msubff - Mathbox for Mario Carneiro

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Theorem msubff 35728

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 7967	. . . . . . . . 9 ⊢ (𝑒 ∈ ((mTC‘𝑇) × 𝑅) → (1^st ‘𝑒) ∈ (mTC‘𝑇))
2		eqid 2737	. . . . . . . . . 10 ⊢ (mTC‘𝑇) = (mTC‘𝑇)
3		msubff.e	. . . . . . . . . 10 ⊢ 𝐸 = (mEx‘𝑇)
4		msubff.r	. . . . . . . . . 10 ⊢ 𝑅 = (mREx‘𝑇)
5	2, 3, 4	mexval 35700	. . . . . . . . 9 ⊢ 𝐸 = ((mTC‘𝑇) × 𝑅)
6	1, 5	eleq2s 2855	. . . . . . . 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 2737	. . . . . . . . . . 11 ⊢ (mRSubst‘𝑇) = (mRSubst‘𝑇)
10	8, 4, 9	mrsubff 35710	. . . . . . . . . 10 ⊢ (𝑇 ∈ 𝑊 → (mRSubst‘𝑇):(𝑅 ↑_pm 𝑉)⟶(𝑅 ↑_m 𝑅))
11	10	ffvelcdmda 7030	. . . . . . . . 9 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ (𝑅 ↑_m 𝑅))
12		elmapi 8789	. . . . . . . . 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 7968	. . . . . . . . 9 ⊢ (𝑒 ∈ ((mTC‘𝑇) × 𝑅) → (2^nd ‘𝑒) ∈ 𝑅)
15	14, 5	eleq2s 2855	. . . . . . . 8 ⊢ (𝑒 ∈ 𝐸 → (2^nd ‘𝑒) ∈ 𝑅)
16		ffvelcdm 7027	. . . . . . . 8 ⊢ ((((mRSubst‘𝑇)‘𝑓):𝑅⟶𝑅 ∧ (2^nd ‘𝑒) ∈ 𝑅) → (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒)) ∈ 𝑅)
17	13, 15, 16	syl2an 597	. . . . . . 7 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) ∧ 𝑒 ∈ 𝐸) → (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒)) ∈ 𝑅)
18		opelxp 5660	. . . . . . 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 584	. . . . . 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 2848	. . . . 5 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) ∧ 𝑒 ∈ 𝐸) → ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩ ∈ 𝐸)
21	20	fmpttd 7061	. . . 4 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩):𝐸⟶𝐸)
22	3	fvexi 6848	. . . . 5 ⊢ 𝐸 ∈ V
23	22, 22	elmap 8812	. . . 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 7061	. 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 35721	. . 3 ⊢ (𝑇 ∈ 𝑊 → 𝑆 = (𝑓 ∈ (𝑅 ↑_pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩)))
28	27	feq1d 6644	. 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 1542 ∈ wcel 2114 ⟨cop 4574 ↦ cmpt 5167 × cxp 5622 ⟶wf 6488 ‘cfv 6492 (class class class)co 7360 1^st c1st 7933 2^nd c2nd 7934 ↑_m cmap 8766 ↑_pm cpm 8767 mVRcmvar 35659 mTCcmtc 35662 mRExcmrex 35664 mExcmex 35665 mRSubstcmrsub 35668 mSubstcmsub 35669

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 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

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-rmo 3343 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-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-er 8636 df-map 8768 df-pm 8769 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-n0 12429 df-z 12516 df-uz 12780 df-fz 13453 df-fzo 13600 df-seq 13955 df-hash 14284 df-word 14467 df-concat 14524 df-s1 14550 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-0g 17395 df-gsum 17396 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-frmd 18808 df-mrex 35684 df-mex 35685 df-mrsub 35688 df-msub 35689

This theorem is referenced by: msubf 35730 msubff1 35754 mclsind 35768

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