Theorem fncnvimaeqv 17903

Description: The inverse images of the universal class V under functions on the universal class V are the universal class V itself. (Proposed by Mario Carneiro, 7-Mar-2020.) (Contributed by AV, 7-Mar-2020.)

Assertion

Ref	Expression
fncnvimaeqv	⊢ (𝐹 Fn V → (◡𝐹 “ V) = V)

Proof of Theorem fncnvimaeqv

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fncnvima2 6975	. 2 ⊢ (𝐹 Fn V → (◡𝐹 “ V) = {𝑦 ∈ V ∣ (𝐹‘𝑦) ∈ V})
2		fveq2 6809	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
3	2	eleq1d 2822	. . . . 5 ⊢ (𝑦 = 𝑥 → ((𝐹‘𝑦) ∈ V ↔ (𝐹‘𝑥) ∈ V))
4	3	elrab 3633	. . . 4 ⊢ (𝑥 ∈ {𝑦 ∈ V ∣ (𝐹‘𝑦) ∈ V} ↔ (𝑥 ∈ V ∧ (𝐹‘𝑥) ∈ V))
5		fvexd 6824	. . . . 5 ⊢ (𝐹 Fn V → (𝐹‘𝑥) ∈ V)
6	5	biantrud 532	. . . 4 ⊢ (𝐹 Fn V → (𝑥 ∈ V ↔ (𝑥 ∈ V ∧ (𝐹‘𝑥) ∈ V)))
7	4, 6	bitr4id 289	. . 3 ⊢ (𝐹 Fn V → (𝑥 ∈ {𝑦 ∈ V ∣ (𝐹‘𝑦) ∈ V} ↔ 𝑥 ∈ V))
8	7	eqrdv 2735	. 2 ⊢ (𝐹 Fn V → {𝑦 ∈ V ∣ (𝐹‘𝑦) ∈ V} = V)
9	1, 8	eqtrd 2777	1 ⊢ (𝐹 Fn V → (◡𝐹 “ V) = V)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1540   ∈ wcel 2105  {crab 3404  Vcvv 3441  ^◡ccnv 5604   “ cima 5608   Fn wfn 6458  ‘cfv 6463

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2170  ax-ext 2708  ax-sep 5236  ax-nul 5243  ax-pr 5365

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3443  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4849  df-br 5086  df-opab 5148  df-id 5505  df-xp 5611  df-rel 5612  df-cnv 5613  df-co 5614  df-dm 5615  df-rn 5616  df-res 5617  df-ima 5618  df-iota 6415  df-fun 6465  df-fn 6466  df-fv 6471

This theorem is referenced by:  bascnvimaeqv  17904