑦𝐶 ↦ (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥))))
106105eqeq2d 2750 . . . . 5 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) ↔ 𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥)))))
10786, 106syl5ibrcom 250 . . . 4 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) → (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → 𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))))
10879, 107impbid 215 . . 3 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) → (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) ↔ 𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))))
109108ex 416 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶))) → (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) ↔ 𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)))))
1101, 2, 19, 37, 109en3d 8605 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴m 𝐵) ↑m 𝐶) ≈ (𝐴m (𝐵 × 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1088   = wceq 1542  wcel 2114  wral 3054  Vcvv 3400   class class class wbr 5040  cmpt 5120   × cxp 5533   Fn wfn 6345  wf 6346  cfv 6350  (class class class)co 7183  cmpo 7185  m cmap 8450  cen 8565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7492
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-iun 4893  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-iota 6308  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-ov 7186  df-oprab 7187  df-mpo 7188  df-1st 7727  df-2nd 7728  df-map 8452  df-en 8569
This theorem is referenced by:  mappwen  9625  cfpwsdom  10097  rpnnen  15685  rexpen  15686  enrelmap  41192
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