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Theorem imasless 17484

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 2724	. . 3 ⊢ (le‘𝑅) = (le‘𝑅)
6		imasless.l	. . 3 ⊢ ≤ = (le‘𝑈)
7	1, 2, 3, 4, 5, 6	imasle 17467	. 2 ⊢ (𝜑 → ≤ = ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹))
8		relco 6097	. . . 4 ⊢ Rel ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹)
9		relssdmrn 6257	. . . 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 6243	. . . . 5 ⊢ dom ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹) = (^◡^◡𝐹 “ dom (𝐹 ∘ (le‘𝑅)))
12		fof 6795	. . . . . . . . 9 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹:𝑉⟶𝐵)
13		frel 6712	. . . . . . . . 9 ⊢ (𝐹:𝑉⟶𝐵 → Rel 𝐹)
14	3, 12, 13	3syl 18	. . . . . . . 8 ⊢ (𝜑 → Rel 𝐹)
15		dfrel2 6178	. . . . . . . 8 ⊢ (Rel 𝐹 ↔ ^◡^◡𝐹 = 𝐹)
16	14, 15	sylib 217	. . . . . . 7 ⊢ (𝜑 → ^◡^◡𝐹 = 𝐹)
17	16	imaeq1d 6048	. . . . . 6 ⊢ (𝜑 → (^◡^◡𝐹 “ dom (𝐹 ∘ (le‘𝑅))) = (𝐹 “ dom (𝐹 ∘ (le‘𝑅))))
18		imassrn 6060	. . . . . . 7 ⊢ (𝐹 “ dom (𝐹 ∘ (le‘𝑅))) ⊆ ran 𝐹
19		forn 6798	. . . . . . . 8 ⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵)
20	3, 19	syl 17	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 = 𝐵)
21	18, 20	sseqtrid 4026	. . . . . 6 ⊢ (𝜑 → (𝐹 “ dom (𝐹 ∘ (le‘𝑅))) ⊆ 𝐵)
22	17, 21	eqsstrd 4012	. . . . 5 ⊢ (𝜑 → (^◡^◡𝐹 “ dom (𝐹 ∘ (le‘𝑅))) ⊆ 𝐵)
23	11, 22	eqsstrid 4022	. . . 4 ⊢ (𝜑 → dom ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹) ⊆ 𝐵)
24		rncoss 5961	. . . . 5 ⊢ ran ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹) ⊆ ran (𝐹 ∘ (le‘𝑅))
25		rnco2 6242	. . . . . 6 ⊢ ran (𝐹 ∘ (le‘𝑅)) = (𝐹 “ ran (le‘𝑅))
26		imassrn 6060	. . . . . . 7 ⊢ (𝐹 “ ran (le‘𝑅)) ⊆ ran 𝐹
27	26, 20	sseqtrid 4026	. . . . . 6 ⊢ (𝜑 → (𝐹 “ ran (le‘𝑅)) ⊆ 𝐵)
28	25, 27	eqsstrid 4022	. . . . 5 ⊢ (𝜑 → ran (𝐹 ∘ (le‘𝑅)) ⊆ 𝐵)
29	24, 28	sstrid 3985	. . . 4 ⊢ (𝜑 → ran ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹) ⊆ 𝐵)
30		xpss12 5681	. . . 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 583	. . 3 ⊢ (𝜑 → (dom ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹) × ran ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹)) ⊆ (𝐵 × 𝐵))
32	10, 31	sstrid 3985	. 2 ⊢ (𝜑 → ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹) ⊆ (𝐵 × 𝐵))
33	7, 32	eqsstrd 4012	1 ⊢ (𝜑 → ≤ ⊆ (𝐵 × 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ⊆ wss 3940 × cxp 5664 ^◡ccnv 5665 dom cdm 5666 ran crn 5667 “ cima 5669 ∘ ccom 5670 Rel wrel 5671 ⟶wf 6529 –onto→wfo 6531 ‘cfv 6533 (class class class)co 7401 Basecbs 17142 lecple 17202 “_s cimas 17448

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-sup 9432 df-inf 9433 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-struct 17078 df-slot 17113 df-ndx 17125 df-base 17143 df-plusg 17208 df-mulr 17209 df-sca 17211 df-vsca 17212 df-ip 17213 df-tset 17214 df-ple 17215 df-ds 17217 df-imas 17452

This theorem is referenced by: xpsless 17522

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