Theorem fvnonrel 39947

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 6691	. . 3 ⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) ∈ (ran (𝐴 ∖ ◡◡𝐴) ∪ {∅})
2		rnnonrel 39941	. . . . 5 ⊢ ran (𝐴 ∖ ◡◡𝐴) = ∅
3		0ss 4348	. . . . 5 ⊢ ∅ ⊆ {∅}
4	2, 3	eqsstri 3999	. . . 4 ⊢ ran (𝐴 ∖ ◡◡𝐴) ⊆ {∅}
5		ssequn1 4154	. . . 4 ⊢ (ran (𝐴 ∖ ◡◡𝐴) ⊆ {∅} ↔ (ran (𝐴 ∖ ◡◡𝐴) ∪ {∅}) = {∅})
6	4, 5	mpbi 232	. . 3 ⊢ (ran (𝐴 ∖ ◡◡𝐴) ∪ {∅}) = {∅}
7	1, 6	eleqtri 2909	. 2 ⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) ∈ {∅}
8		fvex 6676	. . 3 ⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) ∈ V
9	8	elsn 4574	. 2 ⊢ (((𝐴 ∖ ◡◡𝐴)‘𝑋) ∈ {∅} ↔ ((𝐴 ∖ ◡◡𝐴)‘𝑋) = ∅)
10	7, 9	mpbi 232	1 ⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) = ∅

Syntax hints:   = wceq 1531   ∈ wcel 2108   ∖ cdif 3931   ∪ cun 3932   ⊆ wss 3934  ∅c0 4289  {csn 4559  ^◡ccnv 5547  ran crn 5549  ‘cfv 6348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559  df-iota 6307  df-fv 6356

This theorem is referenced by: (None)