pmtrcnel2 - Mathbox for Thierry Arnoux

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Theorem pmtrcnel2 33054

Description: Variation on pmtrcnel 33053. (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 19382	. . . . 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 6241	. . . . . . . 8 ⊢ ((^◡(𝑇‘{𝐼, 𝐽}) ∘ (𝑇‘{𝐼, 𝐽})) ∘ 𝐹) = (^◡(𝑇‘{𝐼, 𝐽}) ∘ ((𝑇‘{𝐼, 𝐽}) ∘ 𝐹))
4		pmtrcnel.d	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
5		difss 4102	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∖ I ) ⊆ 𝐹
6		dmss 5869	. . . . . . . . . . . . . . 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 3947	. . . . . . . . . . . . 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 19312	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ 𝐵 → 𝐹:𝐷–1-1-onto→𝐷)
14		f1of 6803	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝐷–1-1-onto→𝐷 → 𝐹:𝐷⟶𝐷)
15	10, 13, 14	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:𝐷⟶𝐷)
16	15	fdmd 6701	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom 𝐹 = 𝐷)
17	9, 16	eleqtrd 2831	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 ∈ 𝐷)
18		pmtrcnel.j	. . . . . . . . . . . . 13 ⊢ 𝐽 = (𝐹‘𝐼)
19	15, 17	ffvelcdmd 7060	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹‘𝐼) ∈ 𝐷)
20	18, 19	eqeltrid 2833	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ 𝐷)
21	17, 20	prssd 4789	. . . . . . . . . . 11 ⊢ (𝜑 → {𝐼, 𝐽} ⊆ 𝐷)
22	15	ffnd 6692	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 Fn 𝐷)
23		fnelnfp 7154	. . . . . . . . . . . . . . . 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 837	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹‘𝐼) ≠ 𝐼)
26	25	necomd 2981	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐼 ≠ (𝐹‘𝐼))
27	18	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 = (𝐹‘𝐼))
28	26, 27	neeqtrrd 3000	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 ≠ 𝐽)
29		enpr2 9962	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝐷 ∧ 𝐽 ∈ 𝐷 ∧ 𝐼 ≠ 𝐽) → {𝐼, 𝐽} ≈ 2_o)
30	17, 20, 28, 29	syl3anc 1373	. . . . . . . . . . 11 ⊢ (𝜑 → {𝐼, 𝐽} ≈ 2_o)
31		pmtrcnel.t	. . . . . . . . . . . 12 ⊢ 𝑇 = (pmTrsp‘𝐷)
32		eqid 2730	. . . . . . . . . . . 12 ⊢ ran 𝑇 = ran 𝑇
33	31, 32	pmtrrn 19394	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ 𝑉 ∧ {𝐼, 𝐽} ⊆ 𝐷 ∧ {𝐼, 𝐽} ≈ 2_o) → (𝑇‘{𝐼, 𝐽}) ∈ ran 𝑇)
34	4, 21, 30, 33	syl3anc 1373	. . . . . . . . . 10 ⊢ (𝜑 → (𝑇‘{𝐼, 𝐽}) ∈ ran 𝑇)
35	31, 32	pmtrff1o 19400	. . . . . . . . . 10 ⊢ ((𝑇‘{𝐼, 𝐽}) ∈ ran 𝑇 → (𝑇‘{𝐼, 𝐽}):𝐷–1-1-onto→𝐷)
36		f1ococnv1 6832	. . . . . . . . . 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 5828	. . . . . . . 8 ⊢ (𝜑 → ((^◡(𝑇‘{𝐼, 𝐽}) ∘ (𝑇‘{𝐼, 𝐽})) ∘ 𝐹) = (( I ↾ 𝐷) ∘ 𝐹))
39	3, 38	eqtr3id 2779	. . . . . . 7 ⊢ (𝜑 → (^◡(𝑇‘{𝐼, 𝐽}) ∘ ((𝑇‘{𝐼, 𝐽}) ∘ 𝐹)) = (( I ↾ 𝐷) ∘ 𝐹))
40		fcoi2 6738	. . . . . . . 8 ⊢ (𝐹:𝐷⟶𝐷 → (( I ↾ 𝐷) ∘ 𝐹) = 𝐹)
41	15, 40	syl 17	. . . . . . 7 ⊢ (𝜑 → (( I ↾ 𝐷) ∘ 𝐹) = 𝐹)
42	39, 41	eqtrd 2765	. . . . . 6 ⊢ (𝜑 → (^◡(𝑇‘{𝐼, 𝐽}) ∘ ((𝑇‘{𝐼, 𝐽}) ∘ 𝐹)) = 𝐹)
43	42	difeq1d 4091	. . . . 5 ⊢ (𝜑 → ((^◡(𝑇‘{𝐼, 𝐽}) ∘ ((𝑇‘{𝐼, 𝐽}) ∘ 𝐹)) ∖ I ) = (𝐹 ∖ I ))
44	43	dmeqd 5872	. . . 4 ⊢ (𝜑 → dom ((^◡(𝑇‘{𝐼, 𝐽}) ∘ ((𝑇‘{𝐼, 𝐽}) ∘ 𝐹)) ∖ I ) = dom (𝐹 ∖ I ))
45	31, 32	pmtrfcnv 19401	. . . . . . . . . 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 4091	. . . . . . . 8 ⊢ (𝜑 → (^◡(𝑇‘{𝐼, 𝐽}) ∖ I ) = ((𝑇‘{𝐼, 𝐽}) ∖ I ))
48	47	dmeqd 5872	. . . . . . 7 ⊢ (𝜑 → dom (^◡(𝑇‘{𝐼, 𝐽}) ∖ I ) = dom ((𝑇‘{𝐼, 𝐽}) ∖ I ))
49	31	pmtrmvd 19393	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝑉 ∧ {𝐼, 𝐽} ⊆ 𝐷 ∧ {𝐼, 𝐽} ≈ 2_o) → dom ((𝑇‘{𝐼, 𝐽}) ∖ I ) = {𝐼, 𝐽})
50	4, 21, 30, 49	syl3anc 1373	. . . . . . 7 ⊢ (𝜑 → dom ((𝑇‘{𝐼, 𝐽}) ∖ I ) = {𝐼, 𝐽})
51	48, 50	eqtrd 2765	. . . . . 6 ⊢ (𝜑 → dom (^◡(𝑇‘{𝐼, 𝐽}) ∖ I ) = {𝐼, 𝐽})
52	51	uneq1d 4133	. . . . 5 ⊢ (𝜑 → (dom (^◡(𝑇‘{𝐼, 𝐽}) ∖ I ) ∪ dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I )) = ({𝐼, 𝐽} ∪ dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I )))
53		uncom 4124	. . . . 5 ⊢ ({𝐼, 𝐽} ∪ dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I )) = (dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ∪ {𝐼, 𝐽})
54	52, 53	eqtrdi 2781	. . . 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 4004	. . 3 ⊢ (𝜑 → dom (𝐹 ∖ I ) ⊆ (dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ∪ {𝐼, 𝐽}))
56	55	ssdifd 4111	. 2 ⊢ (𝜑 → (dom (𝐹 ∖ I ) ∖ {𝐼, 𝐽}) ⊆ ((dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ∪ {𝐼, 𝐽}) ∖ {𝐼, 𝐽}))
57		difun2 4447	. . 3 ⊢ ((dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ∪ {𝐼, 𝐽}) ∖ {𝐼, 𝐽}) = (dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ∖ {𝐼, 𝐽})
58		difss 4102	. . 3 ⊢ (dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ∖ {𝐼, 𝐽}) ⊆ dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I )
59	57, 58	eqsstri 3996	. 2 ⊢ ((dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ∪ {𝐼, 𝐽}) ∖ {𝐼, 𝐽}) ⊆ dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I )
60	56, 59	sstrdi 3962	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 1540 ∈ wcel 2109 ≠ wne 2926 ∖ cdif 3914 ∪ cun 3915 ⊆ wss 3917 {cpr 4594 class class class wbr 5110 I cid 5535 ^◡ccnv 5640 dom cdm 5641 ran crn 5642 ↾ cres 5643 ∘ ccom 5645 Fn wfn 6509 ⟶wf 6510 –1-1-onto→wf1o 6513 ‘cfv 6514 2_oc2o 8431 ≈ cen 8918 Basecbs 17186 SymGrpcsymg 19306 pmTrspcpmtr 19378

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

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 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-map 8804 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-uz 12801 df-fz 13476 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-tset 17246 df-efmnd 18803 df-symg 19307 df-pmtr 19379

This theorem is referenced by: pmtrcnelor 33055

W3C validator