Theorem xrnresex 36459

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 36457	. . 3 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = ((𝑅 ↾ 𝐴) ⋉ (𝑆 ↾ 𝐴))
2		xrnres2 36456	. . 3 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆 ↾ 𝐴))
3	1, 2	eqtr3i 2768	. 2 ⊢ ((𝑅 ↾ 𝐴) ⋉ (𝑆 ↾ 𝐴)) = (𝑅 ⋉ (𝑆 ↾ 𝐴))
4		dfres4 36355	. . . 4 ⊢ (𝑅 ↾ 𝐴) = (𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴)))
5		dfres4 36355	. . . 4 ⊢ (𝑆 ↾ 𝐴) = (𝑆 ∩ (𝐴 × ran (𝑆 ↾ 𝐴)))
6	4, 5	xrneq12i 36441	. . 3 ⊢ ((𝑅 ↾ 𝐴) ⋉ (𝑆 ↾ 𝐴)) = ((𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴))) ⋉ (𝑆 ∩ (𝐴 × ran (𝑆 ↾ 𝐴))))
7		simp1 1134	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ (𝑆 ↾ 𝐴) ∈ 𝑋) → 𝐴 ∈ 𝑉)
8		resexg 5926	. . . . . 6 ⊢ (𝑅 ∈ 𝑊 → (𝑅 ↾ 𝐴) ∈ V)
9		rnexg 7725	. . . . . 6 ⊢ ((𝑅 ↾ 𝐴) ∈ V → ran (𝑅 ↾ 𝐴) ∈ V)
10	8, 9	syl 17	. . . . 5 ⊢ (𝑅 ∈ 𝑊 → ran (𝑅 ↾ 𝐴) ∈ V)
11	10	3ad2ant2 1132	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ (𝑆 ↾ 𝐴) ∈ 𝑋) → ran (𝑅 ↾ 𝐴) ∈ V)
12		rnexg 7725	. . . . 5 ⊢ ((𝑆 ↾ 𝐴) ∈ 𝑋 → ran (𝑆 ↾ 𝐴) ∈ V)
13	12	3ad2ant3 1133	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ (𝑆 ↾ 𝐴) ∈ 𝑋) → ran (𝑆 ↾ 𝐴) ∈ V)
14		inxpxrn 36448	. . . . 5 ⊢ ((𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴))) ⋉ (𝑆 ∩ (𝐴 × ran (𝑆 ↾ 𝐴)))) = ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (ran (𝑅 ↾ 𝐴) × ran (𝑆 ↾ 𝐴))))
15		xrninxpex 36447	. . . . 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 2843	. . . 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 1369	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ (𝑆 ↾ 𝐴) ∈ 𝑋) → ((𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴))) ⋉ (𝑆 ∩ (𝐴 × ran (𝑆 ↾ 𝐴)))) ∈ V)
18	6, 17	eqeltrid 2843	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ (𝑆 ↾ 𝐴) ∈ 𝑋) → ((𝑅 ↾ 𝐴) ⋉ (𝑆 ↾ 𝐴)) ∈ V)
19	3, 18	eqeltrrid 2844	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ (𝑆 ↾ 𝐴) ∈ 𝑋) → (𝑅 ⋉ (𝑆 ↾ 𝐴)) ∈ V)

Syntax hints:   → wi 4   ∧ w3a 1085   ∈ wcel 2108  Vcvv 3422   ∩ cin 3882   × cxp 5578  ran crn 5581   ↾ cres 5582   ⋉ cxrn 36259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fo 6424  df-fv 6426  df-1st 7804  df-2nd 7805  df-xrn 36428

This theorem is referenced by:  xrnidresex  36460  xrncnvepresex  36461