Theorem xrnres2 34503

Description: Two ways to express restriction of range Cartesian product, cf. xrnres 34502, xrnres3 34504. (Contributed by Peter Mazsa, 6-Sep-2021.)

Assertion

Ref	Expression
xrnres2	⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆 ↾ 𝐴))

Proof of Theorem xrnres2

Step	Hyp	Ref	Expression
1		resco 5783	. . 3 ⊢ ((◡(2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴) = (◡(2nd ↾ (V × V)) ∘ (𝑆 ↾ 𝐴))
2	1	ineq2i 3962	. 2 ⊢ ((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ ((◡(2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴)) = ((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ (𝑆 ↾ 𝐴)))
3		df-xrn 34475	. . . 4 ⊢ (𝑅 ⋉ 𝑆) = ((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆))
4	3	reseq1i 5530	. . 3 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴)
5		inres 5555	. . 3 ⊢ ((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ ((◡(2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴)) = (((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴)
6	4, 5	eqtr4i 2796	. 2 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = ((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ ((◡(2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴))
7		df-xrn 34475	. 2 ⊢ (𝑅 ⋉ (𝑆 ↾ 𝐴)) = ((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ (𝑆 ↾ 𝐴)))
8	2, 6, 7	3eqtr4i 2803	1 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆 ↾ 𝐴))

Syntax hints:   = wceq 1631  Vcvv 3351   ∩ cin 3722   × cxp 5247  ^◡ccnv 5248   ↾ cres 5251   ∘ ccom 5253  1^st c1st 7313  2^nd c2nd 7314   ⋉ cxrn 34314

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-xp 5255  df-rel 5256  df-co 5258  df-res 5261  df-xrn 34475

This theorem is referenced by:  xrnresex  34506  br1cossxrnres  34540