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Theorem imasless 17554

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 2735	. . 3 ⊢ (le‘𝑅) = (le‘𝑅)
6		imasless.l	. . 3 ⊢ ≤ = (le‘𝑈)
7	1, 2, 3, 4, 5, 6	imasle 17537	. 2 ⊢ (𝜑 → ≤ = ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹))
8		relco 6095	. . . 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 6790	. . . . . . . . 9 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹:𝑉⟶𝐵)
13		frel 6711	. . . . . . . . 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 218	. . . . . . 7 ⊢ (𝜑 → ^◡^◡𝐹 = 𝐹)
17	16	imaeq1d 6046	. . . . . 6 ⊢ (𝜑 → (^◡^◡𝐹 “ dom (𝐹 ∘ (le‘𝑅))) = (𝐹 “ dom (𝐹 ∘ (le‘𝑅))))
18		imassrn 6058	. . . . . . 7 ⊢ (𝐹 “ dom (𝐹 ∘ (le‘𝑅))) ⊆ ran 𝐹
19		forn 6793	. . . . . . . 8 ⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵)
20	3, 19	syl 17	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 = 𝐵)
21	18, 20	sseqtrid 4001	. . . . . 6 ⊢ (𝜑 → (𝐹 “ dom (𝐹 ∘ (le‘𝑅))) ⊆ 𝐵)
22	17, 21	eqsstrd 3993	. . . . 5 ⊢ (𝜑 → (^◡^◡𝐹 “ dom (𝐹 ∘ (le‘𝑅))) ⊆ 𝐵)
23	11, 22	eqsstrid 3997	. . . 4 ⊢ (𝜑 → dom ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹) ⊆ 𝐵)
24		rncoss 5955	. . . . 5 ⊢ ran ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹) ⊆ ran (𝐹 ∘ (le‘𝑅))
25		rnco2 6242	. . . . . 6 ⊢ ran (𝐹 ∘ (le‘𝑅)) = (𝐹 “ ran (le‘𝑅))
26		imassrn 6058	. . . . . . 7 ⊢ (𝐹 “ ran (le‘𝑅)) ⊆ ran 𝐹
27	26, 20	sseqtrid 4001	. . . . . 6 ⊢ (𝜑 → (𝐹 “ ran (le‘𝑅)) ⊆ 𝐵)
28	25, 27	eqsstrid 3997	. . . . 5 ⊢ (𝜑 → ran (𝐹 ∘ (le‘𝑅)) ⊆ 𝐵)
29	24, 28	sstrid 3970	. . . 4 ⊢ (𝜑 → ran ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹) ⊆ 𝐵)
30		xpss12 5669	. . . 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 584	. . 3 ⊢ (𝜑 → (dom ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹) × ran ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹)) ⊆ (𝐵 × 𝐵))
32	10, 31	sstrid 3970	. 2 ⊢ (𝜑 → ((𝐹 ∘ (le‘𝑅)) ∘ ^◡𝐹) ⊆ (𝐵 × 𝐵))
33	7, 32	eqsstrd 3993	1 ⊢ (𝜑 → ≤ ⊆ (𝐵 × 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ⊆ wss 3926 × cxp 5652 ^◡ccnv 5653 dom cdm 5654 ran crn 5655 “ cima 5657 ∘ ccom 5658 Rel wrel 5659 ⟶wf 6527 –onto→wfo 6529 ‘cfv 6531 (class class class)co 7405 Basecbs 17228 lecple 17278 “_s cimas 17518

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8719 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-sup 9454 df-inf 9455 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12502 df-z 12589 df-dec 12709 df-uz 12853 df-fz 13525 df-struct 17166 df-slot 17201 df-ndx 17213 df-base 17229 df-plusg 17284 df-mulr 17285 df-sca 17287 df-vsca 17288 df-ip 17289 df-tset 17290 df-ple 17291 df-ds 17293 df-imas 17522

This theorem is referenced by: xpsless 17592

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