Theorem f1ococnv1 2818

Description: The composition of a one-to-one onto function's converse and itself equals the identity relation restricted to the function's domain.

Assertion

Ref	Expression
f1ococnv1	⊢ (F:A–1-1-onto→B → (◡F ∘ F) = (I ↾ A))

Proof of Theorem f1ococnv1

Step	Hyp	Ref	Expression
1		f1orel 2803	. . . 4 ⊢ (F:A–1-1-onto→B → Rel F)
2		dfrel2 2660	. . . 4 ⊢ (Rel F ↔ ◡◡F = F)
3	1, 2	sylib 173	. . 3 ⊢ (F:A–1-1-onto→B → ◡◡F = F)
4	3	coeq2d 2507	. 2 ⊢ (F:A–1-1-onto→B → (◡F ∘ ◡◡F) = (◡F ∘ F))
5		f1ocnv 2811	. . 3 ⊢ (F:A–1-1-onto→B → ◡F:B–1-1-onto→A)
6		f1ococnv2 2817	. . 3 ⊢ (◡F:B–1-1-onto→A → (◡F ∘ ◡◡F) = (I ↾ A))
7	5, 6	syl 12	. 2 ⊢ (F:A–1-1-onto→B → (◡F ∘ ◡◡F) = (I ↾ A))
8	4, 7	eqtr3d 1130	1 ⊢ (F:A–1-1-onto→B → (◡F ∘ F) = (I ↾ A))

Syntax hints:   → wi 2   = wceq 1091  Icid 2057  ^◡ccnv 2409   ↾ cres 2412   ∘ ccom 2414  Rel wrel 2415  –1-1-onto→wf1o 2421

This theorem is referenced by:  f1ocnvfv1 2919  mapenlem1 3384

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-3an 583  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-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437

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