pmtrcnel2 - Mathbox for Thierry Arnoux

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Theorem pmtrcnel2 33750

Description: Variation on pmtrcnel 33749. (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 19710	. . . . 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 6290	. . . . . . . 8 ⊢ ((^◡(𝑇‘{𝐼, 𝐽}) ∘ (𝑇‘{𝐼, 𝐽})) ∘ 𝐹) = (^◡(𝑇‘{𝐼, 𝐽}) ∘ ((𝑇‘{𝐼, 𝐽}) ∘ 𝐹))
4		pmtrcnel.d	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
5		difss 4075	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∖ I ) ⊆ 𝐹
6		dmss 5896	. . . . . . . . . . . . . . 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 3911	. . . . . . . . . . . . 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 19640	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ 𝐵 → 𝐹:𝐷⟶1-1onto𝐷)
14		f1of 6854	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝐷⟶1-1onto𝐷 → 𝐹:𝐷⟶𝐷)
15	10, 13, 14	3syl 19	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:𝐷⟶𝐷)
16	15	fdmd 6750	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom 𝐹 = 𝐷)
17	9, 16	eleqtrd 2866	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 ∈ 𝐷)
18		pmtrcnel.j	. . . . . . . . . . . . 13 ⊢ 𝐽 = (𝐹‘𝐼)
19	15, 17	ffvelcdmd 7117	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹‘𝐼) ∈ 𝐷)
20	18, 19	eqeltrid 2868	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ 𝐷)
21	17, 20	prssd 4783	. . . . . . . . . . 11 ⊢ (𝜑 → {𝐼, 𝐽} ⊆ 𝐷)
22	15	ffnd 6740	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 Fn 𝐷)
23		fnelnfp 7213	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 Fn 𝐷 ∧ 𝐼 ∈ 𝐷) → (𝐼 ∈ dom (𝐹 ∖ I ) ↔ (𝐹‘𝐼) ≠ 𝐼))
24	23	biimpa 482	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 Fn 𝐷 ∧ 𝐼 ∈ 𝐷) ∧ 𝐼 ∈ dom (𝐹 ∖ I )) → (𝐹‘𝐼) ≠ 𝐼)
25	22, 17, 8, 24	syl21anc 851	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹‘𝐼) ≠ 𝐼)
26	25	necomd 3014	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐼 ≠ (𝐹‘𝐼))
27	18	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 = (𝐹‘𝐼))
28	26, 27	neeqtrrd 3033	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 ≠ 𝐽)
29		enpr2 10122	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝐷 ∧ 𝐽 ∈ 𝐷 ∧ 𝐼 ≠ 𝐽) → {𝐼, 𝐽} ≈ 2_o)
30	17, 20, 28, 29	syl3anc 1399	. . . . . . . . . . 11 ⊢ (𝜑 → {𝐼, 𝐽} ≈ 2_o)
31		pmtrcnel.t	. . . . . . . . . . . 12 ⊢ 𝑇 = (pmTrsp‘𝐷)
32		eqid 2763	. . . . . . . . . . . 12 ⊢ ran 𝑇 = ran 𝑇
33	31, 32	pmtrrn 19722	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ 𝑉 ∧ {𝐼, 𝐽} ⊆ 𝐷 ∧ {𝐼, 𝐽} ≈ 2_o) → (𝑇‘{𝐼, 𝐽}) ∈ ran 𝑇)
34	4, 21, 30, 33	syl3anc 1399	. . . . . . . . . 10 ⊢ (𝜑 → (𝑇‘{𝐼, 𝐽}) ∈ ran 𝑇)
35	31, 32	pmtrff1o 19728	. . . . . . . . . 10 ⊢ ((𝑇‘{𝐼, 𝐽}) ∈ ran 𝑇 → (𝑇‘{𝐼, 𝐽}):𝐷⟶1-1onto𝐷)
36		f1ococnv1 6885	. . . . . . . . . 10 ⊢ ((𝑇‘{𝐼, 𝐽}):𝐷⟶1-1onto𝐷 → (^◡(𝑇‘{𝐼, 𝐽}) ∘ (𝑇‘{𝐼, 𝐽})) = ( I ↾ 𝐷))
37	34, 35, 36	3syl 19	. . . . . . . . 9 ⊢ (𝜑 → (^◡(𝑇‘{𝐼, 𝐽}) ∘ (𝑇‘{𝐼, 𝐽})) = ( I ↾ 𝐷))
38	37	coeq1d 5850	. . . . . . . 8 ⊢ (𝜑 → ((^◡(𝑇‘{𝐼, 𝐽}) ∘ (𝑇‘{𝐼, 𝐽})) ∘ 𝐹) = (( I ↾ 𝐷) ∘ 𝐹))
39	3, 38	eqtr3id 2812	. . . . . . 7 ⊢ (𝜑 → (^◡(𝑇‘{𝐼, 𝐽}) ∘ ((𝑇‘{𝐼, 𝐽}) ∘ 𝐹)) = (( I ↾ 𝐷) ∘ 𝐹))
40		fcoi2 6787	. . . . . . . 8 ⊢ (𝐹:𝐷⟶𝐷 → (( I ↾ 𝐷) ∘ 𝐹) = 𝐹)
41	15, 40	syl 18	. . . . . . 7 ⊢ (𝜑 → (( I ↾ 𝐷) ∘ 𝐹) = 𝐹)
42	39, 41	eqtrd 2798	. . . . . 6 ⊢ (𝜑 → (^◡(𝑇‘{𝐼, 𝐽}) ∘ ((𝑇‘{𝐼, 𝐽}) ∘ 𝐹)) = 𝐹)
43	42	difeq1d 4065	. . . . 5 ⊢ (𝜑 → ((^◡(𝑇‘{𝐼, 𝐽}) ∘ ((𝑇‘{𝐼, 𝐽}) ∘ 𝐹)) ∖ I ) = (𝐹 ∖ I ))
44	43	dmeqd 5899	. . . 4 ⊢ (𝜑 → dom ((^◡(𝑇‘{𝐼, 𝐽}) ∘ ((𝑇‘{𝐼, 𝐽}) ∘ 𝐹)) ∖ I ) = dom (𝐹 ∖ I ))
45	31, 32	pmtrfcnv 19729	. . . . . . . . . 10 ⊢ ((𝑇‘{𝐼, 𝐽}) ∈ ran 𝑇 → ^◡(𝑇‘{𝐼, 𝐽}) = (𝑇‘{𝐼, 𝐽}))
46	34, 45	syl 18	. . . . . . . . 9 ⊢ (𝜑 → ^◡(𝑇‘{𝐼, 𝐽}) = (𝑇‘{𝐼, 𝐽}))
47	46	difeq1d 4065	. . . . . . . 8 ⊢ (𝜑 → (^◡(𝑇‘{𝐼, 𝐽}) ∖ I ) = ((𝑇‘{𝐼, 𝐽}) ∖ I ))
48	47	dmeqd 5899	. . . . . . 7 ⊢ (𝜑 → dom (^◡(𝑇‘{𝐼, 𝐽}) ∖ I ) = dom ((𝑇‘{𝐼, 𝐽}) ∖ I ))
49	31	pmtrmvd 19721	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝑉 ∧ {𝐼, 𝐽} ⊆ 𝐷 ∧ {𝐼, 𝐽} ≈ 2_o) → dom ((𝑇‘{𝐼, 𝐽}) ∖ I ) = {𝐼, 𝐽})
50	4, 21, 30, 49	syl3anc 1399	. . . . . . 7 ⊢ (𝜑 → dom ((𝑇‘{𝐼, 𝐽}) ∖ I ) = {𝐼, 𝐽})
51	48, 50	eqtrd 2798	. . . . . 6 ⊢ (𝜑 → dom (^◡(𝑇‘{𝐼, 𝐽}) ∖ I ) = {𝐼, 𝐽})
52	51	uneq1d 4169	. . . . 5 ⊢ (𝜑 → (dom (^◡(𝑇‘{𝐼, 𝐽}) ∖ I ) ∪ dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I )) = ({𝐼, 𝐽} ∪ dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I )))
53		uncom 4160	. . . . 5 ⊢ ({𝐼, 𝐽} ∪ dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I )) = (dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ∪ {𝐼, 𝐽})
54	52, 53	eqtrdi 2814	. . . 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 3967	. . 3 ⊢ (𝜑 → dom (𝐹 ∖ I ) ⊆ (dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ∪ {𝐼, 𝐽}))
56	55	ssdifd 4084	. 2 ⊢ (𝜑 → (dom (𝐹 ∖ I ) ∖ {𝐼, 𝐽}) ⊆ ((dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ∪ {𝐼, 𝐽}) ∖ {𝐼, 𝐽}))
57		difun2 4440	. . 3 ⊢ ((dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ∪ {𝐼, 𝐽}) ∖ {𝐼, 𝐽}) = (dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ∖ {𝐼, 𝐽})
58		difss 4075	. . 3 ⊢ (dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ∖ {𝐼, 𝐽}) ⊆ dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I )
59	57, 58	eqsstri 3959	. 2 ⊢ ((dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ∪ {𝐼, 𝐽}) ∖ {𝐼, 𝐽}) ⊆ dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I )
60	56, 59	sstrdi 3925	1 ⊢ (𝜑 → (dom (𝐹 ∖ I ) ∖ {𝐼, 𝐽}) ⊆ dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ⊆ wss 3900 ∖ cdif 4048 ∪ cun 4152 {cpr 4590 class class class wbr 5108 I cid 5552 ^◡ccnv 5657 dom cdm 5658 ran crn 5659 ↾ cres 5660 ∘ ccom 5662 Fn wfn 6561 ⟶wf 6562 ⟶1-1ontowf1o 6565 ‘cfv 6566 2_oc2o 8508 ≈ cen 9008 Basecbs 17431 SymGrpcsymg 19634 pmTrspcpmtr 19706

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-1st 8035 df-2nd 8036 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-er 8755 df-map 8887 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-z 12734 df-uz 13006 df-fz 13680 df-struct 17369 df-sets 17386 df-slot 17404 df-ndx 17416 df-base 17432 df-ress 17453 df-plusg 17486 df-tset 17492 df-efmnd 19115 df-symg 19635 df-pmtr 19707

This theorem is used by: pmtrcnelor 33751

W3C validator