Theorem resssxp 6258

Description: If the 𝑅-image of a class 𝐴 is a subclass of 𝐵, then the restriction of 𝑅 to 𝐴 is a subset of the Cartesian product of 𝐴 and 𝐵. (Contributed by RP, 24-Dec-2019.)

Assertion

Ref	Expression
resssxp	⊢ ((𝑅 “ 𝐴) ⊆ 𝐵 ↔ (𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵))

Proof of Theorem resssxp

Step	Hyp	Ref	Expression
1		df-ima 5682	. . 3 ⊢ (𝑅 “ 𝐴) = ran (𝑅 ↾ 𝐴)
2	1	sseq1i 4006	. 2 ⊢ ((𝑅 “ 𝐴) ⊆ 𝐵 ↔ ran (𝑅 ↾ 𝐴) ⊆ 𝐵)
3		dmres 5995	. . . 4 ⊢ dom (𝑅 ↾ 𝐴) = (𝐴 ∩ dom 𝑅)
4		inss1 4224	. . . 4 ⊢ (𝐴 ∩ dom 𝑅) ⊆ 𝐴
5	3, 4	eqsstri 4012	. . 3 ⊢ dom (𝑅 ↾ 𝐴) ⊆ 𝐴
6	5	biantrur 531	. 2 ⊢ (ran (𝑅 ↾ 𝐴) ⊆ 𝐵 ↔ (dom (𝑅 ↾ 𝐴) ⊆ 𝐴 ∧ ran (𝑅 ↾ 𝐴) ⊆ 𝐵))
7		relres 6002	. . . . 5 ⊢ Rel (𝑅 ↾ 𝐴)
8		relssdmrn 6256	. . . . 5 ⊢ (Rel (𝑅 ↾ 𝐴) → (𝑅 ↾ 𝐴) ⊆ (dom (𝑅 ↾ 𝐴) × ran (𝑅 ↾ 𝐴)))
9	7, 8	ax-mp 5	. . . 4 ⊢ (𝑅 ↾ 𝐴) ⊆ (dom (𝑅 ↾ 𝐴) × ran (𝑅 ↾ 𝐴))
10		xpss12 5684	. . . 4 ⊢ ((dom (𝑅 ↾ 𝐴) ⊆ 𝐴 ∧ ran (𝑅 ↾ 𝐴) ⊆ 𝐵) → (dom (𝑅 ↾ 𝐴) × ran (𝑅 ↾ 𝐴)) ⊆ (𝐴 × 𝐵))
11	9, 10	sstrid 3989	. . 3 ⊢ ((dom (𝑅 ↾ 𝐴) ⊆ 𝐴 ∧ ran (𝑅 ↾ 𝐴) ⊆ 𝐵) → (𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵))
12		dmss 5894	. . . . 5 ⊢ ((𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵) → dom (𝑅 ↾ 𝐴) ⊆ dom (𝐴 × 𝐵))
13		dmxpss 6159	. . . . 5 ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴
14	12, 13	sstrdi 3990	. . . 4 ⊢ ((𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵) → dom (𝑅 ↾ 𝐴) ⊆ 𝐴)
15		rnss 5930	. . . . 5 ⊢ ((𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵) → ran (𝑅 ↾ 𝐴) ⊆ ran (𝐴 × 𝐵))
16		rnxpss 6160	. . . . 5 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵
17	15, 16	sstrdi 3990	. . . 4 ⊢ ((𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵) → ran (𝑅 ↾ 𝐴) ⊆ 𝐵)
18	14, 17	jca 512	. . 3 ⊢ ((𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵) → (dom (𝑅 ↾ 𝐴) ⊆ 𝐴 ∧ ran (𝑅 ↾ 𝐴) ⊆ 𝐵))
19	11, 18	impbii 208	. 2 ⊢ ((dom (𝑅 ↾ 𝐴) ⊆ 𝐴 ∧ ran (𝑅 ↾ 𝐴) ⊆ 𝐵) ↔ (𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵))
20	2, 6, 19	3bitri 296	1 ⊢ ((𝑅 “ 𝐴) ⊆ 𝐵 ↔ (𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵))

Syntax hints:   ↔ wb 205   ∧ wa 396   ∩ cin 3943   ⊆ wss 3944   × cxp 5667  dom cdm 5669  ran crn 5670   ↾ cres 5671   “ cima 5672  Rel wrel 5674

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682

This theorem is referenced by:  gsumpart  32078  dfhe2  42296