Theorem funimacnv 2711

Description: The image of the converse image of a function.

Assertion

Ref	Expression
funimacnv	⊢ (Fun F → (F “ (◡F “ A)) = (A ∩ ran F))

Proof of Theorem funimacnv

Step	Hyp	Ref	Expression
1		funcnvcnv 2701	. . . . . 6 ⊢ (Fun F → Fun ◡◡F)
2		funcnvres 2710	. . . . . 6 ⊢ (Fun ◡◡F → ◡(◡F ↾ A) = (◡◡F ↾ (◡F “ A)))
3	1, 2	syl 12	. . . . 5 ⊢ (Fun F → ◡(◡F ↾ A) = (◡◡F ↾ (◡F “ A)))
4		funrel 2681	. . . . . . 7 ⊢ (Fun F → Rel F)
5		dfrel2 2660	. . . . . . 7 ⊢ (Rel F ↔ ◡◡F = F)
6	4, 5	sylib 173	. . . . . 6 ⊢ (Fun F → ◡◡F = F)
7		reseq1 2575	. . . . . 6 ⊢ (◡◡F = F → (◡◡F ↾ (◡F “ A)) = (F ↾ (◡F “ A)))
8	6, 7	syl 12	. . . . 5 ⊢ (Fun F → (◡◡F ↾ (◡F “ A)) = (F ↾ (◡F “ A)))
9	3, 8	eqtrd 1128	. . . 4 ⊢ (Fun F → ◡(◡F ↾ A) = (F ↾ (◡F “ A)))
10	9	rneqd 2557	. . 3 ⊢ (Fun F → ran ◡(◡F ↾ A) = ran (F ↾ (◡F “ A)))
11		df-ima 2431	. . 3 ⊢ (F “ (◡F “ A)) = ran (F ↾ (◡F “ A))
12	10, 11	syl6reqr 1143	. 2 ⊢ (Fun F → (F “ (◡F “ A)) = ran ◡(◡F ↾ A))
13		dfdm4 2525	. . 3 ⊢ dom (◡F ↾ A) = ran ◡(◡F ↾ A)
14		dmres 2584	. . . 4 ⊢ dom (◡F ↾ A) = (A ∩ dom ◡F)
15		df-rn 2429	. . . . 5 ⊢ ran F = dom ◡F
16	15	ineq2i 1642	. . . 4 ⊢ (A ∩ ran F) = (A ∩ dom ◡F)
17	14, 16	eqtr4 1122	. . 3 ⊢ dom (◡F ↾ A) = (A ∩ ran F)
18	13, 17	eqtr3 1121	. 2 ⊢ ran ◡(◡F ↾ A) = (A ∩ ran F)
19	12, 18	syl6eq 1140	1 ⊢ (Fun F → (F “ (◡F “ A)) = (A ∩ ran F))

Syntax hints:   → wi 2   = wceq 1091   ∩ cin 1486  ^◡ccnv 2409  dom cdm 2410  ran crn 2411   ↾ cres 2412   “ cima 2413  Rel wrel 2415  Fun wfun 2416

This theorem is referenced by:  funimass1 2712  funimass2 2713

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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