Theorem xrnresex 38738

Description: Sufficient condition for a restricted range Cartesian product to be a set. (Contributed by Peter Mazsa, 16-Dec-2020.) (Revised by Peter Mazsa, 7-Sep-2021.)

Assertion

Ref	Expression
xrnresex	⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ (𝑆 ↾ 𝐴) ∈ 𝑋) → (𝑅 ⋉ (𝑆 ↾ 𝐴)) ∈ V)

Proof of Theorem xrnresex

Step	Hyp	Ref	Expression
1		xrnres3 38736	. . 3 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = ((𝑅 ↾ 𝐴) ⋉ (𝑆 ↾ 𝐴))
2		xrnres2 38735	. . 3 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆 ↾ 𝐴))
3	1, 2	eqtr3i 2760	. 2 ⊢ ((𝑅 ↾ 𝐴) ⋉ (𝑆 ↾ 𝐴)) = (𝑅 ⋉ (𝑆 ↾ 𝐴))
4		dfres4 38608	. . . 4 ⊢ (𝑅 ↾ 𝐴) = (𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴)))
5		dfres4 38608	. . . 4 ⊢ (𝑆 ↾ 𝐴) = (𝑆 ∩ (𝐴 × ran (𝑆 ↾ 𝐴)))
6	4, 5	xrneq12i 38712	. . 3 ⊢ ((𝑅 ↾ 𝐴) ⋉ (𝑆 ↾ 𝐴)) = ((𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴))) ⋉ (𝑆 ∩ (𝐴 × ran (𝑆 ↾ 𝐴))))
7		simp1 1137	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ (𝑆 ↾ 𝐴) ∈ 𝑋) → 𝐴 ∈ 𝑉)
8		resexg 5981	. . . . . 6 ⊢ (𝑅 ∈ 𝑊 → (𝑅 ↾ 𝐴) ∈ V)
9		rnexg 7842	. . . . . 6 ⊢ ((𝑅 ↾ 𝐴) ∈ V → ran (𝑅 ↾ 𝐴) ∈ V)
10	8, 9	syl 17	. . . . 5 ⊢ (𝑅 ∈ 𝑊 → ran (𝑅 ↾ 𝐴) ∈ V)
11	10	3ad2ant2 1135	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ (𝑆 ↾ 𝐴) ∈ 𝑋) → ran (𝑅 ↾ 𝐴) ∈ V)
12		rnexg 7842	. . . . 5 ⊢ ((𝑆 ↾ 𝐴) ∈ 𝑋 → ran (𝑆 ↾ 𝐴) ∈ V)
13	12	3ad2ant3 1136	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ (𝑆 ↾ 𝐴) ∈ 𝑋) → ran (𝑆 ↾ 𝐴) ∈ V)
14		inxpxrn 38727	. . . . 5 ⊢ ((𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴))) ⋉ (𝑆 ∩ (𝐴 × ran (𝑆 ↾ 𝐴)))) = ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (ran (𝑅 ↾ 𝐴) × ran (𝑆 ↾ 𝐴))))
15		xrninxpex 38726	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ran (𝑅 ↾ 𝐴) ∈ V ∧ ran (𝑆 ↾ 𝐴) ∈ V) → ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (ran (𝑅 ↾ 𝐴) × ran (𝑆 ↾ 𝐴)))) ∈ V)
16	14, 15	eqeltrid 2839	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ran (𝑅 ↾ 𝐴) ∈ V ∧ ran (𝑆 ↾ 𝐴) ∈ V) → ((𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴))) ⋉ (𝑆 ∩ (𝐴 × ran (𝑆 ↾ 𝐴)))) ∈ V)
17	7, 11, 13, 16	syl3anc 1374	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ (𝑆 ↾ 𝐴) ∈ 𝑋) → ((𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴))) ⋉ (𝑆 ∩ (𝐴 × ran (𝑆 ↾ 𝐴)))) ∈ V)
18	6, 17	eqeltrid 2839	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ (𝑆 ↾ 𝐴) ∈ 𝑋) → ((𝑅 ↾ 𝐴) ⋉ (𝑆 ↾ 𝐴)) ∈ V)
19	3, 18	eqeltrrid 2840	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ (𝑆 ↾ 𝐴) ∈ 𝑋) → (𝑅 ⋉ (𝑆 ↾ 𝐴)) ∈ V)

Syntax hints:   → wi 4   ∧ w3a 1087   ∈ wcel 2114  Vcvv 3427   ∩ cin 3884   × cxp 5618  ran crn 5621   ↾ cres 5622   ⋉ cxrn 38483

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-rab 3388  df-v 3429  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-fo 6493  df-fv 6495  df-1st 7931  df-2nd 7932  df-xrn 38689

This theorem is referenced by:  xrnidresex  38739  xrncnvepresex  38740