Theorem bascnvimaeqv 16808

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 15910	. . 3 ⊢ Base = Slot 1
2	1	slotfn 15922	. 2 ⊢ Base Fn V
3		fncnvimaeqv 16807	. 2 ⊢ (Base Fn V → (◡Base “ V) = V)
4	2, 3	ax-mp 5	1 ⊢ (◡Base “ V) = V

Syntax hints:   = wceq 1523  Vcvv 3231  ^◡ccnv 5142   “ cima 5146   Fn wfn 5921  1c1 9975  Basecbs 15904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934  df-slot 15908  df-base 15910

This theorem is referenced by: (None)