Theorem bascnvimaeqv 17120

Description: The inverse image of the universal class V under the base function is the universal class V itself. (Proposed by Mario Carneiro, 7-Mar-2020.) (Contributed by AV, 7-Mar-2020.)

Assertion

Ref	Expression
bascnvimaeqv	⊢ (◡Base “ V) = V

Proof of Theorem bascnvimaeqv

Step	Hyp	Ref	Expression
1		df-base 16235	. . 3 ⊢ Base = Slot 1
2	1	slotfn 16247	. 2 ⊢ Base Fn V
3		fncnvimaeqv 17119	. 2 ⊢ (Base Fn V → (◡Base “ V) = V)
4	2, 3	ax-mp 5	1 ⊢ (◡Base “ V) = V

Syntax hints:   = wceq 1656  Vcvv 3414  ^◡ccnv 5345   “ cima 5349   Fn wfn 6122  1c1 10260  Basecbs 16229

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-fv 6135  df-slot 16233  df-base 16235

This theorem is referenced by: (None)