Theorem xrnresex 38809

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 38807	. . 3 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = ((𝑅 ↾ 𝐴) ⋉ (𝑆 ↾ 𝐴))
2		xrnres2 38806	. . 3 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆 ↾ 𝐴))
3	1, 2	eqtr3i 2766	. 2 ⊢ ((𝑅 ↾ 𝐴) ⋉ (𝑆 ↾ 𝐴)) = (𝑅 ⋉ (𝑆 ↾ 𝐴))
4		dfres4 38679	. . . 4 ⊢ (𝑅 ↾ 𝐴) = (𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴)))
5		dfres4 38679	. . . 4 ⊢ (𝑆 ↾ 𝐴) = (𝑆 ∩ (𝐴 × ran (𝑆 ↾ 𝐴)))
6	4, 5	xrneq12i 38783	. . 3 ⊢ ((𝑅 ↾ 𝐴) ⋉ (𝑆 ↾ 𝐴)) = ((𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴))) ⋉ (𝑆 ∩ (𝐴 × ran (𝑆 ↾ 𝐴))))
7		simp1 1143	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ (𝑆 ↾ 𝐴) ∈ 𝑋) → 𝐴 ∈ 𝑉)
8		resexg 5985	. . . . . 6 ⊢ (𝑅 ∈ 𝑊 → (𝑅 ↾ 𝐴) ∈ V)
9		rnexg 7846	. . . . . 6 ⊢ ((𝑅 ↾ 𝐴) ∈ V → ran (𝑅 ↾ 𝐴) ∈ V)
10	8, 9	syl 17	. . . . 5 ⊢ (𝑅 ∈ 𝑊 → ran (𝑅 ↾ 𝐴) ∈ V)
11	10	3ad2ant2 1141	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ (𝑆 ↾ 𝐴) ∈ 𝑋) → ran (𝑅 ↾ 𝐴) ∈ V)
12		rnexg 7846	. . . . 5 ⊢ ((𝑆 ↾ 𝐴) ∈ 𝑋 → ran (𝑆 ↾ 𝐴) ∈ V)
13	12	3ad2ant3 1142	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ (𝑆 ↾ 𝐴) ∈ 𝑋) → ran (𝑆 ↾ 𝐴) ∈ V)
14		inxpxrn 38798	. . . . 5 ⊢ ((𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴))) ⋉ (𝑆 ∩ (𝐴 × ran (𝑆 ↾ 𝐴)))) = ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (ran (𝑅 ↾ 𝐴) × ran (𝑆 ↾ 𝐴))))
15		xrninxpex 38797	. . . . 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 2845	. . . 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 1380	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ (𝑆 ↾ 𝐴) ∈ 𝑋) → ((𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴))) ⋉ (𝑆 ∩ (𝐴 × ran (𝑆 ↾ 𝐴)))) ∈ V)
18	6, 17	eqeltrid 2845	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ (𝑆 ↾ 𝐴) ∈ 𝑋) → ((𝑅 ↾ 𝐴) ⋉ (𝑆 ↾ 𝐴)) ∈ V)
19	3, 18	eqeltrrid 2846	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ (𝑆 ↾ 𝐴) ∈ 𝑋) → (𝑅 ⋉ (𝑆 ↾ 𝐴)) ∈ V)

Syntax hints:   → wi 4   ∧ w3a 1093   ∈ wcel 2121  Vcvv 3433   ∩ cin 3883   × cxp 5618  ran crn 5621   ↾ cres 5622   ⋉ cxrn 38554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7681

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  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 6444  df-fun 6490  df-fn 6491  df-f 6492  df-fo 6494  df-fv 6496  df-1st 7933  df-2nd 7934  df-xrn 38760

This theorem is referenced by:  xrnidresex  38810  xrncnvepresex  38811