Theorem fvnonrel 44024

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 6868	. . 3 ⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) ∈ (ran (𝐴 ∖ ◡◡𝐴) ∪ {∅})
2		rnnonrel 44018	. . . . 5 ⊢ ran (𝐴 ∖ ◡◡𝐴) = ∅
3		0ss 4340	. . . . 5 ⊢ ∅ ⊆ {∅}
4	2, 3	eqsstri 3968	. . . 4 ⊢ ran (𝐴 ∖ ◡◡𝐴) ⊆ {∅}
5		ssequn1 4126	. . . 4 ⊢ (ran (𝐴 ∖ ◡◡𝐴) ⊆ {∅} ↔ (ran (𝐴 ∖ ◡◡𝐴) ∪ {∅}) = {∅})
6	4, 5	mpbi 230	. . 3 ⊢ (ran (𝐴 ∖ ◡◡𝐴) ∪ {∅}) = {∅}
7	1, 6	eleqtri 2834	. 2 ⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) ∈ {∅}
8		fvex 6853	. . 3 ⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) ∈ V
9	8	elsn 4582	. 2 ⊢ (((𝐴 ∖ ◡◡𝐴)‘𝑋) ∈ {∅} ↔ ((𝐴 ∖ ◡◡𝐴)‘𝑋) = ∅)
10	7, 9	mpbi 230	1 ⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) = ∅

Syntax hints:   = wceq 1542   ∈ wcel 2114   ∖ cdif 3886   ∪ cun 3887   ⊆ wss 3889  ∅c0 4273  {csn 4567  ^◡ccnv 5630  ran crn 5632  ‘cfv 6498

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-xp 5637  df-rel 5638  df-cnv 5639  df-dm 5641  df-rn 5642  df-iota 6454  df-fv 6506

This theorem is referenced by: (None)