Theorem fococnv2 6725

Description: The composition of an onto function and its converse. (Contributed by Stefan O'Rear, 12-Feb-2015.)

Assertion

Ref	Expression
fococnv2	⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 ∘ ◡𝐹) = ( I ↾ 𝐵))

Proof of Theorem fococnv2

Step	Hyp	Ref	Expression
1		fofun 6673	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → Fun 𝐹)
2		funcocnv2 6724	. . 3 ⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))
3	1, 2	syl 17	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))
4		forn 6675	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
5	4	reseq2d 5880	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → ( I ↾ ran 𝐹) = ( I ↾ 𝐵))
6	3, 5	eqtrd 2778	1 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 ∘ ◡𝐹) = ( I ↾ 𝐵))

Syntax hints:   → wi 4   = wceq 1539   I cid 5479  ^◡ccnv 5579  ran crn 5581   ↾ cres 5582   ∘ ccom 5584  Fun wfun 6412  –onto→wfo 6416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-fun 6420  df-fn 6421  df-f 6422  df-fo 6424

This theorem is referenced by:  f1ococnv2  6726  foeqcnvco  7152