Theorem fococnv2 6877

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

Assertion

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

Proof of Theorem fococnv2

Step	Hyp	Ref	Expression
1		fofun 6822	. . 3 ⊢ (𝐹:𝐴⟶onto𝐵 → Fun 𝐹)
2		funcocnv2 6875	. . 3 ⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))
3	1, 2	syl 18	. 2 ⊢ (𝐹:𝐴⟶onto𝐵 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))
4		forn 6824	. . 3 ⊢ (𝐹:𝐴⟶onto𝐵 → ran 𝐹 = 𝐵)
5	4	reseq2d 5982	. 2 ⊢ (𝐹:𝐴⟶onto𝐵 → ( I ↾ ran 𝐹) = ( I ↾ 𝐵))
6	3, 5	eqtrd 2798	1 ⊢ (𝐹:𝐴⟶onto𝐵 → (𝐹 ∘ ◡𝐹) = ( I ↾ 𝐵))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   I cid 5550  ^◡ccnv 5655  ran crn 5657   ↾ cres 5658   ∘ ccom 5660  Fun wfun 6555  ⟶ontowfo 6559

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5253  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5172  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567

This theorem is used by:  f1ococnv2  6878  foeqcnvco  7334