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Theorem mapxpen 9087
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 7392 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴m 𝐵) ↑m 𝐶) ∈ V)
2 ovexd 7392 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴m (𝐵 × 𝐶)) ∈ V)
3 elmapi 8787 . . . . . . . . . 10 (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) → 𝑓:𝐶⟶(𝐴m 𝐵))
43ffvelcdmda 7035 . . . . . . . . 9 ((𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑦𝐶) → (𝑓𝑦) ∈ (𝐴m 𝐵))
5 elmapi 8787 . . . . . . . . 9 ((𝑓𝑦) ∈ (𝐴m 𝐵) → (𝑓𝑦):𝐵𝐴)
64, 5syl 17 . . . . . . . 8 ((𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑦𝐶) → (𝑓𝑦):𝐵𝐴)
76ffvelcdmda 7035 . . . . . . 7 (((𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑦𝐶) ∧ 𝑥𝐵) → ((𝑓𝑦)‘𝑥) ∈ 𝐴)
87an32s 650 . . . . . 6 (((𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑥𝐵) ∧ 𝑦𝐶) → ((𝑓𝑦)‘𝑥) ∈ 𝐴)
98ralrimiva 3143 . . . . 5 ((𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑥𝐵) → ∀𝑦𝐶 ((𝑓𝑦)‘𝑥) ∈ 𝐴)
109ralrimiva 3143 . . . 4 (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) → ∀𝑥𝐵𝑦𝐶 ((𝑓𝑦)‘𝑥) ∈ 𝐴)
11 eqid 2736 . . . . 5 (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))
1211fmpo 8000 . . . 4 (∀𝑥𝐵𝑦𝐶 ((𝑓𝑦)‘𝑥) ∈ 𝐴 ↔ (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)):(𝐵 × 𝐶)⟶𝐴)
1310, 12sylib 217 . . 3 (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) → (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)):(𝐵 × 𝐶)⟶𝐴)
14 simp1 1136 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐴𝑉)
15 xpexg 7684 . . . . 5 ((𝐵𝑊𝐶𝑋) → (𝐵 × 𝐶) ∈ V)
16153adant1 1130 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐵 × 𝐶) ∈ V)
17 elmapg 8778 . . . 4 ((𝐴𝑉 ∧ (𝐵 × 𝐶) ∈ V) → ((𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) ∈ (𝐴m (𝐵 × 𝐶)) ↔ (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)):(𝐵 × 𝐶)⟶𝐴))
1814, 16, 17syl2anc 584 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) ∈ (𝐴m (𝐵 × 𝐶)) ↔ (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)):(𝐵 × 𝐶)⟶𝐴))
1913, 18syl5ibr 245 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) → (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) ∈ (𝐴m (𝐵 × 𝐶))))
20 elmapi 8787 . . . . . . . . 9 (𝑔 ∈ (𝐴m (𝐵 × 𝐶)) → 𝑔:(𝐵 × 𝐶)⟶𝐴)
2120adantl 482 . . . . . . . 8 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶))) → 𝑔:(𝐵 × 𝐶)⟶𝐴)
22 fovcdm 7524 . . . . . . . . . 10 ((𝑔:(𝐵 × 𝐶)⟶𝐴𝑥𝐵𝑦𝐶) → (𝑥𝑔𝑦) ∈ 𝐴)
23223expa 1118 . . . . . . . . 9 (((𝑔:(𝐵 × 𝐶)⟶𝐴𝑥𝐵) ∧ 𝑦𝐶) → (𝑥𝑔𝑦) ∈ 𝐴)
2423an32s 650 . . . . . . . 8 (((𝑔:(𝐵 × 𝐶)⟶𝐴𝑦𝐶) ∧ 𝑥𝐵) → (𝑥𝑔𝑦) ∈ 𝐴)
2521, 24sylanl1 678 . . . . . . 7 (((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶))) ∧ 𝑦𝐶) ∧ 𝑥𝐵) → (𝑥𝑔𝑦) ∈ 𝐴)
2625fmpttd 7063 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶))) ∧ 𝑦𝐶) → (𝑥𝐵 ↦ (𝑥𝑔𝑦)):𝐵𝐴)
27 elmapg 8778 . . . . . . . 8 ((𝐴𝑉𝐵𝑊) → ((𝑥𝐵 ↦ (𝑥𝑔𝑦)) ∈ (𝐴m 𝐵) ↔ (𝑥𝐵 ↦ (𝑥𝑔𝑦)):𝐵𝐴))
28273adant3 1132 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝑥𝐵 ↦ (𝑥𝑔𝑦)) ∈ (𝐴m 𝐵) ↔ (𝑥𝐵 ↦ (𝑥𝑔𝑦)):𝐵𝐴))
2928ad2antrr 724 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶))) ∧ 𝑦𝐶) → ((𝑥𝐵 ↦ (𝑥𝑔𝑦)) ∈ (𝐴m 𝐵) ↔ (𝑥𝐵 ↦ (𝑥𝑔𝑦)):𝐵𝐴))
3026, 29mpbird 256 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶))) ∧ 𝑦𝐶) → (𝑥𝐵 ↦ (𝑥𝑔𝑦)) ∈ (𝐴m 𝐵))
3130fmpttd 7063 . . . 4 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶))) → (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))):𝐶⟶(𝐴m 𝐵))
3231ex 413 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝑔 ∈ (𝐴m (𝐵 × 𝐶)) → (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))):𝐶⟶(𝐴m 𝐵)))
33 ovex 7390 . . . 4 (𝐴m 𝐵) ∈ V
34 simp3 1138 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐶𝑋)
35 elmapg 8778 . . . 4 (((𝐴m 𝐵) ∈ V ∧ 𝐶𝑋) → ((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) ∈ ((𝐴m 𝐵) ↑m 𝐶) ↔ (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))):𝐶⟶(𝐴m 𝐵)))
3633, 34, 35sylancr 587 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) ∈ ((𝐴m 𝐵) ↑m 𝐶) ↔ (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))):𝐶⟶(𝐴m 𝐵)))
3732, 36sylibrd 258 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝑔 ∈ (𝐴m (𝐵 × 𝐶)) → (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) ∈ ((𝐴m 𝐵) ↑m 𝐶)))
38 elmapfn 8803 . . . . . . . 8 (𝑔 ∈ (𝐴m (𝐵 × 𝐶)) → 𝑔 Fn (𝐵 × 𝐶))
3938ad2antll 727 . . . . . . 7 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) → 𝑔 Fn (𝐵 × 𝐶))
40 fnov 7487 . . . . . . 7 (𝑔 Fn (𝐵 × 𝐶) ↔ 𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑔𝑦)))
4139, 40sylib 217 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) → 𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑔𝑦)))
42 simp3 1138 . . . . . . . . . 10 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → 𝑦𝐶)
4326adantlrl 718 . . . . . . . . . . . 12 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) ∧ 𝑦𝐶) → (𝑥𝐵 ↦ (𝑥𝑔𝑦)):𝐵𝐴)
44433adant2 1131 . . . . . . . . . . 11 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → (𝑥𝐵 ↦ (𝑥𝑔𝑦)):𝐵𝐴)
45 simp1l2 1267 . . . . . . . . . . 11 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → 𝐵𝑊)
46 simp1l1 1266 . . . . . . . . . . 11 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → 𝐴𝑉)
47 fex2 7870 . . . . . . . . . . 11 (((𝑥𝐵 ↦ (𝑥𝑔𝑦)):𝐵𝐴𝐵𝑊𝐴𝑉) → (𝑥𝐵 ↦ (𝑥𝑔𝑦)) ∈ V)
4844, 45, 46, 47syl3anc 1371 . . . . . . . . . 10 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → (𝑥𝐵 ↦ (𝑥𝑔𝑦)) ∈ V)
49 eqid 2736 . . . . . . . . . . 11 (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))
5049fvmpt2 6959 . . . . . . . . . 10 ((𝑦𝐶 ∧ (𝑥𝐵 ↦ (𝑥𝑔𝑦)) ∈ V) → ((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦) = (𝑥𝐵 ↦ (𝑥𝑔𝑦)))
5142, 48, 50syl2anc 584 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → ((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦) = (𝑥𝐵 ↦ (𝑥𝑔𝑦)))
5251fveq1d 6844 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥) = ((𝑥𝐵 ↦ (𝑥𝑔𝑦))‘𝑥))
53 simp2 1137 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → 𝑥𝐵)
54 ovex 7390 . . . . . . . . 9 (𝑥𝑔𝑦) ∈ V
55 eqid 2736 . . . . . . . . . 10 (𝑥𝐵 ↦ (𝑥𝑔𝑦)) = (𝑥𝐵 ↦ (𝑥𝑔𝑦))
5655fvmpt2 6959 . . . . . . . . 9 ((𝑥𝐵 ∧ (𝑥𝑔𝑦) ∈ V) → ((𝑥𝐵 ↦ (𝑥𝑔𝑦))‘𝑥) = (𝑥𝑔𝑦))
5753, 54, 56sylancl 586 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → ((𝑥𝐵 ↦ (𝑥𝑔𝑦))‘𝑥) = (𝑥𝑔𝑦))
5852, 57eqtrd 2776 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥) = (𝑥𝑔𝑦))
5958mpoeq3dva 7434 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) → (𝑥𝐵, 𝑦𝐶 ↦ (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)) = (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑔𝑦)))
6041, 59eqtr4d 2779 . . . . 5 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) → 𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)))
61 eqid 2736 . . . . . . 7 𝐵 = 𝐵
62 nfcv 2907 . . . . . . . . . 10 𝑥𝐶
63 nfmpt1 5213 . . . . . . . . . 10 𝑥(𝑥𝐵 ↦ (𝑥𝑔𝑦))
6462, 63nfmpt 5212 . . . . . . . . 9 𝑥(𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))
6564nfeq2 2924 . . . . . . . 8 𝑥 𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))
66 nfmpt1 5213 . . . . . . . . . . . 12 𝑦(𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))
6766nfeq2 2924 . . . . . . . . . . 11 𝑦 𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))
68 fveq1 6841 . . . . . . . . . . . . 13 (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) → (𝑓𝑦) = ((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦))
6968fveq1d 6844 . . . . . . . . . . . 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 3240 . . . . . . . . . 10 (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) → ∀𝑦𝐶 ((𝑓𝑦)‘𝑥) = (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥))
72 eqid 2736 . . . . . . . . . 10 𝐶 = 𝐶
7371, 72jctil 520 . . . . . . . . 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 3240 . . . . . . 7 (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) → ∀𝑥𝐵 (𝐶 = 𝐶 ∧ ∀𝑦𝐶 ((𝑓𝑦)‘𝑥) = (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)))
76 mpoeq123 7429 . . . . . . 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 587 . . . . . 6 (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) → (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) = (𝑥𝐵, 𝑦𝐶 ↦ (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)))
7877eqeq2d 2747 . . . . 5 (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) → (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) ↔ 𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥))))
7960, 78syl5ibrcom 246 . . . 4 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) → (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) → 𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))))
803ad2antrl 726 . . . . . . 7 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) → 𝑓:𝐶⟶(𝐴m 𝐵))
8180feqmptd 6910 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) → 𝑓 = (𝑦𝐶 ↦ (𝑓𝑦)))
82 simprl 769 . . . . . . . . 9 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) → 𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶))
8382, 6sylan 580 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) ∧ 𝑦𝐶) → (𝑓𝑦):𝐵𝐴)
8483feqmptd 6910 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) ∧ 𝑦𝐶) → (𝑓𝑦) = (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥)))
8584mpteq2dva 5205 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) → (𝑦𝐶 ↦ (𝑓𝑦)) = (𝑦𝐶 ↦ (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥))))
8681, 85eqtrd 2776 . . . . 5 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) → 𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥))))
87 nfmpo2 7438 . . . . . . . . 9 𝑦(𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))
8887nfeq2 2924 . . . . . . . 8 𝑦 𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))
89 eqidd 2737 . . . . . . . . 9 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → 𝐵 = 𝐵)
90 nfmpo1 7437 . . . . . . . . . . 11 𝑥(𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))
9190nfeq2 2924 . . . . . . . . . 10 𝑥 𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))
92 nfv 1917 . . . . . . . . . 10 𝑥 𝑦𝐶
93 fvex 6855 . . . . . . . . . . . . 13 ((𝑓𝑦)‘𝑥) ∈ V
9411ovmpt4g 7502 . . . . . . . . . . . . 13 ((𝑥𝐵𝑦𝐶 ∧ ((𝑓𝑦)‘𝑥) ∈ V) → (𝑥(𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))𝑦) = ((𝑓𝑦)‘𝑥))
9593, 94mp3an3 1450 . . . . . . . . . . . 12 ((𝑥𝐵𝑦𝐶) → (𝑥(𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))𝑦) = ((𝑓𝑦)‘𝑥))
96 oveq 7363 . . . . . . . . . . . . 13 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → (𝑥𝑔𝑦) = (𝑥(𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))𝑦))
9796eqeq1d 2738 . . . . . . . . . . . 12 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → ((𝑥𝑔𝑦) = ((𝑓𝑦)‘𝑥) ↔ (𝑥(𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))𝑦) = ((𝑓𝑦)‘𝑥)))
9895, 97syl5ibr 245 . . . . . . . . . . 11 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → ((𝑥𝐵𝑦𝐶) → (𝑥𝑔𝑦) = ((𝑓𝑦)‘𝑥)))
9998expcomd 417 . . . . . . . . . 10 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → (𝑦𝐶 → (𝑥𝐵 → (𝑥𝑔𝑦) = ((𝑓𝑦)‘𝑥))))
10091, 92, 99ralrimd 3247 . . . . . . . . 9 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → (𝑦𝐶 → ∀𝑥𝐵 (𝑥𝑔𝑦) = ((𝑓𝑦)‘𝑥)))
101 mpteq12 5197 . . . . . . . . 9 ((𝐵 = 𝐵 ∧ ∀𝑥𝐵 (𝑥𝑔𝑦) = ((𝑓𝑦)‘𝑥)) → (𝑥𝐵 ↦ (𝑥𝑔𝑦)) = (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥)))
10289, 100, 101syl6an 682 . . . . . . . 8 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → (𝑦𝐶 → (𝑥𝐵 ↦ (𝑥𝑔𝑦)) = (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥))))
10388, 102ralrimi 3240 . . . . . . 7 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → ∀𝑦𝐶 (𝑥𝐵 ↦ (𝑥𝑔𝑦)) = (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥)))
104 mpteq12 5197 . . . . . . 7 ((𝐶 = 𝐶 ∧ ∀𝑦𝐶 (𝑥𝐵 ↦ (𝑥𝑔𝑦)) = (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥))) → (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) = (𝑦𝐶 ↦ (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥))))
10572, 103, 104sylancr 587 . . . . . 6 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) = (𝑦𝐶 ↦ (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥))))
106105eqeq2d 2747 . . . . 5 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) ↔ 𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥)))))
10786, 106syl5ibrcom 246 . . . 4 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) → (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → 𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))))
10879, 107impbid 211 . . 3 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴m (𝐵 × 𝐶)))) → (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) ↔ 𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))))
109108ex 413 . 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 8929 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴m 𝐵) ↑m 𝐶) ≈ (𝐴m (𝐵 × 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  Vcvv 3445   class class class wbr 5105  cmpt 5188   × cxp 5631   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  cmpo 7359  m cmap 8765  cen 8880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-map 8767  df-en 8884
This theorem is referenced by:  mappwen  10048  cfpwsdom  10520  rpnnen  16109  rexpen  16110  enrelmap  42259
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