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Theorem imasless 16402

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 2760	. . 3 ⊢ (le‘𝑅) = (le‘𝑅)
6		imasless.l	. . 3 ⊢ ≤ = (le‘𝑈)
7	1, 2, 3, 4, 5, 6	imasle 16385	. 2 ⊢ (𝜑 → ≤ = ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹))
8		relco 5794	. . . 4 ⊢ Rel ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹)
9		relssdmrn 5817	. . . 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 5804	. . . . 5 ⊢ dom ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹) = (^◡^◡𝐹 “ dom (𝐹 ∘ (le‘𝑅)))
12		fof 6276	. . . . . . . . 9 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹:𝑉⟶𝐵)
13		frel 6211	. . . . . . . . 9 ⊢ (𝐹:𝑉⟶𝐵 → Rel 𝐹)
14	3, 12, 13	3syl 18	. . . . . . . 8 ⊢ (𝜑 → Rel 𝐹)
15		dfrel2 5741	. . . . . . . 8 ⊢ (Rel 𝐹 ↔ ^◡^◡𝐹 = 𝐹)
16	14, 15	sylib 208	. . . . . . 7 ⊢ (𝜑 → ^◡^◡𝐹 = 𝐹)
17	16	imaeq1d 5623	. . . . . 6 ⊢ (𝜑 → (^◡^◡𝐹 “ dom (𝐹 ∘ (le‘𝑅))) = (𝐹 “ dom (𝐹 ∘ (le‘𝑅))))
18		imassrn 5635	. . . . . . 7 ⊢ (𝐹 “ dom (𝐹 ∘ (le‘𝑅))) ⊆ ran 𝐹
19		forn 6279	. . . . . . . 8 ⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵)
20	3, 19	syl 17	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 = 𝐵)
21	18, 20	syl5sseq 3794	. . . . . 6 ⊢ (𝜑 → (𝐹 “ dom (𝐹 ∘ (le‘𝑅))) ⊆ 𝐵)
22	17, 21	eqsstrd 3780	. . . . 5 ⊢ (𝜑 → (^◡^◡𝐹 “ dom (𝐹 ∘ (le‘𝑅))) ⊆ 𝐵)
23	11, 22	syl5eqss 3790	. . . 4 ⊢ (𝜑 → dom ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹) ⊆ 𝐵)
24		rncoss 5541	. . . . 5 ⊢ ran ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹) ⊆ ran (𝐹 ∘ (le‘𝑅))
25		rnco2 5803	. . . . . 6 ⊢ ran (𝐹 ∘ (le‘𝑅)) = (𝐹 “ ran (le‘𝑅))
26		imassrn 5635	. . . . . . 7 ⊢ (𝐹 “ ran (le‘𝑅)) ⊆ ran 𝐹
27	26, 20	syl5sseq 3794	. . . . . 6 ⊢ (𝜑 → (𝐹 “ ran (le‘𝑅)) ⊆ 𝐵)
28	25, 27	syl5eqss 3790	. . . . 5 ⊢ (𝜑 → ran (𝐹 ∘ (le‘𝑅)) ⊆ 𝐵)
29	24, 28	syl5ss 3755	. . . 4 ⊢ (𝜑 → ran ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹) ⊆ 𝐵)
30		xpss12 5281	. . . 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 696	. . 3 ⊢ (𝜑 → (dom ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹) × ran ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹)) ⊆ (𝐵 × 𝐵))
32	10, 31	syl5ss 3755	. 2 ⊢ (𝜑 → ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹) ⊆ (𝐵 × 𝐵))
33	7, 32	eqsstrd 3780	1 ⊢ (𝜑 → ≤ ⊆ (𝐵 × 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1632 ∈ wcel 2139 ⊆ wss 3715 × cxp 5264 ^◡ccnv 5265 dom cdm 5266 ran crn 5267 “ cima 5269 ∘ ccom 5270 Rel wrel 5271 ⟶wf 6045 –onto→wfo 6047 ‘cfv 6049 (class class class)co 6813 Basecbs 16059 lecple 16150 “_s cimas 16366

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-1st 7333 df-2nd 7334 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-oadd 7733 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-sup 8513 df-inf 8514 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-nn 11213 df-2 11271 df-3 11272 df-4 11273 df-5 11274 df-6 11275 df-7 11276 df-8 11277 df-9 11278 df-n0 11485 df-z 11570 df-dec 11686 df-uz 11880 df-fz 12520 df-struct 16061 df-ndx 16062 df-slot 16063 df-base 16065 df-plusg 16156 df-mulr 16157 df-sca 16159 df-vsca 16160 df-ip 16161 df-tset 16162 df-ple 16163 df-ds 16166 df-imas 16370

This theorem is referenced by: xpsless 16442

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