Theorem f1ocnvfv1 2919

Description: The converse value of the value of a one-to-one onto function.

Assertion

Ref	Expression
f1ocnvfv1	⊢ ((F:A–1-1-onto→B ∧ C ∈ A) → (◡F ‘(F ‘C)) = C)

Proof of Theorem f1ocnvfv1

Step	Hyp	Ref	Expression
1		f1ococnv1 2818	. . . 4 ⊢ (F:A–1-1-onto→B → (◡F ∘ F) = (I ↾ A))
2	1	fveq1d 2834	. . 3 ⊢ (F:A–1-1-onto→B → ((◡F ∘ F) ‘C) = ((I ↾ A) ‘C))
3	2	adantr 306	. 2 ⊢ ((F:A–1-1-onto→B ∧ C ∈ A) → ((◡F ∘ F) ‘C) = ((I ↾ A) ‘C))
4		fvco3 2867	. . 3 ⊢ (((Fun ◡F ∧ F:A–→B) ∧ C ∈ A) → ((◡F ∘ F) ‘C) = (◡F ‘(F ‘C)))
5		f1ocnv 2811	. . . . 5 ⊢ (F:A–1-1-onto→B → ◡F:B–1-1-onto→A)
6		f1ofun 2802	. . . . 5 ⊢ (◡F:B–1-1-onto→A → Fun ◡F)
7	5, 6	syl 12	. . . 4 ⊢ (F:A–1-1-onto→B → Fun ◡F)
8		f1of 2800	. . . 4 ⊢ (F:A–1-1-onto→B → F:A–→B)
9	7, 8	jca 236	. . 3 ⊢ (F:A–1-1-onto→B → (Fun ◡F ∧ F:A–→B))
10	4, 9	sylan 343	. 2 ⊢ ((F:A–1-1-onto→B ∧ C ∈ A) → ((◡F ∘ F) ‘C) = (◡F ‘(F ‘C)))
11		fvresi 2901	. . 3 ⊢ (C ∈ A → ((I ↾ A) ‘C) = C)
12	11	adantl 305	. 2 ⊢ ((F:A–1-1-onto→B ∧ C ∈ A) → ((I ↾ A) ‘C) = C)
13	3, 10, 12	3eqtr3d 1133	1 ⊢ ((F:A–1-1-onto→B ∧ C ∈ A) → (◡F ‘(F ‘C)) = C)

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Icid 2057  ^◡ccnv 2409   ↾ cres 2412   ∘ ccom 2414  Fun wfun 2416  –→wf 2418  –1-1-onto→wf1o 2421   ‘cfv 2422

This theorem is referenced by:  f1ocnvfv 2921

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-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

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