Theorem funcnvres 2710

Description: The converse of a restricted function.

Assertion

Ref	Expression
funcnvres	⊢ (Fun ◡F → ◡(F ↾ A) = (◡F ↾ (F “ A)))

Proof of Theorem funcnvres

Step	Hyp	Ref	Expression
1		resss 2587	. . . 4 ⊢ (F ↾ A) ⊆ F
2		cnvss 2512	. . . 4 ⊢ ((F ↾ A) ⊆ F → ◡(F ↾ A) ⊆ ◡F)
3	1, 2	ax-mp 6	. . 3 ⊢ ◡(F ↾ A) ⊆ ◡F
4		funssres 2698	. . 3 ⊢ ((Fun ◡F ∧ ◡(F ↾ A) ⊆ ◡F) → (◡F ↾ dom ◡(F ↾ A)) = ◡(F ↾ A))
5	3, 4	mpan2 519	. 2 ⊢ (Fun ◡F → (◡F ↾ dom ◡(F ↾ A)) = ◡(F ↾ A))
6		df-ima 2431	. . . 4 ⊢ (F “ A) = ran (F ↾ A)
7		df-rn 2429	. . . 4 ⊢ ran (F ↾ A) = dom ◡(F ↾ A)
8	6, 7	eqtr 1119	. . 3 ⊢ (F “ A) = dom ◡(F ↾ A)
9		reseq2 2576	. . 3 ⊢ ((F “ A) = dom ◡(F ↾ A) → (◡F ↾ (F “ A)) = (◡F ↾ dom ◡(F ↾ A)))
10	8, 9	ax-mp 6	. 2 ⊢ (◡F ↾ (F “ A)) = (◡F ↾ dom ◡(F ↾ A))
11	5, 10	syl5req 1137	1 ⊢ (Fun ◡F → ◡(F ↾ A) = (◡F ↾ (F “ A)))

Syntax hints:   → wi 2   = wceq 1091   ⊆ wss 1487  ^◡ccnv 2409  dom cdm 2410  ran crn 2411   ↾ cres 2412   “ cima 2413  Fun wfun 2416

This theorem is referenced by:  funimacnv 2711  f1imacnv 2814  sbthlem4 3352

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-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-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-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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