msubff1o - Mathbox for Mario Carneiro

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Theorem msubff1o 34834

Description: When restricted to complete mappings, the substitution-producing function is bijective to the set of all substitutions. (Contributed by Mario Carneiro, 18-Jul-2016.)

**Hypotheses**
Ref	Expression
msubff1.v	⊢ 𝑉 = (mVR‘𝑇)
msubff1.r	⊢ 𝑅 = (mREx‘𝑇)
msubff1.s	⊢ 𝑆 = (mSubst‘𝑇)

**Assertion**
Ref	Expression
msubff1o	⊢ (𝑇 ∈ mFS → (𝑆 ↾ (𝑅 ↑_m 𝑉)):(𝑅 ↑_m 𝑉)–1-1-onto→ran 𝑆)

**Proof of Theorem msubff1o**
Step	Hyp	Ref	Expression
1		msubff1.v	. . . 4 ⊢ 𝑉 = (mVR‘𝑇)
2		msubff1.r	. . . 4 ⊢ 𝑅 = (mREx‘𝑇)
3		msubff1.s	. . . 4 ⊢ 𝑆 = (mSubst‘𝑇)
4		eqid 2732	. . . 4 ⊢ (mEx‘𝑇) = (mEx‘𝑇)
5	1, 2, 3, 4	msubff1 34833	. . 3 ⊢ (𝑇 ∈ mFS → (𝑆 ↾ (𝑅 ↑_m 𝑉)):(𝑅 ↑_m 𝑉)–1-1→((mEx‘𝑇) ↑_m (mEx‘𝑇)))
6		f1f1orn 6844	. . 3 ⊢ ((𝑆 ↾ (𝑅 ↑_m 𝑉)):(𝑅 ↑_m 𝑉)–1-1→((mEx‘𝑇) ↑_m (mEx‘𝑇)) → (𝑆 ↾ (𝑅 ↑_m 𝑉)):(𝑅 ↑_m 𝑉)–1-1-onto→ran (𝑆 ↾ (𝑅 ↑_m 𝑉)))
7	5, 6	syl 17	. 2 ⊢ (𝑇 ∈ mFS → (𝑆 ↾ (𝑅 ↑_m 𝑉)):(𝑅 ↑_m 𝑉)–1-1-onto→ran (𝑆 ↾ (𝑅 ↑_m 𝑉)))
8	1, 2, 3	msubrn 34806	. . . 4 ⊢ ran 𝑆 = (𝑆 “ (𝑅 ↑_m 𝑉))
9		df-ima 5689	. . . 4 ⊢ (𝑆 “ (𝑅 ↑_m 𝑉)) = ran (𝑆 ↾ (𝑅 ↑_m 𝑉))
10	8, 9	eqtri 2760	. . 3 ⊢ ran 𝑆 = ran (𝑆 ↾ (𝑅 ↑_m 𝑉))
11		f1oeq3 6823	. . 3 ⊢ (ran 𝑆 = ran (𝑆 ↾ (𝑅 ↑_m 𝑉)) → ((𝑆 ↾ (𝑅 ↑_m 𝑉)):(𝑅 ↑_m 𝑉)–1-1-onto→ran 𝑆 ↔ (𝑆 ↾ (𝑅 ↑_m 𝑉)):(𝑅 ↑_m 𝑉)–1-1-onto→ran (𝑆 ↾ (𝑅 ↑_m 𝑉))))
12	10, 11	ax-mp 5	. 2 ⊢ ((𝑆 ↾ (𝑅 ↑_m 𝑉)):(𝑅 ↑_m 𝑉)–1-1-onto→ran 𝑆 ↔ (𝑆 ↾ (𝑅 ↑_m 𝑉)):(𝑅 ↑_m 𝑉)–1-1-onto→ran (𝑆 ↾ (𝑅 ↑_m 𝑉)))
13	7, 12	sylibr 233	1 ⊢ (𝑇 ∈ mFS → (𝑆 ↾ (𝑅 ↑_m 𝑉)):(𝑅 ↑_m 𝑉)–1-1-onto→ran 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 = wceq 1541 ∈ wcel 2106 ran crn 5677 ↾ cres 5678 “ cima 5679 –1-1→wf1 6540 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7411 ↑_m cmap 8822 mVRcmvar 34738 mRExcmrex 34743 mExcmex 34744 mSubstcmsub 34748 mFScmfs 34753

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13489 df-fzo 13632 df-seq 13971 df-hash 14295 df-word 14469 df-concat 14525 df-s1 14550 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-0g 17391 df-gsum 17392 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-frmd 18766 df-mrex 34763 df-mex 34764 df-mrsub 34767 df-msub 34768 df-mfs 34773

This theorem is referenced by: (None)

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