Theorem f1ocnvfv1 3884

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 3715	. . . 4 ⊢ (F:A–1-1-onto→B → (◡F ∘ F) = (I ↾ A))
2	1	fveq1d 3732	. . 3 ⊢ (F:A–1-1-onto→B → ((◡F ∘ F) ‘C) = ((I ↾ A) ‘C))
3	2	adantr 391	. 2 ⊢ ((F:A–1-1-onto→B ⋀ C ∈ A) → ((◡F ∘ F) ‘C) = ((I ↾ A) ‘C))
4		fvco3 3782	. . . 4 ⊢ ((Fun ◡F ⋀ F:A–→B ⋀ C ∈ A) → ((◡F ∘ F) ‘C) = (◡F ‘(F ‘C)))
5	4	3expa 835	. . 3 ⊢ (((Fun ◡F ⋀ F:A–→B) ⋀ C ∈ A) → ((◡F ∘ F) ‘C) = (◡F ‘(F ‘C)))
6		f1ocnv 3707	. . . . 5 ⊢ (F:A–1-1-onto→B → ◡F:B–1-1-onto→A)
7		f1ofun 3697	. . . . 5 ⊢ (◡F:B–1-1-onto→A → Fun ◡F)
8	6, 7	syl 10	. . . 4 ⊢ (F:A–1-1-onto→B → Fun ◡F)
9		f1of 3695	. . . 4 ⊢ (F:A–1-1-onto→B → F:A–→B)
10	8, 9	jca 288	. . 3 ⊢ (F:A–1-1-onto→B → (Fun ◡F ⋀ F:A–→B))
11	5, 10	sylan 450	. 2 ⊢ ((F:A–1-1-onto→B ⋀ C ∈ A) → ((◡F ∘ F) ‘C) = (◡F ‘(F ‘C)))
12		fvresi 3849	. . 3 ⊢ (C ∈ A → ((I ↾ A) ‘C) = C)
13	12	adantl 390	. 2 ⊢ ((F:A–1-1-onto→B ⋀ C ∈ A) → ((I ↾ A) ‘C) = C)
14	3, 11, 13	3eqtr3d 1518	1 ⊢ ((F:A–1-1-onto→B ⋀ C ∈ A) → (◡F ‘(F ‘C)) = C)

Syntax hints:   → wi 3   ⋀ wa 223   = wceq 958   ∈ wcel 960  Icid 2837  ^◡ccnv 3175   ↾ cres 3178   ∘ ccom 3180  Fun wfun 3182  –→wf 3184  –1-1-onto→wf1o 3187   ‘cfv 3188

This theorem is referenced by:  f1ocnvfv 3886  logeft 8757  cnvbrabrat 10040

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204

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