Theorem mtyf 32803

Description: The type function maps variables to variable typecodes. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mtyf.v	⊢ 𝑉 = (mVR‘𝑇)
mtyf.f	⊢ 𝐹 = (mVT‘𝑇)
mtyf.y	⊢ 𝑌 = (mType‘𝑇)

Assertion

Ref	Expression
mtyf	⊢ (𝑇 ∈ mFS → 𝑌:𝑉⟶𝐹)

Proof of Theorem mtyf

Step	Hyp	Ref	Expression
1		mtyf.v	. . . 4 ⊢ 𝑉 = (mVR‘𝑇)
2		eqid 2824	. . . 4 ⊢ (mTC‘𝑇) = (mTC‘𝑇)
3		mtyf.y	. . . 4 ⊢ 𝑌 = (mType‘𝑇)
4	1, 2, 3	mtyf2 32802	. . 3 ⊢ (𝑇 ∈ mFS → 𝑌:𝑉⟶(mTC‘𝑇))
5		ffn 6517	. . . 4 ⊢ (𝑌:𝑉⟶(mTC‘𝑇) → 𝑌 Fn 𝑉)
6		dffn4 6599	. . . 4 ⊢ (𝑌 Fn 𝑉 ↔ 𝑌:𝑉–onto→ran 𝑌)
7	5, 6	sylib 220	. . 3 ⊢ (𝑌:𝑉⟶(mTC‘𝑇) → 𝑌:𝑉–onto→ran 𝑌)
8		fof 6593	. . 3 ⊢ (𝑌:𝑉–onto→ran 𝑌 → 𝑌:𝑉⟶ran 𝑌)
9	4, 7, 8	3syl 18	. 2 ⊢ (𝑇 ∈ mFS → 𝑌:𝑉⟶ran 𝑌)
10		mtyf.f	. . . 4 ⊢ 𝐹 = (mVT‘𝑇)
11	10, 3	mvtval 32751	. . 3 ⊢ 𝐹 = ran 𝑌
12		feq3 6500	. . 3 ⊢ (𝐹 = ran 𝑌 → (𝑌:𝑉⟶𝐹 ↔ 𝑌:𝑉⟶ran 𝑌))
13	11, 12	ax-mp 5	. 2 ⊢ (𝑌:𝑉⟶𝐹 ↔ 𝑌:𝑉⟶ran 𝑌)
14	9, 13	sylibr 236	1 ⊢ (𝑇 ∈ mFS → 𝑌:𝑉⟶𝐹)

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1536   ∈ wcel 2113  ran crn 5559   Fn wfn 6353  ⟶wf 6354  –onto→wfo 6356  ‘cfv 6358  mVRcmvar 32712  mTypecmty 32713  mVTcmvt 32714  mTCcmtc 32715  mFScmfs 32727

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-fo 6364  df-fv 6366  df-mvt 32736  df-mfs 32747

This theorem is referenced by: (None)