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Theorem imasless 17348

Description: The order relation defined on an image set is a subset of the base set. (Contributed by Mario Carneiro, 24-Feb-2015.)

**Hypotheses**
Ref	Expression
imasless.u	⊢ (𝜑 → 𝑈 = (𝐹 “_s 𝑅))
imasless.v	⊢ (𝜑 → 𝑉 = (Base‘𝑅))
imasless.f	⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
imasless.r	⊢ (𝜑 → 𝑅 ∈ 𝑍)
imasless.l	⊢ ≤ = (le‘𝑈)

**Assertion**
Ref	Expression
imasless	⊢ (𝜑 → ≤ ⊆ (𝐵 × 𝐵))

**Proof of Theorem imasless**
Step	Hyp	Ref	Expression
1		imasless.u	. . 3 ⊢ (𝜑 → 𝑈 = (𝐹 “_s 𝑅))
2		imasless.v	. . 3 ⊢ (𝜑 → 𝑉 = (Base‘𝑅))
3		imasless.f	. . 3 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
4		imasless.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑍)
5		eqid 2737	. . 3 ⊢ (le‘𝑅) = (le‘𝑅)
6		imasless.l	. . 3 ⊢ ≤ = (le‘𝑈)
7	1, 2, 3, 4, 5, 6	imasle 17331	. 2 ⊢ (𝜑 → ≤ = ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹))
8		relco 6050	. . . 4 ⊢ Rel ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹)
9		relssdmrn 6210	. . . 4 ⊢ (Rel ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹) → ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹) ⊆ (dom ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹) × ran ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹)))
10	8, 9	ax-mp 5	. . 3 ⊢ ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹) ⊆ (dom ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹) × ran ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹))
11		dmco 6196	. . . . 5 ⊢ dom ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹) = (^◡^◡𝐹 “ dom (𝐹 ∘ (le‘𝑅)))
12		fof 6743	. . . . . . . . 9 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹:𝑉⟶𝐵)
13		frel 6660	. . . . . . . . 9 ⊢ (𝐹:𝑉⟶𝐵 → Rel 𝐹)
14	3, 12, 13	3syl 18	. . . . . . . 8 ⊢ (𝜑 → Rel 𝐹)
15		dfrel2 6131	. . . . . . . 8 ⊢ (Rel 𝐹 ↔ ^◡^◡𝐹 = 𝐹)
16	14, 15	sylib 217	. . . . . . 7 ⊢ (𝜑 → ^◡^◡𝐹 = 𝐹)
17	16	imaeq1d 6002	. . . . . 6 ⊢ (𝜑 → (^◡^◡𝐹 “ dom (𝐹 ∘ (le‘𝑅))) = (𝐹 “ dom (𝐹 ∘ (le‘𝑅))))
18		imassrn 6014	. . . . . . 7 ⊢ (𝐹 “ dom (𝐹 ∘ (le‘𝑅))) ⊆ ran 𝐹
19		forn 6746	. . . . . . . 8 ⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵)
20	3, 19	syl 17	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 = 𝐵)
21	18, 20	sseqtrid 3987	. . . . . 6 ⊢ (𝜑 → (𝐹 “ dom (𝐹 ∘ (le‘𝑅))) ⊆ 𝐵)
22	17, 21	eqsstrd 3973	. . . . 5 ⊢ (𝜑 → (^◡^◡𝐹 “ dom (𝐹 ∘ (le‘𝑅))) ⊆ 𝐵)
23	11, 22	eqsstrid 3983	. . . 4 ⊢ (𝜑 → dom ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹) ⊆ 𝐵)
24		rncoss 5917	. . . . 5 ⊢ ran ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹) ⊆ ran (𝐹 ∘ (le‘𝑅))
25		rnco2 6195	. . . . . 6 ⊢ ran (𝐹 ∘ (le‘𝑅)) = (𝐹 “ ran (le‘𝑅))
26		imassrn 6014	. . . . . . 7 ⊢ (𝐹 “ ran (le‘𝑅)) ⊆ ran 𝐹
27	26, 20	sseqtrid 3987	. . . . . 6 ⊢ (𝜑 → (𝐹 “ ran (le‘𝑅)) ⊆ 𝐵)
28	25, 27	eqsstrid 3983	. . . . 5 ⊢ (𝜑 → ran (𝐹 ∘ (le‘𝑅)) ⊆ 𝐵)
29	24, 28	sstrid 3946	. . . 4 ⊢ (𝜑 → ran ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹) ⊆ 𝐵)
30		xpss12 5639	. . . 4 ⊢ ((dom ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹) ⊆ 𝐵 ∧ ran ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹) ⊆ 𝐵) → (dom ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹) × ran ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹)) ⊆ (𝐵 × 𝐵))
31	23, 29, 30	syl2anc 585	. . 3 ⊢ (𝜑 → (dom ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹) × ran ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹)) ⊆ (𝐵 × 𝐵))
32	10, 31	sstrid 3946	. 2 ⊢ (𝜑 → ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹) ⊆ (𝐵 × 𝐵))
33	7, 32	eqsstrd 3973	1 ⊢ (𝜑 → ≤ ⊆ (𝐵 × 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ⊆ wss 3901 × cxp 5622 ^◡ccnv 5623 dom cdm 5624 ran crn 5625 “ cima 5627 ∘ ccom 5628 Rel wrel 5629 ⟶wf 6479 –onto→wfo 6481 ‘cfv 6483 (class class class)co 7341 Basecbs 17009 lecple 17066 “_s cimas 17312

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 2708 ax-rep 5233 ax-sep 5247 ax-nul 5254 ax-pow 5312 ax-pr 5376 ax-un 7654 ax-cnex 11032 ax-resscn 11033 ax-1cn 11034 ax-icn 11035 ax-addcl 11036 ax-addrcl 11037 ax-mulcl 11038 ax-mulrcl 11039 ax-mulcom 11040 ax-addass 11041 ax-mulass 11042 ax-distr 11043 ax-i2m1 11044 ax-1ne0 11045 ax-1rid 11046 ax-rnegex 11047 ax-rrecex 11048 ax-cnre 11049 ax-pre-lttri 11050 ax-pre-lttrn 11051 ax-pre-ltadd 11052 ax-pre-mulgt0 11053

This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3351 df-rab 3405 df-v 3444 df-sbc 3731 df-csb 3847 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3920 df-nul 4274 df-if 4478 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4857 df-iun 4947 df-br 5097 df-opab 5159 df-mpt 5180 df-tr 5214 df-id 5522 df-eprel 5528 df-po 5536 df-so 5537 df-fr 5579 df-we 5581 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6242 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6435 df-fun 6485 df-fn 6486 df-f 6487 df-f1 6488 df-fo 6489 df-f1o 6490 df-fv 6491 df-riota 7297 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7785 df-1st 7903 df-2nd 7904 df-frecs 8171 df-wrecs 8202 df-recs 8276 df-rdg 8315 df-1o 8371 df-er 8573 df-en 8809 df-dom 8810 df-sdom 8811 df-fin 8812 df-sup 9303 df-inf 9304 df-pnf 11116 df-mnf 11117 df-xr 11118 df-ltxr 11119 df-le 11120 df-sub 11312 df-neg 11313 df-nn 12079 df-2 12141 df-3 12142 df-4 12143 df-5 12144 df-6 12145 df-7 12146 df-8 12147 df-9 12148 df-n0 12339 df-z 12425 df-dec 12543 df-uz 12688 df-fz 13345 df-struct 16945 df-slot 16980 df-ndx 16992 df-base 17010 df-plusg 17072 df-mulr 17073 df-sca 17075 df-vsca 17076 df-ip 17077 df-tset 17078 df-ple 17079 df-ds 17081 df-imas 17316

This theorem is referenced by: xpsless 17386

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