Theorem funcnvres2 6572

Description: The converse of a restriction of the converse of a function equals the function restricted to the image of its converse. (Contributed by NM, 4-May-2005.)

Assertion

Ref	Expression
funcnvres2	⊢ (Fun 𝐹 → ◡(◡𝐹 ↾ 𝐴) = (𝐹 ↾ (◡𝐹 “ 𝐴)))

Proof of Theorem funcnvres2

Step	Hyp	Ref	Expression
1		funcnvcnv 6559	. . 3 ⊢ (Fun 𝐹 → Fun ◡◡𝐹)
2		funcnvres 6570	. . 3 ⊢ (Fun ◡◡𝐹 → ◡(◡𝐹 ↾ 𝐴) = (◡◡𝐹 ↾ (◡𝐹 “ 𝐴)))
3	1, 2	syl 17	. 2 ⊢ (Fun 𝐹 → ◡(◡𝐹 ↾ 𝐴) = (◡◡𝐹 ↾ (◡𝐹 “ 𝐴)))
4		funrel 6509	. . . 4 ⊢ (Fun 𝐹 → Rel 𝐹)
5		dfrel2 6147	. . . 4 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹)
6	4, 5	sylib 219	. . 3 ⊢ (Fun 𝐹 → ◡◡𝐹 = 𝐹)
7	6	reseq1d 5937	. 2 ⊢ (Fun 𝐹 → (◡◡𝐹 ↾ (◡𝐹 “ 𝐴)) = (𝐹 ↾ (◡𝐹 “ 𝐴)))
8	3, 7	eqtrd 2775	1 ⊢ (Fun 𝐹 → ◡(◡𝐹 ↾ 𝐴) = (𝐹 ↾ (◡𝐹 “ 𝐴)))

Syntax hints:   → wi 4   = wceq 1547  ^◡ccnv 5624   ↾ cres 5627   “ cima 5628  Rel wrel 5630  Fun wfun 6486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-fun 6494

This theorem is referenced by:  funimacnv  6573  foimacnv  6791  unbenlem  16877  ofco2  22441  dvlog  26640  fresf1o  32730  fressupp  32787