Theorem fncnvimaeqv 17372

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 6833	. 2 ⊢ (𝐹 Fn V → (◡𝐹 “ V) = {𝑦 ∈ V ∣ (𝐹‘𝑦) ∈ V})
2		fvexd 6687	. . . . 5 ⊢ (𝐹 Fn V → (𝐹‘𝑥) ∈ V)
3	2	biantrud 534	. . . 4 ⊢ (𝐹 Fn V → (𝑥 ∈ V ↔ (𝑥 ∈ V ∧ (𝐹‘𝑥) ∈ V)))
4		fveq2 6672	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
5	4	eleq1d 2899	. . . . 5 ⊢ (𝑦 = 𝑥 → ((𝐹‘𝑦) ∈ V ↔ (𝐹‘𝑥) ∈ V))
6	5	elrab 3682	. . . 4 ⊢ (𝑥 ∈ {𝑦 ∈ V ∣ (𝐹‘𝑦) ∈ V} ↔ (𝑥 ∈ V ∧ (𝐹‘𝑥) ∈ V))
7	3, 6	syl6rbbr 292	. . 3 ⊢ (𝐹 Fn V → (𝑥 ∈ {𝑦 ∈ V ∣ (𝐹‘𝑦) ∈ V} ↔ 𝑥 ∈ V))
8	7	eqrdv 2821	. 2 ⊢ (𝐹 Fn V → {𝑦 ∈ V ∣ (𝐹‘𝑦) ∈ V} = V)
9	1, 8	eqtrd 2858	1 ⊢ (𝐹 Fn V → (◡𝐹 “ V) = V)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  {crab 3144  Vcvv 3496  ^◡ccnv 5556   “ cima 5560   Fn wfn 6352  ‘cfv 6357

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-fv 6365

This theorem is referenced by:  bascnvimaeqv  17373