Theorem fvnonrel 42333

Description: The function value of any class under a non-relation is empty. (Contributed by RP, 23-Oct-2020.)

Assertion

Ref	Expression
fvnonrel	⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) = ∅

Proof of Theorem fvnonrel

Step	Hyp	Ref	Expression
1		fvrn0 6918	. . 3 ⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) ∈ (ran (𝐴 ∖ ◡◡𝐴) ∪ {∅})
2		rnnonrel 42327	. . . . 5 ⊢ ran (𝐴 ∖ ◡◡𝐴) = ∅
3		0ss 4395	. . . . 5 ⊢ ∅ ⊆ {∅}
4	2, 3	eqsstri 4015	. . . 4 ⊢ ran (𝐴 ∖ ◡◡𝐴) ⊆ {∅}
5		ssequn1 4179	. . . 4 ⊢ (ran (𝐴 ∖ ◡◡𝐴) ⊆ {∅} ↔ (ran (𝐴 ∖ ◡◡𝐴) ∪ {∅}) = {∅})
6	4, 5	mpbi 229	. . 3 ⊢ (ran (𝐴 ∖ ◡◡𝐴) ∪ {∅}) = {∅}
7	1, 6	eleqtri 2831	. 2 ⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) ∈ {∅}
8		fvex 6901	. . 3 ⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) ∈ V
9	8	elsn 4642	. 2 ⊢ (((𝐴 ∖ ◡◡𝐴)‘𝑋) ∈ {∅} ↔ ((𝐴 ∖ ◡◡𝐴)‘𝑋) = ∅)
10	7, 9	mpbi 229	1 ⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) = ∅

Syntax hints:   = wceq 1541   ∈ wcel 2106   ∖ cdif 3944   ∪ cun 3945   ⊆ wss 3947  ∅c0 4321  {csn 4627  ^◡ccnv 5674  ran crn 5676  ‘cfv 6540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-cnv 5683  df-dm 5685  df-rn 5686  df-iota 6492  df-fv 6548

This theorem is referenced by: (None)