Theorem mapss 3270

Description: Subset inheritance for set exponentiation. Theorem 99 of [Suppes] p. 89.

Hypotheses

Ref	Expression
mapss.1	⊢ B ∈ V
mapss.2	⊢ C ∈ V

Assertion

Ref	Expression
mapss	⊢ (A ⊆ B → (A ↑m C) ⊆ (B ↑m C))

Proof of Theorem mapss

Step	Hyp	Ref	Expression
1		fss 2759	. . . . . 6 ⊢ ((f:C–→A ∧ A ⊆ B) → f:C–→B)
2	1	exp 291	. . . . 5 ⊢ (f:C–→A → (A ⊆ B → f:C–→B))
3	2	com12 13	. . . 4 ⊢ (A ⊆ B → (f:C–→A → f:C–→B))
4	3	19.21aiv 943	. . 3 ⊢ (A ⊆ B → ∀f(f:C–→A → f:C–→B))
5		ss2ab 1551	. . 3 ⊢ ({f∣f:C–→A} ⊆ {f∣f:C–→B} ↔ ∀f(f:C–→A → f:C–→B))
6	4, 5	sylibr 175	. 2 ⊢ (A ⊆ B → {f∣f:C–→A} ⊆ {f∣f:C–→B})
7		mapss.1	. . . 4 ⊢ B ∈ V
8	7	ssex 1700	. . 3 ⊢ (A ⊆ B → A ∈ V)
9		mapss.2	. . . 4 ⊢ C ∈ V
10		mapvalg 3263	. . . 4 ⊢ ((A ∈ V ∧ C ∈ V) → (A ↑m C) = {f∣f:C–→A})
11	9, 10	mpan2 519	. . 3 ⊢ (A ∈ V → (A ↑m C) = {f∣f:C–→A})
12	8, 11	syl 12	. 2 ⊢ (A ⊆ B → (A ↑m C) = {f∣f:C–→A})
13	7, 9	mapval 3264	. . 3 ⊢ (B ↑m C) = {f∣f:C–→B}
14	13	a1i 7	. 2 ⊢ (A ⊆ B → (B ↑m C) = {f∣f:C–→B})
15	6, 12, 14	3sstr4d 1543	1 ⊢ (A ⊆ B → (A ↑m C) ⊆ (B ↑m C))

Syntax hints:   → wi 2  ∀wal 672  {cab 1090   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  –→wf 2418  (class class class)co 3001   ↑_m cm 3258

This theorem is referenced by:  mapdom1 3387

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-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-fv 2438  df-opr 3003  df-oprab 3004  df-map 3259

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