Theorem resssxp 6242

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 5649	. . 3 ⊢ (𝑅 “ 𝐴) = ran (𝑅 ↾ 𝐴)
2	1	sseq1i 3955	. 2 ⊢ ((𝑅 “ 𝐴) ⊆ 𝐵 ↔ ran (𝑅 ↾ 𝐴) ⊆ 𝐵)
3		dmres 5987	. . . 4 ⊢ dom (𝑅 ↾ 𝐴) = (𝐴 ∩ dom 𝑅)
4		inss1 4179	. . . 4 ⊢ (𝐴 ∩ dom 𝑅) ⊆ 𝐴
5	3, 4	eqsstri 3973	. . 3 ⊢ dom (𝑅 ↾ 𝐴) ⊆ 𝐴
6	5	biantrur 537	. 2 ⊢ (ran (𝑅 ↾ 𝐴) ⊆ 𝐵 ↔ (dom (𝑅 ↾ 𝐴) ⊆ 𝐴 ∧ ran (𝑅 ↾ 𝐴) ⊆ 𝐵))
7		relres 5980	. . . . 5 ⊢ Rel (𝑅 ↾ 𝐴)
8		relssdmrn 6241	. . . . 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 5651	. . . 4 ⊢ ((dom (𝑅 ↾ 𝐴) ⊆ 𝐴 ∧ ran (𝑅 ↾ 𝐴) ⊆ 𝐵) → (dom (𝑅 ↾ 𝐴) × ran (𝑅 ↾ 𝐴)) ⊆ (𝐴 × 𝐵))
11	9, 10	sstrid 3938	. . 3 ⊢ ((dom (𝑅 ↾ 𝐴) ⊆ 𝐴 ∧ ran (𝑅 ↾ 𝐴) ⊆ 𝐵) → (𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵))
12		dmss 5867	. . . . 5 ⊢ ((𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵) → dom (𝑅 ↾ 𝐴) ⊆ dom (𝐴 × 𝐵))
13		dmxpss 6142	. . . . 5 ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴
14	12, 13	sstrdi 3939	. . . 4 ⊢ ((𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵) → dom (𝑅 ↾ 𝐴) ⊆ 𝐴)
15		rnss 5904	. . . . 5 ⊢ ((𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵) → ran (𝑅 ↾ 𝐴) ⊆ ran (𝐴 × 𝐵))
16		rnxpss 6143	. . . . 5 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵
17	15, 16	sstrdi 3939	. . . 4 ⊢ ((𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵) → ran (𝑅 ↾ 𝐴) ⊆ 𝐵)
18	14, 17	jca 518	. . 3 ⊢ ((𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵) → (dom (𝑅 ↾ 𝐴) ⊆ 𝐴 ∧ ran (𝑅 ↾ 𝐴) ⊆ 𝐵))
19	11, 18	impbii 211	. 2 ⊢ ((dom (𝑅 ↾ 𝐴) ⊆ 𝐴 ∧ ran (𝑅 ↾ 𝐴) ⊆ 𝐵) ↔ (𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵))
20	2, 6, 19	3bitri 299	1 ⊢ ((𝑅 “ 𝐴) ⊆ 𝐵 ↔ (𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵))

Syntax hints:   ↔ wb 208   ∧ wa 398   ∩ cin 3894   ⊆ wss 3895   × cxp 5634  dom cdm 5636  ran crn 5637   ↾ cres 5638   “ cima 5639  Rel wrel 5641

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-11 2181  ax-ext 2724  ax-sep 5236  ax-pr 5380

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-sb 2081  df-clab 2731  df-cleq 2744  df-clel 2827  df-ne 2948  df-ral 3067  df-rex 3077  df-rab 3405  df-v 3446  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-sn 4573  df-pr 4575  df-op 4579  df-br 5091  df-opab 5153  df-xp 5642  df-rel 5643  df-cnv 5644  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649

This theorem is referenced by:  gsumpart  33193  dfhe2  44288