Theorem bascnvimaeqv 16968

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 16070	. . 3 ⊢ Base = Slot 1
2	1	slotfn 16082	. 2 ⊢ Base Fn V
3		fncnvimaeqv 16967	. 2 ⊢ (Base Fn V → (◡Base “ V) = V)
4	2, 3	ax-mp 5	1 ⊢ (◡Base “ V) = V

Syntax hints:   = wceq 1631  Vcvv 3351  ^◡ccnv 5248   “ cima 5252   Fn wfn 6026  1c1 10139  Basecbs 16064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039  df-slot 16068  df-base 16070

This theorem is referenced by: (None)