Theorem msubff 30681

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 7089	. . . . . . . . 9 ⊢ (𝑒 ∈ ((mTC‘𝑇) × 𝑅) → (1st ‘𝑒) ∈ (mTC‘𝑇))
2		eqid 2610	. . . . . . . . . 10 ⊢ (mTC‘𝑇) = (mTC‘𝑇)
3		msubff.e	. . . . . . . . . 10 ⊢ 𝐸 = (mEx‘𝑇)
4		msubff.r	. . . . . . . . . 10 ⊢ 𝑅 = (mREx‘𝑇)
5	2, 3, 4	mexval 30653	. . . . . . . . 9 ⊢ 𝐸 = ((mTC‘𝑇) × 𝑅)
6	1, 5	eleq2s 2706	. . . . . . . 8 ⊢ (𝑒 ∈ 𝐸 → (1st ‘𝑒) ∈ (mTC‘𝑇))
7	6	adantl 481	. . . . . . 7 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑒 ∈ 𝐸) → (1st ‘𝑒) ∈ (mTC‘𝑇))
8		msubff.v	. . . . . . . . . . 11 ⊢ 𝑉 = (mVR‘𝑇)
9		eqid 2610	. . . . . . . . . . 11 ⊢ (mRSubst‘𝑇) = (mRSubst‘𝑇)
10	8, 4, 9	mrsubff 30663	. . . . . . . . . 10 ⊢ (𝑇 ∈ 𝑊 → (mRSubst‘𝑇):(𝑅 ↑pm 𝑉)⟶(𝑅 ↑𝑚 𝑅))
11	10	ffvelrnda 6267	. . . . . . . . 9 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ (𝑅 ↑𝑚 𝑅))
12		elmapi 7765	. . . . . . . . 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 7090	. . . . . . . . 9 ⊢ (𝑒 ∈ ((mTC‘𝑇) × 𝑅) → (2nd ‘𝑒) ∈ 𝑅)
15	14, 5	eleq2s 2706	. . . . . . . 8 ⊢ (𝑒 ∈ 𝐸 → (2nd ‘𝑒) ∈ 𝑅)
16		ffvelrn 6265	. . . . . . . 8 ⊢ ((((mRSubst‘𝑇)‘𝑓):𝑅⟶𝑅 ∧ (2nd ‘𝑒) ∈ 𝑅) → (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒)) ∈ 𝑅)
17	13, 15, 16	syl2an 493	. . . . . . 7 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑒 ∈ 𝐸) → (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒)) ∈ 𝑅)
18		opelxp 5070	. . . . . . 7 ⊢ (⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))⟩ ∈ ((mTC‘𝑇) × 𝑅) ↔ ((1st ‘𝑒) ∈ (mTC‘𝑇) ∧ (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒)) ∈ 𝑅))
19	7, 17, 18	sylanbrc 695	. . . . . 6 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑒 ∈ 𝐸) → ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))⟩ ∈ ((mTC‘𝑇) × 𝑅))
20	19, 5	syl6eleqr 2699	. . . . 5 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑒 ∈ 𝐸) → ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))⟩ ∈ 𝐸)
21		eqid 2610	. . . . 5 ⊢ (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))⟩) = (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))⟩)
22	20, 21	fmptd 6292	. . . 4 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))⟩):𝐸⟶𝐸)
23		fvex 6113	. . . . . 6 ⊢ (mEx‘𝑇) ∈ V
24	3, 23	eqeltri 2684	. . . . 5 ⊢ 𝐸 ∈ V
25	24, 24	elmap 7772	. . . 4 ⊢ ((𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))⟩) ∈ (𝐸 ↑𝑚 𝐸) ↔ (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))⟩):𝐸⟶𝐸)
26	22, 25	sylibr 223	. . 3 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))⟩) ∈ (𝐸 ↑𝑚 𝐸))
27		eqid 2610	. . 3 ⊢ (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))⟩)) = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))⟩))
28	26, 27	fmptd 6292	. 2 ⊢ (𝑇 ∈ 𝑊 → (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))⟩)):(𝑅 ↑pm 𝑉)⟶(𝐸 ↑𝑚 𝐸))
29		msubff.s	. . . 4 ⊢ 𝑆 = (mSubst‘𝑇)
30	8, 4, 29, 3, 9	msubffval 30674	. . 3 ⊢ (𝑇 ∈ 𝑊 → 𝑆 = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))⟩)))
31	30	feq1d 5943	. 2 ⊢ (𝑇 ∈ 𝑊 → (𝑆:(𝑅 ↑pm 𝑉)⟶(𝐸 ↑𝑚 𝐸) ↔ (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))⟩)):(𝑅 ↑pm 𝑉)⟶(𝐸 ↑𝑚 𝐸)))
32	28, 31	mpbird 246	1 ⊢ (𝑇 ∈ 𝑊 → 𝑆:(𝑅 ↑pm 𝑉)⟶(𝐸 ↑𝑚 𝐸))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131   ↦ cmpt 4643   × cxp 5036  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1^st c1st 7057  2^nd c2nd 7058   ↑_𝑚 cmap 7744   ↑_pm cpm 7745  mVRcmvar 30612  mTCcmtc 30615  mRExcmrex 30617  mExcmex 30618  mRSubstcmrsub 30621  mSubstcmsub 30622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-word 13154  df-concat 13156  df-s1 13157  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-gsum 15926  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-frmd 17209  df-mrex 30637  df-mex 30638  df-mrsub 30641  df-msub 30642

This theorem is referenced by:  msubf  30683  msubff1  30707  mclsind  30721