pmtrcnel2 - Mathbox for Thierry Arnoux

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Theorem pmtrcnel2 33093

Description: Variation on pmtrcnel 33092. (Contributed by Thierry Arnoux, 16-Nov-2023.)

**Hypotheses**
Ref	Expression
pmtrcnel.s	⊢ 𝑆 = (SymGrp‘𝐷)
pmtrcnel.t	⊢ 𝑇 = (pmTrsp‘𝐷)
pmtrcnel.b	⊢ 𝐵 = (Base‘𝑆)
pmtrcnel.j	⊢ 𝐽 = (𝐹‘𝐼)
pmtrcnel.d	⊢ (𝜑 → 𝐷 ∈ 𝑉)
pmtrcnel.f	⊢ (𝜑 → 𝐹 ∈ 𝐵)
pmtrcnel.i	⊢ (𝜑 → 𝐼 ∈ dom (𝐹 ∖ I ))

**Assertion**
Ref	Expression
pmtrcnel2	⊢ (𝜑 → (dom (𝐹 ∖ I ) ∖ {𝐼, 𝐽}) ⊆ dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ))

**Proof of Theorem pmtrcnel2**
Step	Hyp	Ref	Expression
1		mvdco 19478	. . . . 5 ⊢ dom ((^◡(𝑇‘{𝐼, 𝐽}) ∘ ((𝑇‘{𝐼, 𝐽}) ∘ 𝐹)) ∖ I ) ⊆ (dom (^◡(𝑇‘{𝐼, 𝐽}) ∖ I ) ∪ dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ))
2	1	a1i 11	. . . 4 ⊢ (𝜑 → dom ((^◡(𝑇‘{𝐼, 𝐽}) ∘ ((𝑇‘{𝐼, 𝐽}) ∘ 𝐹)) ∖ I ) ⊆ (dom (^◡(𝑇‘{𝐼, 𝐽}) ∖ I ) ∪ dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I )))
3		coass 6287	. . . . . . . 8 ⊢ ((^◡(𝑇‘{𝐼, 𝐽}) ∘ (𝑇‘{𝐼, 𝐽})) ∘ 𝐹) = (^◡(𝑇‘{𝐼, 𝐽}) ∘ ((𝑇‘{𝐼, 𝐽}) ∘ 𝐹))
4		pmtrcnel.d	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
5		difss 4146	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∖ I ) ⊆ 𝐹
6		dmss 5916	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∖ I ) ⊆ 𝐹 → dom (𝐹 ∖ I ) ⊆ dom 𝐹)
7	5, 6	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ dom (𝐹 ∖ I ) ⊆ dom 𝐹
8		pmtrcnel.i	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐼 ∈ dom (𝐹 ∖ I ))
9	7, 8	sselid 3993	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐼 ∈ dom 𝐹)
10		pmtrcnel.f	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
11		pmtrcnel.s	. . . . . . . . . . . . . . . 16 ⊢ 𝑆 = (SymGrp‘𝐷)
12		pmtrcnel.b	. . . . . . . . . . . . . . . 16 ⊢ 𝐵 = (Base‘𝑆)
13	11, 12	symgbasf1o 19407	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ 𝐵 → 𝐹:𝐷–1-1-onto→𝐷)
14		f1of 6849	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝐷–1-1-onto→𝐷 → 𝐹:𝐷⟶𝐷)
15	10, 13, 14	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:𝐷⟶𝐷)
16	15	fdmd 6747	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom 𝐹 = 𝐷)
17	9, 16	eleqtrd 2841	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 ∈ 𝐷)
18		pmtrcnel.j	. . . . . . . . . . . . 13 ⊢ 𝐽 = (𝐹‘𝐼)
19	15, 17	ffvelcdmd 7105	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹‘𝐼) ∈ 𝐷)
20	18, 19	eqeltrid 2843	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ 𝐷)
21	17, 20	prssd 4827	. . . . . . . . . . 11 ⊢ (𝜑 → {𝐼, 𝐽} ⊆ 𝐷)
22	15	ffnd 6738	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 Fn 𝐷)
23		fnelnfp 7197	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 Fn 𝐷 ∧ 𝐼 ∈ 𝐷) → (𝐼 ∈ dom (𝐹 ∖ I ) ↔ (𝐹‘𝐼) ≠ 𝐼))
24	23	biimpa 476	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 Fn 𝐷 ∧ 𝐼 ∈ 𝐷) ∧ 𝐼 ∈ dom (𝐹 ∖ I )) → (𝐹‘𝐼) ≠ 𝐼)
25	22, 17, 8, 24	syl21anc 838	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹‘𝐼) ≠ 𝐼)
26	25	necomd 2994	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐼 ≠ (𝐹‘𝐼))
27	18	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 = (𝐹‘𝐼))
28	26, 27	neeqtrrd 3013	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 ≠ 𝐽)
29		enpr2 10040	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝐷 ∧ 𝐽 ∈ 𝐷 ∧ 𝐼 ≠ 𝐽) → {𝐼, 𝐽} ≈ 2_o)
30	17, 20, 28, 29	syl3anc 1370	. . . . . . . . . . 11 ⊢ (𝜑 → {𝐼, 𝐽} ≈ 2_o)
31		pmtrcnel.t	. . . . . . . . . . . 12 ⊢ 𝑇 = (pmTrsp‘𝐷)
32		eqid 2735	. . . . . . . . . . . 12 ⊢ ran 𝑇 = ran 𝑇
33	31, 32	pmtrrn 19490	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ 𝑉 ∧ {𝐼, 𝐽} ⊆ 𝐷 ∧ {𝐼, 𝐽} ≈ 2_o) → (𝑇‘{𝐼, 𝐽}) ∈ ran 𝑇)
34	4, 21, 30, 33	syl3anc 1370	. . . . . . . . . 10 ⊢ (𝜑 → (𝑇‘{𝐼, 𝐽}) ∈ ran 𝑇)
35	31, 32	pmtrff1o 19496	. . . . . . . . . 10 ⊢ ((𝑇‘{𝐼, 𝐽}) ∈ ran 𝑇 → (𝑇‘{𝐼, 𝐽}):𝐷–1-1-onto→𝐷)
36		f1ococnv1 6878	. . . . . . . . . 10 ⊢ ((𝑇‘{𝐼, 𝐽}):𝐷–1-1-onto→𝐷 → (^◡(𝑇‘{𝐼, 𝐽}) ∘ (𝑇‘{𝐼, 𝐽})) = ( I ↾ 𝐷))
37	34, 35, 36	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → (^◡(𝑇‘{𝐼, 𝐽}) ∘ (𝑇‘{𝐼, 𝐽})) = ( I ↾ 𝐷))
38	37	coeq1d 5875	. . . . . . . 8 ⊢ (𝜑 → ((^◡(𝑇‘{𝐼, 𝐽}) ∘ (𝑇‘{𝐼, 𝐽})) ∘ 𝐹) = (( I ↾ 𝐷) ∘ 𝐹))
39	3, 38	eqtr3id 2789	. . . . . . 7 ⊢ (𝜑 → (^◡(𝑇‘{𝐼, 𝐽}) ∘ ((𝑇‘{𝐼, 𝐽}) ∘ 𝐹)) = (( I ↾ 𝐷) ∘ 𝐹))
40		fcoi2 6784	. . . . . . . 8 ⊢ (𝐹:𝐷⟶𝐷 → (( I ↾ 𝐷) ∘ 𝐹) = 𝐹)
41	15, 40	syl 17	. . . . . . 7 ⊢ (𝜑 → (( I ↾ 𝐷) ∘ 𝐹) = 𝐹)
42	39, 41	eqtrd 2775	. . . . . 6 ⊢ (𝜑 → (^◡(𝑇‘{𝐼, 𝐽}) ∘ ((𝑇‘{𝐼, 𝐽}) ∘ 𝐹)) = 𝐹)
43	42	difeq1d 4135	. . . . 5 ⊢ (𝜑 → ((^◡(𝑇‘{𝐼, 𝐽}) ∘ ((𝑇‘{𝐼, 𝐽}) ∘ 𝐹)) ∖ I ) = (𝐹 ∖ I ))
44	43	dmeqd 5919	. . . 4 ⊢ (𝜑 → dom ((^◡(𝑇‘{𝐼, 𝐽}) ∘ ((𝑇‘{𝐼, 𝐽}) ∘ 𝐹)) ∖ I ) = dom (𝐹 ∖ I ))
45	31, 32	pmtrfcnv 19497	. . . . . . . . . 10 ⊢ ((𝑇‘{𝐼, 𝐽}) ∈ ran 𝑇 → ^◡(𝑇‘{𝐼, 𝐽}) = (𝑇‘{𝐼, 𝐽}))
46	34, 45	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ^◡(𝑇‘{𝐼, 𝐽}) = (𝑇‘{𝐼, 𝐽}))
47	46	difeq1d 4135	. . . . . . . 8 ⊢ (𝜑 → (^◡(𝑇‘{𝐼, 𝐽}) ∖ I ) = ((𝑇‘{𝐼, 𝐽}) ∖ I ))
48	47	dmeqd 5919	. . . . . . 7 ⊢ (𝜑 → dom (^◡(𝑇‘{𝐼, 𝐽}) ∖ I ) = dom ((𝑇‘{𝐼, 𝐽}) ∖ I ))
49	31	pmtrmvd 19489	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝑉 ∧ {𝐼, 𝐽} ⊆ 𝐷 ∧ {𝐼, 𝐽} ≈ 2_o) → dom ((𝑇‘{𝐼, 𝐽}) ∖ I ) = {𝐼, 𝐽})
50	4, 21, 30, 49	syl3anc 1370	. . . . . . 7 ⊢ (𝜑 → dom ((𝑇‘{𝐼, 𝐽}) ∖ I ) = {𝐼, 𝐽})
51	48, 50	eqtrd 2775	. . . . . 6 ⊢ (𝜑 → dom (^◡(𝑇‘{𝐼, 𝐽}) ∖ I ) = {𝐼, 𝐽})
52	51	uneq1d 4177	. . . . 5 ⊢ (𝜑 → (dom (^◡(𝑇‘{𝐼, 𝐽}) ∖ I ) ∪ dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I )) = ({𝐼, 𝐽} ∪ dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I )))
53		uncom 4168	. . . . 5 ⊢ ({𝐼, 𝐽} ∪ dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I )) = (dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ∪ {𝐼, 𝐽})
54	52, 53	eqtrdi 2791	. . . 4 ⊢ (𝜑 → (dom (^◡(𝑇‘{𝐼, 𝐽}) ∖ I ) ∪ dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I )) = (dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ∪ {𝐼, 𝐽}))
55	2, 44, 54	3sstr3d 4042	. . 3 ⊢ (𝜑 → dom (𝐹 ∖ I ) ⊆ (dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ∪ {𝐼, 𝐽}))
56	55	ssdifd 4155	. 2 ⊢ (𝜑 → (dom (𝐹 ∖ I ) ∖ {𝐼, 𝐽}) ⊆ ((dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ∪ {𝐼, 𝐽}) ∖ {𝐼, 𝐽}))
57		difun2 4487	. . 3 ⊢ ((dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ∪ {𝐼, 𝐽}) ∖ {𝐼, 𝐽}) = (dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ∖ {𝐼, 𝐽})
58		difss 4146	. . 3 ⊢ (dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ∖ {𝐼, 𝐽}) ⊆ dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I )
59	57, 58	eqsstri 4030	. 2 ⊢ ((dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ∪ {𝐼, 𝐽}) ∖ {𝐼, 𝐽}) ⊆ dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I )
60	56, 59	sstrdi 4008	1 ⊢ (𝜑 → (dom (𝐹 ∖ I ) ∖ {𝐼, 𝐽}) ⊆ dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∖ cdif 3960 ∪ cun 3961 ⊆ wss 3963 {cpr 4633 class class class wbr 5148 I cid 5582 ^◡ccnv 5688 dom cdm 5689 ran crn 5690 ↾ cres 5691 ∘ ccom 5693 Fn wfn 6558 ⟶wf 6559 –1-1-onto→wf1o 6562 ‘cfv 6563 2_oc2o 8499 ≈ cen 8981 Basecbs 17245 SymGrpcsymg 19401 pmTrspcpmtr 19474

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-tset 17317 df-efmnd 18895 df-symg 19402 df-pmtr 19475

This theorem is referenced by: pmtrcnelor 33094

W3C validator