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Theorem imasless 17483

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 2733	. . 3 ⊢ (le‘𝑅) = (le‘𝑅)
6		imasless.l	. . 3 ⊢ ≤ = (le‘𝑈)
7	1, 2, 3, 4, 5, 6	imasle 17466	. 2 ⊢ (𝜑 → ≤ = ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹))
8		relco 6105	. . . 4 ⊢ Rel ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹)
9		relssdmrn 6265	. . . 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 6251	. . . . 5 ⊢ dom ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹) = (^◡^◡𝐹 “ dom (𝐹 ∘ (le‘𝑅)))
12		fof 6803	. . . . . . . . 9 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹:𝑉⟶𝐵)
13		frel 6720	. . . . . . . . 9 ⊢ (𝐹:𝑉⟶𝐵 → Rel 𝐹)
14	3, 12, 13	3syl 18	. . . . . . . 8 ⊢ (𝜑 → Rel 𝐹)
15		dfrel2 6186	. . . . . . . 8 ⊢ (Rel 𝐹 ↔ ^◡^◡𝐹 = 𝐹)
16	14, 15	sylib 217	. . . . . . 7 ⊢ (𝜑 → ^◡^◡𝐹 = 𝐹)
17	16	imaeq1d 6057	. . . . . 6 ⊢ (𝜑 → (^◡^◡𝐹 “ dom (𝐹 ∘ (le‘𝑅))) = (𝐹 “ dom (𝐹 ∘ (le‘𝑅))))
18		imassrn 6069	. . . . . . 7 ⊢ (𝐹 “ dom (𝐹 ∘ (le‘𝑅))) ⊆ ran 𝐹
19		forn 6806	. . . . . . . 8 ⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵)
20	3, 19	syl 17	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 = 𝐵)
21	18, 20	sseqtrid 4034	. . . . . 6 ⊢ (𝜑 → (𝐹 “ dom (𝐹 ∘ (le‘𝑅))) ⊆ 𝐵)
22	17, 21	eqsstrd 4020	. . . . 5 ⊢ (𝜑 → (^◡^◡𝐹 “ dom (𝐹 ∘ (le‘𝑅))) ⊆ 𝐵)
23	11, 22	eqsstrid 4030	. . . 4 ⊢ (𝜑 → dom ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹) ⊆ 𝐵)
24		rncoss 5970	. . . . 5 ⊢ ran ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹) ⊆ ran (𝐹 ∘ (le‘𝑅))
25		rnco2 6250	. . . . . 6 ⊢ ran (𝐹 ∘ (le‘𝑅)) = (𝐹 “ ran (le‘𝑅))
26		imassrn 6069	. . . . . . 7 ⊢ (𝐹 “ ran (le‘𝑅)) ⊆ ran 𝐹
27	26, 20	sseqtrid 4034	. . . . . 6 ⊢ (𝜑 → (𝐹 “ ran (le‘𝑅)) ⊆ 𝐵)
28	25, 27	eqsstrid 4030	. . . . 5 ⊢ (𝜑 → ran (𝐹 ∘ (le‘𝑅)) ⊆ 𝐵)
29	24, 28	sstrid 3993	. . . 4 ⊢ (𝜑 → ran ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹) ⊆ 𝐵)
30		xpss12 5691	. . . 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 3993	. 2 ⊢ (𝜑 → ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹) ⊆ (𝐵 × 𝐵))
33	7, 32	eqsstrd 4020	1 ⊢ (𝜑 → ≤ ⊆ (𝐵 × 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ⊆ wss 3948 × cxp 5674 ^◡ccnv 5675 dom cdm 5676 ran crn 5677 “ cima 5679 ∘ ccom 5680 Rel wrel 5681 ⟶wf 6537 –onto→wfo 6539 ‘cfv 6541 (class class class)co 7406 Basecbs 17141 lecple 17201 “_s cimas 17447

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-inf 9435 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-fz 13482 df-struct 17077 df-slot 17112 df-ndx 17124 df-base 17142 df-plusg 17207 df-mulr 17208 df-sca 17210 df-vsca 17211 df-ip 17212 df-tset 17213 df-ple 17214 df-ds 17216 df-imas 17451

This theorem is referenced by: xpsless 17521

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