Theorem xrnresex 38386

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 38384	. . 3 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = ((𝑅 ↾ 𝐴) ⋉ (𝑆 ↾ 𝐴))
2		xrnres2 38383	. . 3 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆 ↾ 𝐴))
3	1, 2	eqtr3i 2754	. 2 ⊢ ((𝑅 ↾ 𝐴) ⋉ (𝑆 ↾ 𝐴)) = (𝑅 ⋉ (𝑆 ↾ 𝐴))
4		dfres4 38275	. . . 4 ⊢ (𝑅 ↾ 𝐴) = (𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴)))
5		dfres4 38275	. . . 4 ⊢ (𝑆 ↾ 𝐴) = (𝑆 ∩ (𝐴 × ran (𝑆 ↾ 𝐴)))
6	4, 5	xrneq12i 38364	. . 3 ⊢ ((𝑅 ↾ 𝐴) ⋉ (𝑆 ↾ 𝐴)) = ((𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴))) ⋉ (𝑆 ∩ (𝐴 × ran (𝑆 ↾ 𝐴))))
7		simp1 1136	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ (𝑆 ↾ 𝐴) ∈ 𝑋) → 𝐴 ∈ 𝑉)
8		resexg 5987	. . . . . 6 ⊢ (𝑅 ∈ 𝑊 → (𝑅 ↾ 𝐴) ∈ V)
9		rnexg 7858	. . . . . 6 ⊢ ((𝑅 ↾ 𝐴) ∈ V → ran (𝑅 ↾ 𝐴) ∈ V)
10	8, 9	syl 17	. . . . 5 ⊢ (𝑅 ∈ 𝑊 → ran (𝑅 ↾ 𝐴) ∈ V)
11	10	3ad2ant2 1134	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ (𝑆 ↾ 𝐴) ∈ 𝑋) → ran (𝑅 ↾ 𝐴) ∈ V)
12		rnexg 7858	. . . . 5 ⊢ ((𝑆 ↾ 𝐴) ∈ 𝑋 → ran (𝑆 ↾ 𝐴) ∈ V)
13	12	3ad2ant3 1135	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ (𝑆 ↾ 𝐴) ∈ 𝑋) → ran (𝑆 ↾ 𝐴) ∈ V)
14		inxpxrn 38375	. . . . 5 ⊢ ((𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴))) ⋉ (𝑆 ∩ (𝐴 × ran (𝑆 ↾ 𝐴)))) = ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (ran (𝑅 ↾ 𝐴) × ran (𝑆 ↾ 𝐴))))
15		xrninxpex 38374	. . . . 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 2832	. . . 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 1373	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ (𝑆 ↾ 𝐴) ∈ 𝑋) → ((𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴))) ⋉ (𝑆 ∩ (𝐴 × ran (𝑆 ↾ 𝐴)))) ∈ V)
18	6, 17	eqeltrid 2832	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ (𝑆 ↾ 𝐴) ∈ 𝑋) → ((𝑅 ↾ 𝐴) ⋉ (𝑆 ↾ 𝐴)) ∈ V)
19	3, 18	eqeltrrid 2833	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ (𝑆 ↾ 𝐴) ∈ 𝑋) → (𝑅 ⋉ (𝑆 ↾ 𝐴)) ∈ V)

Syntax hints:   → wi 4   ∧ w3a 1086   ∈ wcel 2109  Vcvv 3444   ∩ cin 3910   × cxp 5629  ran crn 5632   ↾ cres 5633   ⋉ cxrn 38162

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fo 6505  df-fv 6507  df-1st 7947  df-2nd 7948  df-xrn 38347

This theorem is referenced by:  xrnidresex  38387  xrncnvepresex  38388