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Theorem mapxpen 6824
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 6631 . . 3 𝑚 Fn (V × V)
2 elex 2741 . . . . 5 (𝐴𝑉𝐴 ∈ V)
323ad2ant1 1013 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐴 ∈ V)
4 elex 2741 . . . . 5 (𝐵𝑊𝐵 ∈ V)
543ad2ant2 1014 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐵 ∈ V)
6 fnovex 5884 . . . 4 (( ↑𝑚 Fn (V × V) ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑚 𝐵) ∈ V)
71, 3, 5, 6mp3an2i 1337 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴𝑚 𝐵) ∈ V)
8 elex 2741 . . . 4 (𝐶𝑋𝐶 ∈ V)
983ad2ant3 1015 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐶 ∈ V)
10 fnovex 5884 . . 3 (( ↑𝑚 Fn (V × V) ∧ (𝐴𝑚 𝐵) ∈ V ∧ 𝐶 ∈ V) → ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∈ V)
111, 7, 9, 10mp3an2i 1337 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∈ V)
12 xpexg 4723 . . . 4 ((𝐵𝑊𝐶𝑋) → (𝐵 × 𝐶) ∈ V)
13123adant1 1010 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐵 × 𝐶) ∈ V)
14 fnovex 5884 . . 3 (( ↑𝑚 Fn (V × V) ∧ 𝐴 ∈ V ∧ (𝐵 × 𝐶) ∈ V) → (𝐴𝑚 (𝐵 × 𝐶)) ∈ V)
151, 3, 13, 14mp3an2i 1337 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴𝑚 (𝐵 × 𝐶)) ∈ V)
16 elmapi 6646 . . . . . . . . . 10 (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) → 𝑓:𝐶⟶(𝐴𝑚 𝐵))
1716ffvelrnda 5629 . . . . . . . . 9 ((𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑦𝐶) → (𝑓𝑦) ∈ (𝐴𝑚 𝐵))
18 elmapi 6646 . . . . . . . . 9 ((𝑓𝑦) ∈ (𝐴𝑚 𝐵) → (𝑓𝑦):𝐵𝐴)
1917, 18syl 14 . . . . . . . 8 ((𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑦𝐶) → (𝑓𝑦):𝐵𝐴)
2019ffvelrnda 5629 . . . . . . 7 (((𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑦𝐶) ∧ 𝑥𝐵) → ((𝑓𝑦)‘𝑥) ∈ 𝐴)
2120an32s 563 . . . . . 6 (((𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑥𝐵) ∧ 𝑦𝐶) → ((𝑓𝑦)‘𝑥) ∈ 𝐴)
2221ralrimiva 2543 . . . . 5 ((𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑥𝐵) → ∀𝑦𝐶 ((𝑓𝑦)‘𝑥) ∈ 𝐴)
2322ralrimiva 2543 . . . 4 (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) → ∀𝑥𝐵𝑦𝐶 ((𝑓𝑦)‘𝑥) ∈ 𝐴)
24 eqid 2170 . . . . 5 (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))
2524fmpo 6178 . . . 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 992 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐴𝑉)
2827, 13elmapd 6638 . . 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 6646 . . . . . . . . 9 (𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)) → 𝑔:(𝐵 × 𝐶)⟶𝐴)
3130adantl 275 . . . . . . . 8 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶))) → 𝑔:(𝐵 × 𝐶)⟶𝐴)
32 fovrn 5993 . . . . . . . . . 10 ((𝑔:(𝐵 × 𝐶)⟶𝐴𝑥𝐵𝑦𝐶) → (𝑥𝑔𝑦) ∈ 𝐴)
33323expa 1198 . . . . . . . . 9 (((𝑔:(𝐵 × 𝐶)⟶𝐴𝑥𝐵) ∧ 𝑦𝐶) → (𝑥𝑔𝑦) ∈ 𝐴)
3433an32s 563 . . . . . . . 8 (((𝑔:(𝐵 × 𝐶)⟶𝐴𝑦𝐶) ∧ 𝑥𝐵) → (𝑥𝑔𝑦) ∈ 𝐴)
3531, 34sylanl1 400 . . . . . . 7 (((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶))) ∧ 𝑦𝐶) ∧ 𝑥𝐵) → (𝑥𝑔𝑦) ∈ 𝐴)
36 eqid 2170 . . . . . . 7 (𝑥𝐵 ↦ (𝑥𝑔𝑦)) = (𝑥𝐵 ↦ (𝑥𝑔𝑦))
3735, 36fmptd 5648 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶))) ∧ 𝑦𝐶) → (𝑥𝐵 ↦ (𝑥𝑔𝑦)):𝐵𝐴)
38 elmapg 6637 . . . . . . . 8 ((𝐴𝑉𝐵𝑊) → ((𝑥𝐵 ↦ (𝑥𝑔𝑦)) ∈ (𝐴𝑚 𝐵) ↔ (𝑥𝐵 ↦ (𝑥𝑔𝑦)):𝐵𝐴))
39383adant3 1012 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝑥𝐵 ↦ (𝑥𝑔𝑦)) ∈ (𝐴𝑚 𝐵) ↔ (𝑥𝐵 ↦ (𝑥𝑔𝑦)):𝐵𝐴))
4039ad2antrr 485 . . . . . 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 2170 . . . . 5 (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))
4341, 42fmptd 5648 . . . 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 994 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐶𝑋)
467, 45elmapd 6638 . . 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 6647 . . . . . . . 8 (𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)) → 𝑔 Fn (𝐵 × 𝐶))
4948ad2antll 488 . . . . . . 7 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) → 𝑔 Fn (𝐵 × 𝐶))
50 fnovim 5959 . . . . . . 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 994 . . . . . . . . . 10 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → 𝑦𝐶)
5337adantlrl 479 . . . . . . . . . . . 12 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) ∧ 𝑦𝐶) → (𝑥𝐵 ↦ (𝑥𝑔𝑦)):𝐵𝐴)
54533adant2 1011 . . . . . . . . . . 11 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → (𝑥𝐵 ↦ (𝑥𝑔𝑦)):𝐵𝐴)
55 simp1l2 1086 . . . . . . . . . . 11 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → 𝐵𝑊)
56 simp1l1 1085 . . . . . . . . . . 11 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → 𝐴𝑉)
57 fex2 5364 . . . . . . . . . . 11 (((𝑥𝐵 ↦ (𝑥𝑔𝑦)):𝐵𝐴𝐵𝑊𝐴𝑉) → (𝑥𝐵 ↦ (𝑥𝑔𝑦)) ∈ V)
5854, 55, 56, 57syl3anc 1233 . . . . . . . . . 10 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → (𝑥𝐵 ↦ (𝑥𝑔𝑦)) ∈ V)
5942fvmpt2 5577 . . . . . . . . . 10 ((𝑦𝐶 ∧ (𝑥𝐵 ↦ (𝑥𝑔𝑦)) ∈ V) → ((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦) = (𝑥𝐵 ↦ (𝑥𝑔𝑦)))
6052, 58, 59syl2anc 409 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → ((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦) = (𝑥𝐵 ↦ (𝑥𝑔𝑦)))
6160fveq1d 5496 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥) = ((𝑥𝐵 ↦ (𝑥𝑔𝑦))‘𝑥))
62 simp2 993 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → 𝑥𝐵)
63 vex 2733 . . . . . . . . . 10 𝑥 ∈ V
64 vex 2733 . . . . . . . . . 10 𝑔 ∈ V
65 vex 2733 . . . . . . . . . 10 𝑦 ∈ V
66 ovexg 5885 . . . . . . . . . 10 ((𝑥 ∈ V ∧ 𝑔 ∈ V ∧ 𝑦 ∈ V) → (𝑥𝑔𝑦) ∈ V)
6763, 64, 65, 66mp3an 1332 . . . . . . . . 9 (𝑥𝑔𝑦) ∈ V
6836fvmpt2 5577 . . . . . . . . 9 ((𝑥𝐵 ∧ (𝑥𝑔𝑦) ∈ V) → ((𝑥𝐵 ↦ (𝑥𝑔𝑦))‘𝑥) = (𝑥𝑔𝑦))
6962, 67, 68sylancl 411 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → ((𝑥𝐵 ↦ (𝑥𝑔𝑦))‘𝑥) = (𝑥𝑔𝑦))
7061, 69eqtrd 2203 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) ∧ 𝑥𝐵𝑦𝐶) → (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥) = (𝑥𝑔𝑦))
7170mpoeq3dva 5915 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) → (𝑥𝐵, 𝑦𝐶 ↦ (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)) = (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑔𝑦)))
7251, 71eqtr4d 2206 . . . . 5 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) → 𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)))
73 eqid 2170 . . . . . . 7 𝐵 = 𝐵
74 nfcv 2312 . . . . . . . . . 10 𝑥𝐶
75 nfmpt1 4080 . . . . . . . . . 10 𝑥(𝑥𝐵 ↦ (𝑥𝑔𝑦))
7674, 75nfmpt 4079 . . . . . . . . 9 𝑥(𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))
7776nfeq2 2324 . . . . . . . 8 𝑥 𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))
78 nfmpt1 4080 . . . . . . . . . . . 12 𝑦(𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))
7978nfeq2 2324 . . . . . . . . . . 11 𝑦 𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))
80 fveq1 5493 . . . . . . . . . . . . 13 (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) → (𝑓𝑦) = ((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦))
8180fveq1d 5496 . . . . . . . . . . . 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 2541 . . . . . . . . . 10 (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) → ∀𝑦𝐶 ((𝑓𝑦)‘𝑥) = (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥))
84 eqid 2170 . . . . . . . . . 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 2541 . . . . . . 7 (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) → ∀𝑥𝐵 (𝐶 = 𝐶 ∧ ∀𝑦𝐶 ((𝑓𝑦)‘𝑥) = (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)))
88 mpoeq123 5910 . . . . . . 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 412 . . . . . 6 (𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) → (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) = (𝑥𝐵, 𝑦𝐶 ↦ (((𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)))
9089eqeq2d 2182 . . . . 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 487 . . . . . . 7 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) → 𝑓:𝐶⟶(𝐴𝑚 𝐵))
9392feqmptd 5547 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) → 𝑓 = (𝑦𝐶 ↦ (𝑓𝑦)))
94 simprl 526 . . . . . . . . 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 5547 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) ∧ 𝑦𝐶) → (𝑓𝑦) = (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥)))
9796mpteq2dva 4077 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) → (𝑦𝐶 ↦ (𝑓𝑦)) = (𝑦𝐶 ↦ (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥))))
9893, 97eqtrd 2203 . . . . 5 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝑓 ∈ ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ∧ 𝑔 ∈ (𝐴𝑚 (𝐵 × 𝐶)))) → 𝑓 = (𝑦𝐶 ↦ (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥))))
99 nfmpo2 5919 . . . . . . . . 9 𝑦(𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))
10099nfeq2 2324 . . . . . . . 8 𝑦 𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))
101 eqidd 2171 . . . . . . . . 9 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → 𝐵 = 𝐵)
102 nfmpo1 5918 . . . . . . . . . . 11 𝑥(𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))
103102nfeq2 2324 . . . . . . . . . 10 𝑥 𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))
104 nfv 1521 . . . . . . . . . 10 𝑥 𝑦𝐶
105 vex 2733 . . . . . . . . . . . . . . 15 𝑓 ∈ V
106105, 65fvex 5514 . . . . . . . . . . . . . 14 (𝑓𝑦) ∈ V
107106, 63fvex 5514 . . . . . . . . . . . . 13 ((𝑓𝑦)‘𝑥) ∈ V
10824ovmpt4g 5973 . . . . . . . . . . . . 13 ((𝑥𝐵𝑦𝐶 ∧ ((𝑓𝑦)‘𝑥) ∈ V) → (𝑥(𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))𝑦) = ((𝑓𝑦)‘𝑥))
109107, 108mp3an3 1321 . . . . . . . . . . . 12 ((𝑥𝐵𝑦𝐶) → (𝑥(𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))𝑦) = ((𝑓𝑦)‘𝑥))
110 oveq 5857 . . . . . . . . . . . . 13 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → (𝑥𝑔𝑦) = (𝑥(𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥))𝑦))
111110eqeq1d 2179 . . . . . . . . . . . 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 1434 . . . . . . . . . 10 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → (𝑦𝐶 → (𝑥𝐵 → (𝑥𝑔𝑦) = ((𝑓𝑦)‘𝑥))))
114103, 104, 113ralrimd 2548 . . . . . . . . 9 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → (𝑦𝐶 → ∀𝑥𝐵 (𝑥𝑔𝑦) = ((𝑓𝑦)‘𝑥)))
115 mpteq12 4070 . . . . . . . . 9 ((𝐵 = 𝐵 ∧ ∀𝑥𝐵 (𝑥𝑔𝑦) = ((𝑓𝑦)‘𝑥)) → (𝑥𝐵 ↦ (𝑥𝑔𝑦)) = (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥)))
116101, 114, 115syl6an 1427 . . . . . . . 8 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → (𝑦𝐶 → (𝑥𝐵 ↦ (𝑥𝑔𝑦)) = (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥))))
117100, 116ralrimi 2541 . . . . . . 7 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → ∀𝑦𝐶 (𝑥𝐵 ↦ (𝑥𝑔𝑦)) = (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥)))
118 mpteq12 4070 . . . . . . 7 ((𝐶 = 𝐶 ∧ ∀𝑦𝐶 (𝑥𝐵 ↦ (𝑥𝑔𝑦)) = (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥))) → (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) = (𝑦𝐶 ↦ (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥))))
11984, 117, 118sylancr 412 . . . . . 6 (𝑔 = (𝑥𝐵, 𝑦𝐶 ↦ ((𝑓𝑦)‘𝑥)) → (𝑦𝐶 ↦ (𝑥𝐵 ↦ (𝑥𝑔𝑦))) = (𝑦𝐶 ↦ (𝑥𝐵 ↦ ((𝑓𝑦)‘𝑥))))
120119eqeq2d 2182 . . . . 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 6745 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ≈ (𝐴𝑚 (𝐵 × 𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 973   = wceq 1348  wcel 2141  wral 2448  Vcvv 2730   class class class wbr 3987  cmpt 4048   × cxp 4607   Fn wfn 5191  wf 5192  cfv 5196  (class class class)co 5851  cmpo 5853  𝑚 cmap 6624  cen 6714
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-map 6626  df-en 6717
This theorem is referenced by: (None)
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