pmtrcnel2 - Mathbox for Thierry Arnoux

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Theorem pmtrcnel2 33267

Description: Variation on pmtrcnel 33266. (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 19485	. . . . 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 6253	. . . . . . . 8 ⊢ ((^◡(𝑇‘{𝐼, 𝐽}) ∘ (𝑇‘{𝐼, 𝐽})) ∘ 𝐹) = (^◡(𝑇‘{𝐼, 𝐽}) ∘ ((𝑇‘{𝐼, 𝐽}) ∘ 𝐹))
4		pmtrcnel.d	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
5		difss 4089	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∖ I ) ⊆ 𝐹
6		dmss 5878	. . . . . . . . . . . . . . 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 3934	. . . . . . . . . . . . 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 19415	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ 𝐵 → 𝐹:𝐷–1-1-onto→𝐷)
14		f1of 6806	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝐷–1-1-onto→𝐷 → 𝐹:𝐷⟶𝐷)
15	10, 13, 14	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:𝐷⟶𝐷)
16	15	fdmd 6702	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom 𝐹 = 𝐷)
17	9, 16	eleqtrd 2864	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 ∈ 𝐷)
18		pmtrcnel.j	. . . . . . . . . . . . 13 ⊢ 𝐽 = (𝐹‘𝐼)
19	15, 17	ffvelcdmd 7066	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹‘𝐼) ∈ 𝐷)
20	18, 19	eqeltrid 2866	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ 𝐷)
21	17, 20	prssd 4780	. . . . . . . . . . 11 ⊢ (𝜑 → {𝐼, 𝐽} ⊆ 𝐷)
22	15	ffnd 6692	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 Fn 𝐷)
23		fnelnfp 7161	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 Fn 𝐷 ∧ 𝐼 ∈ 𝐷) → (𝐼 ∈ dom (𝐹 ∖ I ) ↔ (𝐹‘𝐼) ≠ 𝐼))
24	23	biimpa 480	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 Fn 𝐷 ∧ 𝐼 ∈ 𝐷) ∧ 𝐼 ∈ dom (𝐹 ∖ I )) → (𝐹‘𝐼) ≠ 𝐼)
25	22, 17, 8, 24	syl21anc 848	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹‘𝐼) ≠ 𝐼)
26	25	necomd 3012	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐼 ≠ (𝐹‘𝐼))
27	18	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 = (𝐹‘𝐼))
28	26, 27	neeqtrrd 3031	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 ≠ 𝐽)
29		enpr2 9960	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝐷 ∧ 𝐽 ∈ 𝐷 ∧ 𝐼 ≠ 𝐽) → {𝐼, 𝐽} ≈ 2_o)
30	17, 20, 28, 29	syl3anc 1390	. . . . . . . . . . 11 ⊢ (𝜑 → {𝐼, 𝐽} ≈ 2_o)
31		pmtrcnel.t	. . . . . . . . . . . 12 ⊢ 𝑇 = (pmTrsp‘𝐷)
32		eqid 2762	. . . . . . . . . . . 12 ⊢ ran 𝑇 = ran 𝑇
33	31, 32	pmtrrn 19497	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ 𝑉 ∧ {𝐼, 𝐽} ⊆ 𝐷 ∧ {𝐼, 𝐽} ≈ 2_o) → (𝑇‘{𝐼, 𝐽}) ∈ ran 𝑇)
34	4, 21, 30, 33	syl3anc 1390	. . . . . . . . . 10 ⊢ (𝜑 → (𝑇‘{𝐼, 𝐽}) ∈ ran 𝑇)
35	31, 32	pmtrff1o 19503	. . . . . . . . . 10 ⊢ ((𝑇‘{𝐼, 𝐽}) ∈ ran 𝑇 → (𝑇‘{𝐼, 𝐽}):𝐷–1-1-onto→𝐷)
36		f1ococnv1 6836	. . . . . . . . . 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 5833	. . . . . . . 8 ⊢ (𝜑 → ((^◡(𝑇‘{𝐼, 𝐽}) ∘ (𝑇‘{𝐼, 𝐽})) ∘ 𝐹) = (( I ↾ 𝐷) ∘ 𝐹))
39	3, 38	eqtr3id 2811	. . . . . . 7 ⊢ (𝜑 → (^◡(𝑇‘{𝐼, 𝐽}) ∘ ((𝑇‘{𝐼, 𝐽}) ∘ 𝐹)) = (( I ↾ 𝐷) ∘ 𝐹))
40		fcoi2 6739	. . . . . . . 8 ⊢ (𝐹:𝐷⟶𝐷 → (( I ↾ 𝐷) ∘ 𝐹) = 𝐹)
41	15, 40	syl 17	. . . . . . 7 ⊢ (𝜑 → (( I ↾ 𝐷) ∘ 𝐹) = 𝐹)
42	39, 41	eqtrd 2797	. . . . . 6 ⊢ (𝜑 → (^◡(𝑇‘{𝐼, 𝐽}) ∘ ((𝑇‘{𝐼, 𝐽}) ∘ 𝐹)) = 𝐹)
43	42	difeq1d 4079	. . . . 5 ⊢ (𝜑 → ((^◡(𝑇‘{𝐼, 𝐽}) ∘ ((𝑇‘{𝐼, 𝐽}) ∘ 𝐹)) ∖ I ) = (𝐹 ∖ I ))
44	43	dmeqd 5881	. . . 4 ⊢ (𝜑 → dom ((^◡(𝑇‘{𝐼, 𝐽}) ∘ ((𝑇‘{𝐼, 𝐽}) ∘ 𝐹)) ∖ I ) = dom (𝐹 ∖ I ))
45	31, 32	pmtrfcnv 19504	. . . . . . . . . 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 4079	. . . . . . . 8 ⊢ (𝜑 → (^◡(𝑇‘{𝐼, 𝐽}) ∖ I ) = ((𝑇‘{𝐼, 𝐽}) ∖ I ))
48	47	dmeqd 5881	. . . . . . 7 ⊢ (𝜑 → dom (^◡(𝑇‘{𝐼, 𝐽}) ∖ I ) = dom ((𝑇‘{𝐼, 𝐽}) ∖ I ))
49	31	pmtrmvd 19496	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝑉 ∧ {𝐼, 𝐽} ⊆ 𝐷 ∧ {𝐼, 𝐽} ≈ 2_o) → dom ((𝑇‘{𝐼, 𝐽}) ∖ I ) = {𝐼, 𝐽})
50	4, 21, 30, 49	syl3anc 1390	. . . . . . 7 ⊢ (𝜑 → dom ((𝑇‘{𝐼, 𝐽}) ∖ I ) = {𝐼, 𝐽})
51	48, 50	eqtrd 2797	. . . . . 6 ⊢ (𝜑 → dom (^◡(𝑇‘{𝐼, 𝐽}) ∖ I ) = {𝐼, 𝐽})
52	51	uneq1d 4120	. . . . 5 ⊢ (𝜑 → (dom (^◡(𝑇‘{𝐼, 𝐽}) ∖ I ) ∪ dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I )) = ({𝐼, 𝐽} ∪ dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I )))
53		uncom 4111	. . . . 5 ⊢ ({𝐼, 𝐽} ∪ dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I )) = (dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ∪ {𝐼, 𝐽})
54	52, 53	eqtrdi 2813	. . . 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 3990	. . 3 ⊢ (𝜑 → dom (𝐹 ∖ I ) ⊆ (dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ∪ {𝐼, 𝐽}))
56	55	ssdifd 4098	. 2 ⊢ (𝜑 → (dom (𝐹 ∖ I ) ∖ {𝐼, 𝐽}) ⊆ ((dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ∪ {𝐼, 𝐽}) ∖ {𝐼, 𝐽}))
57		difun2 4435	. . 3 ⊢ ((dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ∪ {𝐼, 𝐽}) ∖ {𝐼, 𝐽}) = (dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ∖ {𝐼, 𝐽})
58		difss 4089	. . 3 ⊢ (dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ∖ {𝐼, 𝐽}) ⊆ dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I )
59	57, 58	eqsstri 3982	. 2 ⊢ ((dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ∪ {𝐼, 𝐽}) ∖ {𝐼, 𝐽}) ⊆ dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I )
60	56, 59	sstrdi 3948	1 ⊢ (𝜑 → (dom (𝐹 ∖ I ) ∖ {𝐼, 𝐽}) ⊆ dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1560 ∈ wcel 2142 ≠ wne 2957 ∖ cdif 3901 ∪ cun 3902 ⊆ wss 3904 {cpr 4584 class class class wbr 5100 I cid 5541 ^◡ccnv 5646 dom cdm 5647 ran crn 5648 ↾ cres 5649 ∘ ccom 5651 Fn wfn 6516 ⟶wf 6517 –1-1-onto→wf1o 6520 ‘cfv 6521 2_oc2o 8431 ≈ cen 8924 Basecbs 17245 SymGrpcsymg 19409 pmTrspcpmtr 19481

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7847 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8678 df-map 8810 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-nn 12211 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12482 df-z 12569 df-uz 12840 df-fz 13513 df-struct 17183 df-sets 17200 df-slot 17218 df-ndx 17230 df-base 17246 df-ress 17267 df-plusg 17299 df-tset 17305 df-efmnd 18903 df-symg 19410 df-pmtr 19482

This theorem is referenced by: pmtrcnelor 33268

W3C validator