Theorem resdmdfsnOLD 6021

Description: Obsolete version of resdmdfsn 6020 as of 16-Jun-2026. (Contributed by AV, 2-Dec-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
resdmdfsnOLD	⊢ (Rel 𝑅 → (𝑅 ↾ (V ∖ {𝑋})) = (𝑅 ↾ (dom 𝑅 ∖ {𝑋})))

Proof of Theorem resdmdfsnOLD

Step	Hyp	Ref	Expression
1		resindmOLD 6019	. 2 ⊢ (Rel 𝑅 → (𝑅 ↾ ((V ∖ {𝑋}) ∩ dom 𝑅)) = (𝑅 ↾ (V ∖ {𝑋})))
2		indif1 4227	. . . 4 ⊢ ((V ∖ {𝑋}) ∩ dom 𝑅) = ((V ∩ dom 𝑅) ∖ {𝑋})
3		incom 4106	. . . . . 6 ⊢ (V ∩ dom 𝑅) = (dom 𝑅 ∩ V)
4		inv1 4347	. . . . . 6 ⊢ (dom 𝑅 ∩ V) = dom 𝑅
5	3, 4	eqtri 2783	. . . . 5 ⊢ (V ∩ dom 𝑅) = dom 𝑅
6	5	difeq1i 4059	. . . 4 ⊢ ((V ∩ dom 𝑅) ∖ {𝑋}) = (dom 𝑅 ∖ {𝑋})
7	2, 6	eqtri 2783	. . 3 ⊢ ((V ∖ {𝑋}) ∩ dom 𝑅) = (dom 𝑅 ∖ {𝑋})
8	7	reseq2i 5966	. 2 ⊢ (𝑅 ↾ ((V ∖ {𝑋}) ∩ dom 𝑅)) = (𝑅 ↾ (dom 𝑅 ∖ {𝑋}))
9	1, 8	eqtr3di 2810	1 ⊢ (Rel 𝑅 → (𝑅 ↾ (V ∖ {𝑋})) = (𝑅 ↾ (dom 𝑅 ∖ {𝑋})))

This proof depends on syntax axioms:   → wi 4   = wceq 1569  Vcvv 3450   ∖ cdif 4045   ∩ cin 4089  {csn 4584  dom cdm 5650   ↾ cres 5652  Rel wrel 5655

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-dm 5660  df-res 5662

This theorem is used by: (None)