Theorem xrnres4 39280

Description: Two ways to express restriction of range Cartesian product. (Contributed by Peter Mazsa, 29-Dec-2020.)

Assertion

Ref	Expression
xrnres4	⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (ran (𝑅 ↾ 𝐴) × ran (𝑆 ↾ 𝐴))))

Proof of Theorem xrnres4

Step	Hyp	Ref	Expression
1		xrnres3 39279	. 2 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = ((𝑅 ↾ 𝐴) ⋉ (𝑆 ↾ 𝐴))
2		dfres4 39151	. . 3 ⊢ (𝑅 ↾ 𝐴) = (𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴)))
3		dfres4 39151	. . 3 ⊢ (𝑆 ↾ 𝐴) = (𝑆 ∩ (𝐴 × ran (𝑆 ↾ 𝐴)))
4	2, 3	xrneq12i 39255	. 2 ⊢ ((𝑅 ↾ 𝐴) ⋉ (𝑆 ↾ 𝐴)) = ((𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴))) ⋉ (𝑆 ∩ (𝐴 × ran (𝑆 ↾ 𝐴))))
5		inxpxrn 39270	. 2 ⊢ ((𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴))) ⋉ (𝑆 ∩ (𝐴 × ran (𝑆 ↾ 𝐴)))) = ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (ran (𝑅 ↾ 𝐴) × ran (𝑆 ↾ 𝐴))))
6	1, 4, 5	3eqtri 2787	1 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (ran (𝑅 ↾ 𝐴) × ran (𝑆 ↾ 𝐴))))

This proof depends on syntax axioms:   = wceq 1570   ∩ cin 3897   × cxp 5645  ran crn 5648   ↾ cres 5649   ⋉ cxrn 39026

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7734

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-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-fo 6533  df-fv 6535  df-1st 7984  df-2nd 7985  df-xrn 39232

This theorem is used by: (None)