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Theorem mapxpen 6742
 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 6549 . . 3 𝑚 Fn (V × V)
2 elex 2697 . . . . 5 (𝐴𝑉𝐴 ∈ V)
323ad2ant1 1002 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐴 ∈ V)
4 elex 2697 . . . . 5 (𝐵𝑊𝐵 ∈ V)
543ad2ant2 1003 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐵 ∈ V)
6 fnovex 5804 . . . 4 (( ↑𝑚 Fn (V × V) ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑚 𝐵) ∈ V)
71, 3, 5, 6mp3an2i 1320 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴𝑚 𝐵) ∈ V)
8 elex 2697 . . . 4 (𝐶𝑋𝐶 ∈ V)
983ad2ant3 1004 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐶 ∈ V)
10 fnovex 5804 . . 3 (( ↑𝑚 Fn (V × V) ∧ (𝐴𝑚 𝐵) ∈ V ∧ 𝐶 ∈ V) → ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∈ V)
111, 7, 9, 10mp3an2i 1320 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∈ V)
12 xpexg 4653 . . . 4 ((𝐵𝑊𝐶𝑋) → (𝐵 × 𝐶) ∈ V)
13123adant1 999 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐵 × 𝐶) ∈ V)
14 fnovex 5804 . . 3 (( ↑𝑚 Fn (V × V) ∧ 𝐴 ∈ V ∧ (𝐵 × 𝐶) ∈ V) → (𝐴𝑚 (𝐵 × 𝐶)) ∈ V)
151, 3, 13, 14mp3an2i 1320 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴𝑚 (𝐵 × 𝐶)) ∈ V)
16 elmapi 6564 . . . . . . . . . 10 (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) → 𝑓:𝐶⟶(𝐴𝑚 𝐵))
1716ffvelrnda 5555 . . . . . . . . 9 ((𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑦𝐶) → (𝑓𝑦) ∈ (𝐴𝑚 𝐵))
18 elmapi 6564 . . . . . . . . 9 ((𝑓𝑦) ∈ (𝐴𝑚 𝐵) → (𝑓𝑦):𝐵𝐴)
1917, 18syl 14 . . . . . . . 8 ((𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑦𝐶) → (𝑓𝑦):𝐵𝐴)
2019ffvelrnda 5555 . . . . . . 7 (((𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑦𝐶) ∧ 𝑥𝐵) → ((𝑓𝑦)‘𝑥) ∈ 𝐴)
2120an32s 557 . . . . . 6 (((𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑥𝐵) ∧ 𝑦𝐶) → ((𝑓𝑦)‘𝑥) ∈ 𝐴)
2221ralrimiva 2505 . . . . 5 ((𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑥𝐵) → ∀𝑦𝐶 ((𝑓𝑦)‘𝑥) ∈ 𝐴)
2322ralrimiva 2505 . . . 4 (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) → ∀𝑥𝐵𝑦𝐶 ((𝑓𝑦)‘𝑥) ∈ 𝐴)
24 eqid 2139 . . . . 5 (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))
2524fmpo 6099 . . . 4 (∀𝑥𝐵𝑦𝐶 ((𝑓𝑦)‘𝑥) ∈ 𝐴 ↔ (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)):(𝐵 × 𝐶)⟶𝐴)
2623, 25sylib 121 . . 3 (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) → (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)):(𝐵 × 𝐶)⟶𝐴)
27 simp1 981 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐴𝑉)
2827, 13elmapd 6556 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) ∈ (𝐴𝑚 (𝐵 × 𝐶)) ↔ (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)):(𝐵 × 𝐶)⟶𝐴))
2926, 28syl5ibr 155 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) → (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) ∈ (𝐴𝑚 (𝐵 × 𝐶))))
30 elmapi 6564 . . . . . . . . 9 (𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)) → 𝑔:(𝐵 × 𝐶)⟶𝐴)
3130adantl 275 . . . . . . . 8 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶))) → 𝑔:(𝐵 × 𝐶)⟶𝐴)
32 fovrn 5913 . . . . . . . . . 10 ((𝑔:(𝐵 × 𝐶)⟶𝐴𝑥𝐵𝑦𝐶) → (𝑥𝑔𝑦) ∈ 𝐴)
33323expa 1181 . . . . . . . . 9 (((𝑔:(𝐵 × 𝐶)⟶𝐴𝑥𝐵) ∧ 𝑦𝐶) → (𝑥𝑔𝑦) ∈ 𝐴)
3433an32s 557 . . . . . . . 8 (((𝑔:(𝐵 × 𝐶)⟶𝐴𝑦𝐶) ∧ 𝑥𝐵) → (𝑥𝑔𝑦) ∈ 𝐴)
3531, 34sylanl1 399 . . . . . . 7 (((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶))) ∧ 𝑦𝐶) ∧ 𝑥𝐵) → (𝑥𝑔𝑦) ∈ 𝐴)
36 eqid 2139 . . . . . . 7 (𝑥𝐵 ↦ (𝑥𝑔𝑦)) = (𝑥𝐵 ↦ (𝑥𝑔𝑦))
3735, 36fmptd 5574 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶))) ∧ 𝑦𝐶) → (𝑥𝐵 ↦ (𝑥𝑔𝑦)):𝐵𝐴)
38 elmapg 6555 . . . . . . . 8 ((𝐴𝑉𝐵𝑊) → ((𝑥𝐵 ↦ (𝑥𝑔𝑦)) ∈ (𝐴𝑚 𝐵) ↔ (𝑥𝐵 ↦ (𝑥𝑔𝑦)):𝐵𝐴))
39383adant3 1001 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝑥𝐵 ↦ (𝑥𝑔𝑦)) ∈ (𝐴𝑚 𝐵) ↔ (𝑥𝐵 ↦ (𝑥𝑔𝑦)):𝐵𝐴))
4039ad2antrr 479 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶))) ∧ 𝑦𝐶) → ((𝑥𝐵 ↦ (𝑥𝑔𝑦)) ∈ (𝐴𝑚 𝐵) ↔ (𝑥𝐵 ↦ (𝑥𝑔𝑦)):𝐵𝐴))
4137, 40mpbird 166 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶))) ∧ 𝑦𝐶) → (𝑥𝐵 ↦ (𝑥𝑔𝑦)) ∈ (𝐴𝑚 𝐵))
42 eqid 2139 . . . . 5 (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))
4341, 42fmptd 5574 . . . 4 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶))) → (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))):𝐶⟶(𝐴𝑚 𝐵))
4443ex 114 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)) → (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))):𝐶⟶(𝐴𝑚 𝐵)))
45 simp3 983 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐶𝑋)
467, 45elmapd 6556 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ↔ (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))):𝐶⟶(𝐴𝑚 𝐵)))
4744, 46sylibrd 168 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)) → (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶)))
48 elmapfn 6565 . . . . . . . 8 (𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)) → 𝑔 Fn (𝐵 × 𝐶))
4948ad2antll 482 . . . . . . 7 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) → 𝑔 Fn (𝐵 × 𝐶))
50 fnovim 5879 . . . . . . 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 983 . . . . . . . . . 10 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → 𝑦𝐶)
5337adantlrl 473 . . . . . . . . . . . 12 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) ∧ 𝑦𝐶) → (𝑥𝐵 ↦ (𝑥𝑔𝑦)):𝐵𝐴)
54533adant2 1000 . . . . . . . . . . 11 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → (𝑥𝐵 ↦ (𝑥𝑔𝑦)):𝐵𝐴)
55 simp1l2 1075 . . . . . . . . . . 11 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → 𝐵𝑊)
56 simp1l1 1074 . . . . . . . . . . 11 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → 𝐴𝑉)
57 fex2 5291 . . . . . . . . . . 11 (((𝑥𝐵 ↦ (𝑥𝑔𝑦)):𝐵𝐴𝐵𝑊𝐴𝑉) → (𝑥𝐵 ↦ (𝑥𝑔𝑦)) ∈ V)
5854, 55, 56, 57syl3anc 1216 . . . . . . . . . 10 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → (𝑥𝐵 ↦ (𝑥𝑔𝑦)) ∈ V)
5942fvmpt2 5504 . . . . . . . . . 10 ((𝑦𝐶 ∧ (𝑥𝐵 ↦ (𝑥𝑔𝑦)) ∈ V) → ((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦) = (𝑥𝐵 ↦ (𝑥𝑔𝑦)))
6052, 58, 59syl2anc 408 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → ((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦) = (𝑥𝐵 ↦ (𝑥𝑔𝑦)))
6160fveq1d 5423 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥) = ((𝑥𝐵 ↦ (𝑥𝑔𝑦))‘𝑥))
62 simp2 982 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → 𝑥𝐵)
63 vex 2689 . . . . . . . . . 10 𝑥 ∈ V
64 vex 2689 . . . . . . . . . 10 𝑔 ∈ V
65 vex 2689 . . . . . . . . . 10 𝑦 ∈ V
66 ovexg 5805 . . . . . . . . . 10 ((𝑥 ∈ V ∧ 𝑔 ∈ V ∧ 𝑦 ∈ V) → (𝑥𝑔𝑦) ∈ V)
6763, 64, 65, 66mp3an 1315 . . . . . . . . 9 (𝑥𝑔𝑦) ∈ V
6836fvmpt2 5504 . . . . . . . . 9 ((𝑥𝐵 ∧ (𝑥𝑔𝑦) ∈ V) → ((𝑥𝐵 ↦ (𝑥𝑔𝑦))‘𝑥) = (𝑥𝑔𝑦))
6962, 67, 68sylancl 409 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → ((𝑥𝐵 ↦ (𝑥𝑔𝑦))‘𝑥) = (𝑥𝑔𝑦))
7061, 69eqtrd 2172 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥) = (𝑥𝑔𝑦))
7170mpoeq3dva 5835 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) → (𝑥𝐵, 𝑦𝐶 ↦ (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)) = (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑔𝑦)))
7251, 71eqtr4d 2175 . . . . 5 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) → 𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)))
73 eqid 2139 . . . . . . 7 𝐵 = 𝐵
74 nfcv 2281 . . . . . . . . . 10 𝑥𝐶
75 nfmpt1 4021 . . . . . . . . . 10 𝑥(𝑥𝐵 ↦ (𝑥𝑔𝑦))
7674, 75nfmpt 4020 . . . . . . . . 9 𝑥(𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))
7776nfeq2 2293 . . . . . . . 8 𝑥 𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))
78 nfmpt1 4021 . . . . . . . . . . . 12 𝑦(𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))
7978nfeq2 2293 . . . . . . . . . . 11 𝑦 𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))
80 fveq1 5420 . . . . . . . . . . . . 13 (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) → (𝑓𝑦) = ((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦))
8180fveq1d 5423 . . . . . . . . . . . 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 2503 . . . . . . . . . 10 (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) → ∀𝑦𝐶 ((𝑓𝑦)‘𝑥) = (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥))
84 eqid 2139 . . . . . . . . . 10 𝐶 = 𝐶
8583, 84jctil 310 . . . . . . . . 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 2503 . . . . . . 7 (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) → ∀𝑥𝐵 (𝐶 = 𝐶 ∧ ∀𝑦𝐶 ((𝑓𝑦)‘𝑥) = (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)))
88 mpoeq123 5830 . . . . . . 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 410 . . . . . 6 (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) → (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) = (𝑥𝐵, 𝑦𝐶 ↦ (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)))
9089eqeq2d 2151 . . . . 5 (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) → (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) ↔ 𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥))))
9172, 90syl5ibrcom 156 . . . 4 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) → (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) → 𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))))
9216ad2antrl 481 . . . . . . 7 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) → 𝑓:𝐶⟶(𝐴𝑚 𝐵))
9392feqmptd 5474 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) → 𝑓 = (𝑦𝐶 ↦ (𝑓𝑦)))
94 simprl 520 . . . . . . . . 9 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) → 𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶))
9594, 19sylan 281 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) ∧ 𝑦𝐶) → (𝑓𝑦):𝐵𝐴)
9695feqmptd 5474 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) ∧ 𝑦𝐶) → (𝑓𝑦) = (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥)))
9796mpteq2dva 4018 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) → (𝑦𝐶 ↦ (𝑓𝑦)) = (𝑦𝐶 ↦ (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥))))
9893, 97eqtrd 2172 . . . . 5 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) → 𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥))))
99 nfmpo2 5839 . . . . . . . . 9 𝑦(𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))
10099nfeq2 2293 . . . . . . . 8 𝑦 𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))
101 eqidd 2140 . . . . . . . . 9 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → 𝐵 = 𝐵)
102 nfmpo1 5838 . . . . . . . . . . 11 𝑥(𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))
103102nfeq2 2293 . . . . . . . . . 10 𝑥 𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))
104 nfv 1508 . . . . . . . . . 10 𝑥 𝑦𝐶
105 vex 2689 . . . . . . . . . . . . . . 15 𝑓 ∈ V
106105, 65fvex 5441 . . . . . . . . . . . . . 14 (𝑓𝑦) ∈ V
107106, 63fvex 5441 . . . . . . . . . . . . 13 ((𝑓𝑦)‘𝑥) ∈ V
10824ovmpt4g 5893 . . . . . . . . . . . . 13 ((𝑥𝐵𝑦𝐶 ∧ ((𝑓𝑦)‘𝑥) ∈ V) → (𝑥(𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))𝑦) = ((𝑓𝑦)‘𝑥))
109107, 108mp3an3 1304 . . . . . . . . . . . 12 ((𝑥𝐵𝑦𝐶) → (𝑥(𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))𝑦) = ((𝑓𝑦)‘𝑥))
110 oveq 5780 . . . . . . . . . . . . 13 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → (𝑥𝑔𝑦) = (𝑥(𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))𝑦))
111110eqeq1d 2148 . . . . . . . . . . . 12 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → ((𝑥𝑔𝑦) = ((𝑓𝑦)‘𝑥) ↔ (𝑥(𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))𝑦) = ((𝑓𝑦)‘𝑥)))
112109, 111syl5ibr 155 . . . . . . . . . . 11 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → ((𝑥𝐵𝑦𝐶) → (𝑥𝑔𝑦) = ((𝑓𝑦)‘𝑥)))
113112expcomd 1417 . . . . . . . . . 10 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → (𝑦𝐶 → (𝑥𝐵 → (𝑥𝑔𝑦) = ((𝑓𝑦)‘𝑥))))
114103, 104, 113ralrimd 2510 . . . . . . . . 9 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → (𝑦𝐶 → ∀𝑥𝐵 (𝑥𝑔𝑦) = ((𝑓𝑦)‘𝑥)))
115 mpteq12 4011 . . . . . . . . 9 ((𝐵 = 𝐵 ∧ ∀𝑥𝐵 (𝑥𝑔𝑦) = ((𝑓𝑦)‘𝑥)) → (𝑥𝐵 ↦ (𝑥𝑔𝑦)) = (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥)))
116101, 114, 115syl6an 1410 . . . . . . . 8 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → (𝑦𝐶 → (𝑥𝐵 ↦ (𝑥𝑔𝑦)) = (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥))))
117100, 116ralrimi 2503 . . . . . . 7 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → ∀𝑦𝐶 (𝑥𝐵 ↦ (𝑥𝑔𝑦)) = (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥)))
118 mpteq12 4011 . . . . . . 7 ((𝐶 = 𝐶 ∧ ∀𝑦𝐶 (𝑥𝐵 ↦ (𝑥𝑔𝑦)) = (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥))) → (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) = (𝑦𝐶 ↦ (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥))))
11984, 117, 118sylancr 410 . . . . . 6 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) = (𝑦𝐶 ↦ (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥))))
120119eqeq2d 2151 . . . . 5 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) ↔ 𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥)))))
12198, 120syl5ibrcom 156 . . . 4 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) → (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → 𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))))
12291, 121impbid 128 . . 3 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) → (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) ↔ 𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))))
123122ex 114 . 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 6663 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ≈ (𝐴𝑚 (𝐵 × 𝐶)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 962   = wceq 1331   ∈ wcel 1480  ∀wral 2416  Vcvv 2686   class class class wbr 3929   ↦ cmpt 3989   × cxp 4537   Fn wfn 5118  ⟶wf 5119  ‘cfv 5123  (class class class)co 5774   ∈ cmpo 5776   ↑𝑚 cmap 6542   ≈ cen 6632 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-map 6544  df-en 6635 This theorem is referenced by: (None)
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