Theorem mapen 3386

Description: Two set exponentiations are equinumerous when their bases and exponents are equinumerous. Theorem 6H(c) of [Enderton] p. 139.

Hypotheses

Ref	Expression
mapen.1	⊢ A ∈ V
mapen.2	⊢ B ∈ V
mapen.3	⊢ C ∈ V
mapen.4	⊢ D ∈ V

Assertion

Ref	Expression
mapen	⊢ ((A ≈ B ∧ C ≈ D) → (A ↑m C) ≈ (B ↑m D))

Proof of Theorem mapen

Step	Hyp	Ref	Expression
1		mapen.1	. . . . . . . 8 ⊢ A ∈ V
2		mapen.2	. . . . . . . 8 ⊢ B ∈ V
3		mapen.3	. . . . . . . 8 ⊢ C ∈ V
4		mapen.4	. . . . . . . 8 ⊢ D ∈ V
5		cleqid 1102	. . . . . . . 8 ⊢ {⟨x, y⟩∣(x ∈ (A ↑m C) ∧ y = ((f ∘ x) ∘ ◡g))} = {⟨x, y⟩∣(x ∈ (A ↑m C) ∧ y = ((f ∘ x) ∘ ◡g))}
6	1, 2, 3, 4, 5	mapenlem2 3385	. . . . . . 7 ⊢ ((f:A–1-1-onto→B ∧ g:C–1-1-onto→D) → {⟨x, y⟩∣(x ∈ (A ↑m C) ∧ y = ((f ∘ x) ∘ ◡g))}:(A ↑m C)–1-1-onto→(B ↑m D))
7		oprex 3018	. . . . . . . 8 ⊢ (A ↑m C) ∈ V
8	7	f1oen 3301	. . . . . . 7 ⊢ ({⟨x, y⟩∣(x ∈ (A ↑m C) ∧ y = ((f ∘ x) ∘ ◡g))}:(A ↑m C)–1-1-onto→(B ↑m D) → (A ↑m C) ≈ (B ↑m D))
9	6, 8	syl 12	. . . . . 6 ⊢ ((f:A–1-1-onto→B ∧ g:C–1-1-onto→D) → (A ↑m C) ≈ (B ↑m D))
10	9	exp 291	. . . . 5 ⊢ (f:A–1-1-onto→B → (g:C–1-1-onto→D → (A ↑m C) ≈ (B ↑m D)))
11	10	19.23aiv 952	. . . 4 ⊢ (∃f f:A–1-1-onto→B → (g:C–1-1-onto→D → (A ↑m C) ≈ (B ↑m D)))
12	11	19.23adv 954	. . 3 ⊢ (∃f f:A–1-1-onto→B → (∃g g:C–1-1-onto→D → (A ↑m C) ≈ (B ↑m D)))
13	12	imp 277	. 2 ⊢ ((∃f f:A–1-1-onto→B ∧ ∃g g:C–1-1-onto→D) → (A ↑m C) ≈ (B ↑m D))
14	2	bren 3282	. 2 ⊢ (A ≈ B ↔ ∃f f:A–1-1-onto→B)
15	4	bren 3282	. 2 ⊢ (C ≈ D ↔ ∃g g:C–1-1-onto→D)
16	13, 14, 15	syl2anb 350	1 ⊢ ((A ≈ B ∧ C ≈ D) → (A ↑m C) ≈ (B ↑m D))

Syntax hints:   → wi 2   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  Vcvv 1348   class class class wbr 2054  {copab 2055  ^◡ccnv 2409   ∘ ccom 2414  –1-1-onto→wf1o 2421  (class class class)co 3001   ↑_m cm 3258   ≈ cen 3271

This theorem is referenced by:  mapdom1 3387  mapdom2 3389  pwen 3398  infmap1 4950

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-opr 3003  df-oprab 3004  df-map 3259  df-en 3274

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