Theorem bascnvimaeqv 16584

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 15700	. . 3 ⊢ Base = Slot 1
2	1	slotfn 15709	. 2 ⊢ Base Fn V
3		fncnvimaeqv 16583	. 2 ⊢ (Base Fn V → (◡Base “ V) = V)
4	2, 3	ax-mp 5	1 ⊢ (◡Base “ V) = V

Syntax hints:   = wceq 1475  Vcvv 3173  ^◡ccnv 5037   “ cima 5041   Fn wfn 5799  1c1 9816  Basecbs 15695

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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  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-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  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-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-slot 15699  df-base 15700

This theorem is referenced by: (None)