msubff - Mathbox for Mario Carneiro

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Theorem msubff 35513

Description: A substitution is a function from 𝐸 to 𝐸. (Contributed by Mario Carneiro, 18-Jul-2016.)

**Hypotheses**
Ref	Expression
msubff.v	⊢ 𝑉 = (mVR‘𝑇)
msubff.r	⊢ 𝑅 = (mREx‘𝑇)
msubff.s	⊢ 𝑆 = (mSubst‘𝑇)
msubff.e	⊢ 𝐸 = (mEx‘𝑇)

**Assertion**
Ref	Expression
msubff	⊢ (𝑇 ∈ 𝑊 → 𝑆:(𝑅 ↑_pm 𝑉)⟶(𝐸 ↑_m 𝐸))

Proof of Theorem msubff Dummy variables 𝑒 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xp1st 7956	. . . . . . . . 9 ⊢ (𝑒 ∈ ((mTC‘𝑇) × 𝑅) → (1^st ‘𝑒) ∈ (mTC‘𝑇))
2		eqid 2729	. . . . . . . . . 10 ⊢ (mTC‘𝑇) = (mTC‘𝑇)
3		msubff.e	. . . . . . . . . 10 ⊢ 𝐸 = (mEx‘𝑇)
4		msubff.r	. . . . . . . . . 10 ⊢ 𝑅 = (mREx‘𝑇)
5	2, 3, 4	mexval 35485	. . . . . . . . 9 ⊢ 𝐸 = ((mTC‘𝑇) × 𝑅)
6	1, 5	eleq2s 2846	. . . . . . . 8 ⊢ (𝑒 ∈ 𝐸 → (1^st ‘𝑒) ∈ (mTC‘𝑇))
7	6	adantl 481	. . . . . . 7 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) ∧ 𝑒 ∈ 𝐸) → (1^st ‘𝑒) ∈ (mTC‘𝑇))
8		msubff.v	. . . . . . . . . . 11 ⊢ 𝑉 = (mVR‘𝑇)
9		eqid 2729	. . . . . . . . . . 11 ⊢ (mRSubst‘𝑇) = (mRSubst‘𝑇)
10	8, 4, 9	mrsubff 35495	. . . . . . . . . 10 ⊢ (𝑇 ∈ 𝑊 → (mRSubst‘𝑇):(𝑅 ↑_pm 𝑉)⟶(𝑅 ↑_m 𝑅))
11	10	ffvelcdmda 7018	. . . . . . . . 9 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ (𝑅 ↑_m 𝑅))
12		elmapi 8776	. . . . . . . . 9 ⊢ (((mRSubst‘𝑇)‘𝑓) ∈ (𝑅 ↑_m 𝑅) → ((mRSubst‘𝑇)‘𝑓):𝑅⟶𝑅)
13	11, 12	syl 17	. . . . . . . 8 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓):𝑅⟶𝑅)
14		xp2nd 7957	. . . . . . . . 9 ⊢ (𝑒 ∈ ((mTC‘𝑇) × 𝑅) → (2^nd ‘𝑒) ∈ 𝑅)
15	14, 5	eleq2s 2846	. . . . . . . 8 ⊢ (𝑒 ∈ 𝐸 → (2^nd ‘𝑒) ∈ 𝑅)
16		ffvelcdm 7015	. . . . . . . 8 ⊢ ((((mRSubst‘𝑇)‘𝑓):𝑅⟶𝑅 ∧ (2^nd ‘𝑒) ∈ 𝑅) → (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒)) ∈ 𝑅)
17	13, 15, 16	syl2an 596	. . . . . . 7 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) ∧ 𝑒 ∈ 𝐸) → (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒)) ∈ 𝑅)
18		opelxp 5655	. . . . . . 7 ⊢ (⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩ ∈ ((mTC‘𝑇) × 𝑅) ↔ ((1^st ‘𝑒) ∈ (mTC‘𝑇) ∧ (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒)) ∈ 𝑅))
19	7, 17, 18	sylanbrc 583	. . . . . 6 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) ∧ 𝑒 ∈ 𝐸) → ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩ ∈ ((mTC‘𝑇) × 𝑅))
20	19, 5	eleqtrrdi 2839	. . . . 5 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) ∧ 𝑒 ∈ 𝐸) → ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩ ∈ 𝐸)
21	20	fmpttd 7049	. . . 4 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩):𝐸⟶𝐸)
22	3	fvexi 6836	. . . . 5 ⊢ 𝐸 ∈ V
23	22, 22	elmap 8798	. . . 4 ⊢ ((𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩) ∈ (𝐸 ↑_m 𝐸) ↔ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩):𝐸⟶𝐸)
24	21, 23	sylibr 234	. . 3 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩) ∈ (𝐸 ↑_m 𝐸))
25	24	fmpttd 7049	. 2 ⊢ (𝑇 ∈ 𝑊 → (𝑓 ∈ (𝑅 ↑_pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩)):(𝑅 ↑_pm 𝑉)⟶(𝐸 ↑_m 𝐸))
26		msubff.s	. . . 4 ⊢ 𝑆 = (mSubst‘𝑇)
27	8, 4, 26, 3, 9	msubffval 35506	. . 3 ⊢ (𝑇 ∈ 𝑊 → 𝑆 = (𝑓 ∈ (𝑅 ↑_pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩)))
28	27	feq1d 6634	. 2 ⊢ (𝑇 ∈ 𝑊 → (𝑆:(𝑅 ↑_pm 𝑉)⟶(𝐸 ↑_m 𝐸) ↔ (𝑓 ∈ (𝑅 ↑_pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩)):(𝑅 ↑_pm 𝑉)⟶(𝐸 ↑_m 𝐸)))
29	25, 28	mpbird 257	1 ⊢ (𝑇 ∈ 𝑊 → 𝑆:(𝑅 ↑_pm 𝑉)⟶(𝐸 ↑_m 𝐸))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ⟨cop 4583 ↦ cmpt 5173 × cxp 5617 ⟶wf 6478 ‘cfv 6482 (class class class)co 7349 1^st c1st 7922 2^nd c2nd 7923 ↑_m cmap 8753 ↑_pm cpm 8754 mVRcmvar 35444 mTCcmtc 35447 mRExcmrex 35449 mExcmex 35450 mRSubstcmrsub 35453 mSubstcmsub 35454

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 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-er 8625 df-map 8755 df-pm 8756 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-card 9835 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-nn 12129 df-2 12191 df-n0 12385 df-z 12472 df-uz 12736 df-fz 13411 df-fzo 13558 df-seq 13909 df-hash 14238 df-word 14421 df-concat 14478 df-s1 14503 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-0g 17345 df-gsum 17346 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-submnd 18658 df-frmd 18723 df-mrex 35469 df-mex 35470 df-mrsub 35473 df-msub 35474

This theorem is referenced by: msubf 35515 msubff1 35539 mclsind 35553

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