Theorem resfunexg 2717

Description: The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28.

Assertion

Ref	Expression
resfunexg	⊢ (B ∈ C → (Fun A → (A ↾ B) ∈ V))

Proof of Theorem resfunexg

Step	Hyp	Ref	Expression
1		dmresexg 2586	. . . . 5 ⊢ (B ∈ C → dom (A ↾ B) ∈ V)
2	1	adantr 306	. . . 4 ⊢ ((B ∈ C ∧ Fun A) → dom (A ↾ B) ∈ V)
3		funimaexg 2715	. . . . . 6 ⊢ (B ∈ C → (Fun A → (A “ B) ∈ V))
4		df-ima 2431	. . . . . . 7 ⊢ (A “ B) = ran (A ↾ B)
5	4	eleq1i 1152	. . . . . 6 ⊢ ((A “ B) ∈ V ↔ ran (A ↾ B) ∈ V)
6	3, 5	syl6ib 185	. . . . 5 ⊢ (B ∈ C → (Fun A → ran (A ↾ B) ∈ V))
7	6	imp 277	. . . 4 ⊢ ((B ∈ C ∧ Fun A) → ran (A ↾ B) ∈ V)
8	2, 7	jca 236	. . 3 ⊢ ((B ∈ C ∧ Fun A) → (dom (A ↾ B) ∈ V ∧ ran (A ↾ B) ∈ V))
9		xpexg 2489	. . 3 ⊢ ((dom (A ↾ B) ∈ V ∧ ran (A ↾ B) ∈ V) → (dom (A ↾ B) × ran (A ↾ B)) ∈ V)
10		relres 2591	. . . . 5 ⊢ Rel (A ↾ B)
11		relssdr 2668	. . . . 5 ⊢ (Rel (A ↾ B) → (A ↾ B) ⊆ (dom (A ↾ B) × ran (A ↾ B)))
12	10, 11	ax-mp 6	. . . 4 ⊢ (A ↾ B) ⊆ (dom (A ↾ B) × ran (A ↾ B))
13		ssexg 1702	. . . 4 ⊢ ((dom (A ↾ B) × ran (A ↾ B)) ∈ V → ((A ↾ B) ⊆ (dom (A ↾ B) × ran (A ↾ B)) → (A ↾ B) ∈ V))
14	12, 13	mpi 44	. . 3 ⊢ ((dom (A ↾ B) × ran (A ↾ B)) ∈ V → (A ↾ B) ∈ V)
15	8, 9, 14	3syl 21	. 2 ⊢ ((B ∈ C ∧ Fun A) → (A ↾ B) ∈ V)
16	15	exp 291	1 ⊢ (B ∈ C → (Fun A → (A ↾ B) ∈ V))

Syntax hints:   → wi 2   ∧ wa 196   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487   × cxp 2408  dom cdm 2410  ran crn 2411   ↾ cres 2412   “ cima 2413  Rel wrel 2415  Fun wfun 2416

This theorem is referenced by:  fvresex 2909  tz7.44-2 2967  tz7.44-3 2968  enrefg 3294  numthlem 3598  zornlem1 3603  imadomg 3616  fac0 4871  fac1 4872  facp1t 4873

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-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

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