Theorem mapdom1 3387

Description: Order-preserving property of set exponentiation. Theorem 6L(c) of [Enderton] p. 149.

Hypotheses

Ref	Expression
mapdom1.1	⊢ A ∈ V
mapdom1.2	⊢ B ∈ V
mapdom1.3	⊢ C ∈ V

Assertion

Ref	Expression
mapdom1	⊢ (A ≼ B → (A ↑m C) ≼ (B ↑m C))

Proof of Theorem mapdom1

Step	Hyp	Ref	Expression
1		mapdom1.2	. . 3 ⊢ B ∈ V
2	1	domen 3284	. 2 ⊢ (A ≼ B ↔ ∃x(A ≈ x ∧ x ⊆ B))
3		endomtr 3325	. . . 4 ⊢ (((A ↑m C) ≈ (x ↑m C) ∧ (x ↑m C) ≼ (B ↑m C)) → (A ↑m C) ≼ (B ↑m C))
4		mapdom1.3	. . . . . 6 ⊢ C ∈ V
5	4	enref 3295	. . . . 5 ⊢ C ≈ C
6		mapdom1.1	. . . . . 6 ⊢ A ∈ V
7		visset 1350	. . . . . 6 ⊢ x ∈ V
8	6, 7, 4, 4	mapen 3386	. . . . 5 ⊢ ((A ≈ x ∧ C ≈ C) → (A ↑m C) ≈ (x ↑m C))
9	5, 8	mpan2 519	. . . 4 ⊢ (A ≈ x → (A ↑m C) ≈ (x ↑m C))
10	1, 4	mapss 3270	. . . . 5 ⊢ (x ⊆ B → (x ↑m C) ⊆ (B ↑m C))
11		oprex 3018	. . . . . 6 ⊢ (x ↑m C) ∈ V
12		ssdomg 3311	. . . . . 6 ⊢ ((x ↑m C) ∈ V → ((x ↑m C) ⊆ (B ↑m C) → (x ↑m C) ≼ (B ↑m C)))
13	11, 12	ax-mp 6	. . . . 5 ⊢ ((x ↑m C) ⊆ (B ↑m C) → (x ↑m C) ≼ (B ↑m C))
14	10, 13	syl 12	. . . 4 ⊢ (x ⊆ B → (x ↑m C) ≼ (B ↑m C))
15	3, 9, 14	syl2an 349	. . 3 ⊢ ((A ≈ x ∧ x ⊆ B) → (A ↑m C) ≼ (B ↑m C))
16	15	19.23aiv 952	. 2 ⊢ (∃x(A ≈ x ∧ x ⊆ B) → (A ↑m C) ≼ (B ↑m C))
17	2, 16	sylbi 174	1 ⊢ (A ≼ B → (A ↑m C) ≼ (B ↑m C))

Syntax hints:   → wi 2   ∧ wa 196  ∃wex 678   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487   class class class wbr 2054  (class class class)co 3001   ↑_m cm 3258   ≈ cen 3271   ≼ cdom 3272

This theorem is referenced by:  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  df-dom 3275

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