pmtrcnel2 - Mathbox for Thierry Arnoux

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Theorem pmtrcnel2 33169

Description: Variation on pmtrcnel 33168. (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 19414	. . . . 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 6225	. . . . . . . 8 ⊢ ((^◡(𝑇‘{𝐼, 𝐽}) ∘ (𝑇‘{𝐼, 𝐽})) ∘ 𝐹) = (^◡(𝑇‘{𝐼, 𝐽}) ∘ ((𝑇‘{𝐼, 𝐽}) ∘ 𝐹))
4		pmtrcnel.d	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
5		difss 4077	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∖ I ) ⊆ 𝐹
6		dmss 5852	. . . . . . . . . . . . . . 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 3920	. . . . . . . . . . . . 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 19344	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ 𝐵 → 𝐹:𝐷–1-1-onto→𝐷)
14		f1of 6775	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝐷–1-1-onto→𝐷 → 𝐹:𝐷⟶𝐷)
15	10, 13, 14	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:𝐷⟶𝐷)
16	15	fdmd 6673	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom 𝐹 = 𝐷)
17	9, 16	eleqtrd 2839	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 ∈ 𝐷)
18		pmtrcnel.j	. . . . . . . . . . . . 13 ⊢ 𝐽 = (𝐹‘𝐼)
19	15, 17	ffvelcdmd 7032	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹‘𝐼) ∈ 𝐷)
20	18, 19	eqeltrid 2841	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ 𝐷)
21	17, 20	prssd 4766	. . . . . . . . . . 11 ⊢ (𝜑 → {𝐼, 𝐽} ⊆ 𝐷)
22	15	ffnd 6664	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 Fn 𝐷)
23		fnelnfp 7126	. . . . . . . . . . . . . . . 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 2988	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐼 ≠ (𝐹‘𝐼))
27	18	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 = (𝐹‘𝐼))
28	26, 27	neeqtrrd 3007	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 ≠ 𝐽)
29		enpr2 9920	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝐷 ∧ 𝐽 ∈ 𝐷 ∧ 𝐼 ≠ 𝐽) → {𝐼, 𝐽} ≈ 2_o)
30	17, 20, 28, 29	syl3anc 1374	. . . . . . . . . . 11 ⊢ (𝜑 → {𝐼, 𝐽} ≈ 2_o)
31		pmtrcnel.t	. . . . . . . . . . . 12 ⊢ 𝑇 = (pmTrsp‘𝐷)
32		eqid 2737	. . . . . . . . . . . 12 ⊢ ran 𝑇 = ran 𝑇
33	31, 32	pmtrrn 19426	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ 𝑉 ∧ {𝐼, 𝐽} ⊆ 𝐷 ∧ {𝐼, 𝐽} ≈ 2_o) → (𝑇‘{𝐼, 𝐽}) ∈ ran 𝑇)
34	4, 21, 30, 33	syl3anc 1374	. . . . . . . . . 10 ⊢ (𝜑 → (𝑇‘{𝐼, 𝐽}) ∈ ran 𝑇)
35	31, 32	pmtrff1o 19432	. . . . . . . . . 10 ⊢ ((𝑇‘{𝐼, 𝐽}) ∈ ran 𝑇 → (𝑇‘{𝐼, 𝐽}):𝐷–1-1-onto→𝐷)
36		f1ococnv1 6804	. . . . . . . . . 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 5811	. . . . . . . 8 ⊢ (𝜑 → ((^◡(𝑇‘{𝐼, 𝐽}) ∘ (𝑇‘{𝐼, 𝐽})) ∘ 𝐹) = (( I ↾ 𝐷) ∘ 𝐹))
39	3, 38	eqtr3id 2786	. . . . . . 7 ⊢ (𝜑 → (^◡(𝑇‘{𝐼, 𝐽}) ∘ ((𝑇‘{𝐼, 𝐽}) ∘ 𝐹)) = (( I ↾ 𝐷) ∘ 𝐹))
40		fcoi2 6710	. . . . . . . 8 ⊢ (𝐹:𝐷⟶𝐷 → (( I ↾ 𝐷) ∘ 𝐹) = 𝐹)
41	15, 40	syl 17	. . . . . . 7 ⊢ (𝜑 → (( I ↾ 𝐷) ∘ 𝐹) = 𝐹)
42	39, 41	eqtrd 2772	. . . . . 6 ⊢ (𝜑 → (^◡(𝑇‘{𝐼, 𝐽}) ∘ ((𝑇‘{𝐼, 𝐽}) ∘ 𝐹)) = 𝐹)
43	42	difeq1d 4066	. . . . 5 ⊢ (𝜑 → ((^◡(𝑇‘{𝐼, 𝐽}) ∘ ((𝑇‘{𝐼, 𝐽}) ∘ 𝐹)) ∖ I ) = (𝐹 ∖ I ))
44	43	dmeqd 5855	. . . 4 ⊢ (𝜑 → dom ((^◡(𝑇‘{𝐼, 𝐽}) ∘ ((𝑇‘{𝐼, 𝐽}) ∘ 𝐹)) ∖ I ) = dom (𝐹 ∖ I ))
45	31, 32	pmtrfcnv 19433	. . . . . . . . . 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 4066	. . . . . . . 8 ⊢ (𝜑 → (^◡(𝑇‘{𝐼, 𝐽}) ∖ I ) = ((𝑇‘{𝐼, 𝐽}) ∖ I ))
48	47	dmeqd 5855	. . . . . . 7 ⊢ (𝜑 → dom (^◡(𝑇‘{𝐼, 𝐽}) ∖ I ) = dom ((𝑇‘{𝐼, 𝐽}) ∖ I ))
49	31	pmtrmvd 19425	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝑉 ∧ {𝐼, 𝐽} ⊆ 𝐷 ∧ {𝐼, 𝐽} ≈ 2_o) → dom ((𝑇‘{𝐼, 𝐽}) ∖ I ) = {𝐼, 𝐽})
50	4, 21, 30, 49	syl3anc 1374	. . . . . . 7 ⊢ (𝜑 → dom ((𝑇‘{𝐼, 𝐽}) ∖ I ) = {𝐼, 𝐽})
51	48, 50	eqtrd 2772	. . . . . 6 ⊢ (𝜑 → dom (^◡(𝑇‘{𝐼, 𝐽}) ∖ I ) = {𝐼, 𝐽})
52	51	uneq1d 4108	. . . . 5 ⊢ (𝜑 → (dom (^◡(𝑇‘{𝐼, 𝐽}) ∖ I ) ∪ dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I )) = ({𝐼, 𝐽} ∪ dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I )))
53		uncom 4099	. . . . 5 ⊢ ({𝐼, 𝐽} ∪ dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I )) = (dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ∪ {𝐼, 𝐽})
54	52, 53	eqtrdi 2788	. . . 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 3977	. . 3 ⊢ (𝜑 → dom (𝐹 ∖ I ) ⊆ (dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ∪ {𝐼, 𝐽}))
56	55	ssdifd 4086	. 2 ⊢ (𝜑 → (dom (𝐹 ∖ I ) ∖ {𝐼, 𝐽}) ⊆ ((dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ∪ {𝐼, 𝐽}) ∖ {𝐼, 𝐽}))
57		difun2 4422	. . 3 ⊢ ((dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ∪ {𝐼, 𝐽}) ∖ {𝐼, 𝐽}) = (dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ∖ {𝐼, 𝐽})
58		difss 4077	. . 3 ⊢ (dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ∖ {𝐼, 𝐽}) ⊆ dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I )
59	57, 58	eqsstri 3969	. 2 ⊢ ((dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ∪ {𝐼, 𝐽}) ∖ {𝐼, 𝐽}) ⊆ dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I )
60	56, 59	sstrdi 3935	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 1542 ∈ wcel 2114 ≠ wne 2933 ∖ cdif 3887 ∪ cun 3888 ⊆ wss 3890 {cpr 4570 class class class wbr 5086 I cid 5519 ^◡ccnv 5624 dom cdm 5625 ran crn 5626 ↾ cres 5627 ∘ ccom 5629 Fn wfn 6488 ⟶wf 6489 –1-1-onto→wf1o 6492 ‘cfv 6493 2_oc2o 8393 ≈ cen 8884 Basecbs 17173 SymGrpcsymg 19338 pmTrspcpmtr 19410

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7812 df-1st 7936 df-2nd 7937 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-nn 12169 df-2 12238 df-3 12239 df-4 12240 df-5 12241 df-6 12242 df-7 12243 df-8 12244 df-9 12245 df-n0 12432 df-z 12519 df-uz 12783 df-fz 13456 df-struct 17111 df-sets 17128 df-slot 17146 df-ndx 17158 df-base 17174 df-ress 17195 df-plusg 17227 df-tset 17233 df-efmnd 18831 df-symg 19339 df-pmtr 19411

This theorem is referenced by: pmtrcnelor 33170

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