Theorem fvnonrel 44306

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 6911	. . 3 ⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) ∈ (ran (𝐴 ∖ ◡◡𝐴) ∪ {∅})
2		rnnonrel 44300	. . . . 5 ⊢ ran (𝐴 ∖ ◡◡𝐴) = ∅
3		0ss 4358	. . . . 5 ⊢ ∅ ⊆ {∅}
4	2, 3	eqsstri 3984	. . . 4 ⊢ ran (𝐴 ∖ ◡◡𝐴) ⊆ {∅}
5		ssequn1 4140	. . . 4 ⊢ (ran (𝐴 ∖ ◡◡𝐴) ⊆ {∅} ↔ (ran (𝐴 ∖ ◡◡𝐴) ∪ {∅}) = {∅})
6	4, 5	mpbi 233	. . 3 ⊢ (ran (𝐴 ∖ ◡◡𝐴) ∪ {∅}) = {∅}
7	1, 6	eleqtri 2861	. 2 ⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) ∈ {∅}
8		fvex 6896	. . 3 ⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) ∈ V
9	8	elsn 4605	. 2 ⊢ (((𝐴 ∖ ◡◡𝐴)‘𝑋) ∈ {∅} ↔ ((𝐴 ∖ ◡◡𝐴)‘𝑋) = ∅)
10	7, 9	mpbi 233	1 ⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) = ∅

Syntax hints:   = wceq 1570   ∈ wcel 2143   ∖ cdif 3903   ∪ cun 3904   ⊆ wss 3906  ∅c0 4287  {csn 4590  ^◡ccnv 5662  ran crn 5664  ‘cfv 6538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-xp 5669  df-rel 5670  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-iota 6494  df-fv 6546

This theorem is referenced by: (None)