Theorem cnvresidOLD 6641

Description: Obsolete version of cnvresid 6281 as of 1-Nov-2026. (Contributed by FL, 4-Mar-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
cnvresidOLD	⊢ ◡( I ↾ 𝐴) = ( I ↾ 𝐴)

Proof of Theorem cnvresidOLD

Step	Hyp	Ref	Expression
1		cnvi 5870	. . 3 ⊢ ◡ I = I
2	1	eqcomi 2772	. 2 ⊢ I = ◡ I
3		funi 6592	. . 3 ⊢ Fun I
4		funeq 6579	. . 3 ⊢ ( I = ◡ I → (Fun I ↔ Fun ◡ I ))
5	3, 4	mpbii 236	. 2 ⊢ ( I = ◡ I → Fun ◡ I )
6		funcnvres 6640	. . 3 ⊢ (Fun ◡ I → ◡( I ↾ 𝐴) = (◡ I ↾ ( I “ 𝐴)))
7		imai 6084	. . . 4 ⊢ ( I “ 𝐴) = 𝐴
8	1, 7	reseq12i 5979	. . 3 ⊢ (◡ I ↾ ( I “ 𝐴)) = ( I ↾ 𝐴)
9	6, 8	eqtrdi 2814	. 2 ⊢ (Fun ◡ I → ◡( I ↾ 𝐴) = ( I ↾ 𝐴))
10	2, 5, 9	mp2b 10	1 ⊢ ◡( I ↾ 𝐴) = ( I ↾ 𝐴)

This proof depends on syntax axioms:   = wceq 1570   I cid 5549  ^◡ccnv 5654   ↾ cres 5657   “ cima 5658  Fun wfun 6552

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-12 2216  ax-ext 2735  ax-sep 5252  ax-pr 5395

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-mo 2567  df-eu 2597  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 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-fun 6560

This theorem is used by: (None)