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Theorem mapxpen 6842
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 fnmap 6649 . . 3 𝑚 Fn (V × V)
2 elex 2748 . . . . 5 (𝐴𝑉𝐴 ∈ V)
323ad2ant1 1018 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐴 ∈ V)
4 elex 2748 . . . . 5 (𝐵𝑊𝐵 ∈ V)
543ad2ant2 1019 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐵 ∈ V)
6 fnovex 5902 . . . 4 (( ↑𝑚 Fn (V × V) ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑚 𝐵) ∈ V)
71, 3, 5, 6mp3an2i 1342 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴𝑚 𝐵) ∈ V)
8 elex 2748 . . . 4 (𝐶𝑋𝐶 ∈ V)
983ad2ant3 1020 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐶 ∈ V)
10 fnovex 5902 . . 3 (( ↑𝑚 Fn (V × V) ∧ (𝐴𝑚 𝐵) ∈ V ∧ 𝐶 ∈ V) → ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∈ V)
111, 7, 9, 10mp3an2i 1342 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∈ V)
12 xpexg 4737 . . . 4 ((𝐵𝑊𝐶𝑋) → (𝐵 × 𝐶) ∈ V)
13123adant1 1015 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐵 × 𝐶) ∈ V)
14 fnovex 5902 . . 3 (( ↑𝑚 Fn (V × V) ∧ 𝐴 ∈ V ∧ (𝐵 × 𝐶) ∈ V) → (𝐴𝑚 (𝐵 × 𝐶)) ∈ V)
151, 3, 13, 14mp3an2i 1342 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴𝑚 (𝐵 × 𝐶)) ∈ V)
16 elmapi 6664 . . . . . . . . . 10 (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) → 𝑓:𝐶⟶(𝐴𝑚 𝐵))
1716ffvelcdmda 5647 . . . . . . . . 9 ((𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑦𝐶) → (𝑓𝑦) ∈ (𝐴𝑚 𝐵))
18 elmapi 6664 . . . . . . . . 9 ((𝑓𝑦) ∈ (𝐴𝑚 𝐵) → (𝑓𝑦):𝐵𝐴)
1917, 18syl 14 . . . . . . . 8 ((𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑦𝐶) → (𝑓𝑦):𝐵𝐴)
2019ffvelcdmda 5647 . . . . . . 7 (((𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑦𝐶) ∧ 𝑥𝐵) → ((𝑓𝑦)‘𝑥) ∈ 𝐴)
2120an32s 568 . . . . . 6 (((𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑥𝐵) ∧ 𝑦𝐶) → ((𝑓𝑦)‘𝑥) ∈ 𝐴)
2221ralrimiva 2550 . . . . 5 ((𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑥𝐵) → ∀𝑦𝐶 ((𝑓𝑦)‘𝑥) ∈ 𝐴)
2322ralrimiva 2550 . . . 4 (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) → ∀𝑥𝐵𝑦𝐶 ((𝑓𝑦)‘𝑥) ∈ 𝐴)
24 eqid 2177 . . . . 5 (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))
2524fmpo 6196 . . . 4 (∀𝑥𝐵𝑦𝐶 ((𝑓𝑦)‘𝑥) ∈ 𝐴 ↔ (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)):(𝐵 × 𝐶)⟶𝐴)
2623, 25sylib 122 . . 3 (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) → (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)):(𝐵 × 𝐶)⟶𝐴)
27 simp1 997 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐴𝑉)
2827, 13elmapd 6656 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) ∈ (𝐴𝑚 (𝐵 × 𝐶)) ↔ (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)):(𝐵 × 𝐶)⟶𝐴))
2926, 28syl5ibr 156 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) → (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) ∈ (𝐴𝑚 (𝐵 × 𝐶))))
30 elmapi 6664 . . . . . . . . 9 (𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)) → 𝑔:(𝐵 × 𝐶)⟶𝐴)
3130adantl 277 . . . . . . . 8 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶))) → 𝑔:(𝐵 × 𝐶)⟶𝐴)
32 fovcdm 6011 . . . . . . . . . 10 ((𝑔:(𝐵 × 𝐶)⟶𝐴𝑥𝐵𝑦𝐶) → (𝑥𝑔𝑦) ∈ 𝐴)
33323expa 1203 . . . . . . . . 9 (((𝑔:(𝐵 × 𝐶)⟶𝐴𝑥𝐵) ∧ 𝑦𝐶) → (𝑥𝑔𝑦) ∈ 𝐴)
3433an32s 568 . . . . . . . 8 (((𝑔:(𝐵 × 𝐶)⟶𝐴𝑦𝐶) ∧ 𝑥𝐵) → (𝑥𝑔𝑦) ∈ 𝐴)
3531, 34sylanl1 402 . . . . . . 7 (((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶))) ∧ 𝑦𝐶) ∧ 𝑥𝐵) → (𝑥𝑔𝑦) ∈ 𝐴)
36 eqid 2177 . . . . . . 7 (𝑥𝐵 ↦ (𝑥𝑔𝑦)) = (𝑥𝐵 ↦ (𝑥𝑔𝑦))
3735, 36fmptd 5666 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶))) ∧ 𝑦𝐶) → (𝑥𝐵 ↦ (𝑥𝑔𝑦)):𝐵𝐴)
38 elmapg 6655 . . . . . . . 8 ((𝐴𝑉𝐵𝑊) → ((𝑥𝐵 ↦ (𝑥𝑔𝑦)) ∈ (𝐴𝑚 𝐵) ↔ (𝑥𝐵 ↦ (𝑥𝑔𝑦)):𝐵𝐴))
39383adant3 1017 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝑥𝐵 ↦ (𝑥𝑔𝑦)) ∈ (𝐴𝑚 𝐵) ↔ (𝑥𝐵 ↦ (𝑥𝑔𝑦)):𝐵𝐴))
4039ad2antrr 488 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶))) ∧ 𝑦𝐶) → ((𝑥𝐵 ↦ (𝑥𝑔𝑦)) ∈ (𝐴𝑚 𝐵) ↔ (𝑥𝐵 ↦ (𝑥𝑔𝑦)):𝐵𝐴))
4137, 40mpbird 167 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶))) ∧ 𝑦𝐶) → (𝑥𝐵 ↦ (𝑥𝑔𝑦)) ∈ (𝐴𝑚 𝐵))
42 eqid 2177 . . . . 5 (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))
4341, 42fmptd 5666 . . . 4 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶))) → (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))):𝐶⟶(𝐴𝑚 𝐵))
4443ex 115 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)) → (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))):𝐶⟶(𝐴𝑚 𝐵)))
45 simp3 999 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐶𝑋)
467, 45elmapd 6656 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ↔ (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))):𝐶⟶(𝐴𝑚 𝐵)))
4744, 46sylibrd 169 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)) → (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶)))
48 elmapfn 6665 . . . . . . . 8 (𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)) → 𝑔 Fn (𝐵 × 𝐶))
4948ad2antll 491 . . . . . . 7 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) → 𝑔 Fn (𝐵 × 𝐶))
50 fnovim 5977 . . . . . . 7 (𝑔 Fn (𝐵 × 𝐶) → 𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑔𝑦)))
5149, 50syl 14 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) → 𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑔𝑦)))
52 simp3 999 . . . . . . . . . 10 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → 𝑦𝐶)
5337adantlrl 482 . . . . . . . . . . . 12 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) ∧ 𝑦𝐶) → (𝑥𝐵 ↦ (𝑥𝑔𝑦)):𝐵𝐴)
54533adant2 1016 . . . . . . . . . . 11 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → (𝑥𝐵 ↦ (𝑥𝑔𝑦)):𝐵𝐴)
55 simp1l2 1091 . . . . . . . . . . 11 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → 𝐵𝑊)
56 simp1l1 1090 . . . . . . . . . . 11 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → 𝐴𝑉)
57 fex2 5380 . . . . . . . . . . 11 (((𝑥𝐵 ↦ (𝑥𝑔𝑦)):𝐵𝐴𝐵𝑊𝐴𝑉) → (𝑥𝐵 ↦ (𝑥𝑔𝑦)) ∈ V)
5854, 55, 56, 57syl3anc 1238 . . . . . . . . . 10 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → (𝑥𝐵 ↦ (𝑥𝑔𝑦)) ∈ V)
5942fvmpt2 5595 . . . . . . . . . 10 ((𝑦𝐶 ∧ (𝑥𝐵 ↦ (𝑥𝑔𝑦)) ∈ V) → ((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦) = (𝑥𝐵 ↦ (𝑥𝑔𝑦)))
6052, 58, 59syl2anc 411 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → ((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦) = (𝑥𝐵 ↦ (𝑥𝑔𝑦)))
6160fveq1d 5513 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥) = ((𝑥𝐵 ↦ (𝑥𝑔𝑦))‘𝑥))
62 simp2 998 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → 𝑥𝐵)
63 vex 2740 . . . . . . . . . 10 𝑥 ∈ V
64 vex 2740 . . . . . . . . . 10 𝑔 ∈ V
65 vex 2740 . . . . . . . . . 10 𝑦 ∈ V
66 ovexg 5903 . . . . . . . . . 10 ((𝑥 ∈ V ∧ 𝑔 ∈ V ∧ 𝑦 ∈ V) → (𝑥𝑔𝑦) ∈ V)
6763, 64, 65, 66mp3an 1337 . . . . . . . . 9 (𝑥𝑔𝑦) ∈ V
6836fvmpt2 5595 . . . . . . . . 9 ((𝑥𝐵 ∧ (𝑥𝑔𝑦) ∈ V) → ((𝑥𝐵 ↦ (𝑥𝑔𝑦))‘𝑥) = (𝑥𝑔𝑦))
6962, 67, 68sylancl 413 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → ((𝑥𝐵 ↦ (𝑥𝑔𝑦))‘𝑥) = (𝑥𝑔𝑦))
7061, 69eqtrd 2210 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥) = (𝑥𝑔𝑦))
7170mpoeq3dva 5933 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) → (𝑥𝐵, 𝑦𝐶 ↦ (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)) = (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑔𝑦)))
7251, 71eqtr4d 2213 . . . . 5 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) → 𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)))
73 eqid 2177 . . . . . . 7 𝐵 = 𝐵
74 nfcv 2319 . . . . . . . . . 10 𝑥𝐶
75 nfmpt1 4093 . . . . . . . . . 10 𝑥(𝑥𝐵 ↦ (𝑥𝑔𝑦))
7674, 75nfmpt 4092 . . . . . . . . 9 𝑥(𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))
7776nfeq2 2331 . . . . . . . 8 𝑥 𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))
78 nfmpt1 4093 . . . . . . . . . . . 12 𝑦(𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))
7978nfeq2 2331 . . . . . . . . . . 11 𝑦 𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))
80 fveq1 5510 . . . . . . . . . . . . 13 (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) → (𝑓𝑦) = ((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦))
8180fveq1d 5513 . . . . . . . . . . . 12 (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) → ((𝑓𝑦)‘𝑥) = (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥))
8281a1d 22 . . . . . . . . . . 11 (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) → (𝑦𝐶 → ((𝑓𝑦)‘𝑥) = (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)))
8379, 82ralrimi 2548 . . . . . . . . . 10 (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) → ∀𝑦𝐶 ((𝑓𝑦)‘𝑥) = (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥))
84 eqid 2177 . . . . . . . . . 10 𝐶 = 𝐶
8583, 84jctil 312 . . . . . . . . 9 (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) → (𝐶 = 𝐶 ∧ ∀𝑦𝐶 ((𝑓𝑦)‘𝑥) = (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)))
8685a1d 22 . . . . . . . 8 (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) → (𝑥𝐵 → (𝐶 = 𝐶 ∧ ∀𝑦𝐶 ((𝑓𝑦)‘𝑥) = (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥))))
8777, 86ralrimi 2548 . . . . . . 7 (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) → ∀𝑥𝐵 (𝐶 = 𝐶 ∧ ∀𝑦𝐶 ((𝑓𝑦)‘𝑥) = (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)))
88 mpoeq123 5928 . . . . . . 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)))
8973, 87, 88sylancr 414 . . . . . 6 (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) → (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) = (𝑥𝐵, 𝑦𝐶 ↦ (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)))
9089eqeq2d 2189 . . . . 5 (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) → (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) ↔ 𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥))))
9172, 90syl5ibrcom 157 . . . 4 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) → (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) → 𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))))
9216ad2antrl 490 . . . . . . 7 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) → 𝑓:𝐶⟶(𝐴𝑚 𝐵))
9392feqmptd 5565 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) → 𝑓 = (𝑦𝐶 ↦ (𝑓𝑦)))
94 simprl 529 . . . . . . . . 9 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) → 𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶))
9594, 19sylan 283 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) ∧ 𝑦𝐶) → (𝑓𝑦):𝐵𝐴)
9695feqmptd 5565 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) ∧ 𝑦𝐶) → (𝑓𝑦) = (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥)))
9796mpteq2dva 4090 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) → (𝑦𝐶 ↦ (𝑓𝑦)) = (𝑦𝐶 ↦ (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥))))
9893, 97eqtrd 2210 . . . . 5 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) → 𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥))))
99 nfmpo2 5937 . . . . . . . . 9 𝑦(𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))
10099nfeq2 2331 . . . . . . . 8 𝑦 𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))
101 eqidd 2178 . . . . . . . . 9 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → 𝐵 = 𝐵)
102 nfmpo1 5936 . . . . . . . . . . 11 𝑥(𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))
103102nfeq2 2331 . . . . . . . . . 10 𝑥 𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))
104 nfv 1528 . . . . . . . . . 10 𝑥 𝑦𝐶
105 vex 2740 . . . . . . . . . . . . . . 15 𝑓 ∈ V
106105, 65fvex 5531 . . . . . . . . . . . . . 14 (𝑓𝑦) ∈ V
107106, 63fvex 5531 . . . . . . . . . . . . 13 ((𝑓𝑦)‘𝑥) ∈ V
10824ovmpt4g 5991 . . . . . . . . . . . . 13 ((𝑥𝐵𝑦𝐶 ∧ ((𝑓𝑦)‘𝑥) ∈ V) → (𝑥(𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))𝑦) = ((𝑓𝑦)‘𝑥))
109107, 108mp3an3 1326 . . . . . . . . . . . 12 ((𝑥𝐵𝑦𝐶) → (𝑥(𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))𝑦) = ((𝑓𝑦)‘𝑥))
110 oveq 5875 . . . . . . . . . . . . 13 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → (𝑥𝑔𝑦) = (𝑥(𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))𝑦))
111110eqeq1d 2186 . . . . . . . . . . . 12 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → ((𝑥𝑔𝑦) = ((𝑓𝑦)‘𝑥) ↔ (𝑥(𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))𝑦) = ((𝑓𝑦)‘𝑥)))
112109, 111syl5ibr 156 . . . . . . . . . . 11 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → ((𝑥𝐵𝑦𝐶) → (𝑥𝑔𝑦) = ((𝑓𝑦)‘𝑥)))
113112expcomd 1441 . . . . . . . . . 10 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → (𝑦𝐶 → (𝑥𝐵 → (𝑥𝑔𝑦) = ((𝑓𝑦)‘𝑥))))
114103, 104, 113ralrimd 2555 . . . . . . . . 9 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → (𝑦𝐶 → ∀𝑥𝐵 (𝑥𝑔𝑦) = ((𝑓𝑦)‘𝑥)))
115 mpteq12 4083 . . . . . . . . 9 ((𝐵 = 𝐵 ∧ ∀𝑥𝐵 (𝑥𝑔𝑦) = ((𝑓𝑦)‘𝑥)) → (𝑥𝐵 ↦ (𝑥𝑔𝑦)) = (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥)))
116101, 114, 115syl6an 1434 . . . . . . . 8 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → (𝑦𝐶 → (𝑥𝐵 ↦ (𝑥𝑔𝑦)) = (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥))))
117100, 116ralrimi 2548 . . . . . . 7 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → ∀𝑦𝐶 (𝑥𝐵 ↦ (𝑥𝑔𝑦)) = (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥)))
118 mpteq12 4083 . . . . . . 7 ((𝐶 = 𝐶 ∧ ∀𝑦𝐶 (𝑥𝐵 ↦ (𝑥𝑔𝑦)) = (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥))) → (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) = (𝑦𝐶 ↦ (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥))))
11984, 117, 118sylancr 414 . . . . . 6 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) = (𝑦𝐶 ↦ (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥))))
120119eqeq2d 2189 . . . . 5 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) ↔ 𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥)))))
12198, 120syl5ibrcom 157 . . . 4 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) → (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → 𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))))
12291, 121impbid 129 . . 3 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) → (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) ↔ 𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))))
123122ex 115 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶))) → (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) ↔ 𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)))))
12411, 15, 29, 47, 123en3d 6763 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ≈ (𝐴𝑚 (𝐵 × 𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 978   = wceq 1353  wcel 2148  wral 2455  Vcvv 2737   class class class wbr 4000  cmpt 4061   × cxp 4621   Fn wfn 5207  wf 5208  cfv 5212  (class class class)co 5869  cmpo 5871  𝑚 cmap 6642  cen 6732
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-map 6644  df-en 6735
This theorem is referenced by: (None)
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