![]() |
Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > fvnonrel | Structured version Visualization version GIF version |
Description: The function value of any class under a non-relation is empty. (Contributed by RP, 23-Oct-2020.) |
Ref | Expression |
---|---|
fvnonrel | ⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvrn0 6932 | . . 3 ⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) ∈ (ran (𝐴 ∖ ◡◡𝐴) ∪ {∅}) | |
2 | rnnonrel 43070 | . . . . 5 ⊢ ran (𝐴 ∖ ◡◡𝐴) = ∅ | |
3 | 0ss 4400 | . . . . 5 ⊢ ∅ ⊆ {∅} | |
4 | 2, 3 | eqsstri 4016 | . . . 4 ⊢ ran (𝐴 ∖ ◡◡𝐴) ⊆ {∅} |
5 | ssequn1 4182 | . . . 4 ⊢ (ran (𝐴 ∖ ◡◡𝐴) ⊆ {∅} ↔ (ran (𝐴 ∖ ◡◡𝐴) ∪ {∅}) = {∅}) | |
6 | 4, 5 | mpbi 229 | . . 3 ⊢ (ran (𝐴 ∖ ◡◡𝐴) ∪ {∅}) = {∅} |
7 | 1, 6 | eleqtri 2827 | . 2 ⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) ∈ {∅} |
8 | fvex 6915 | . . 3 ⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) ∈ V | |
9 | 8 | elsn 4647 | . 2 ⊢ (((𝐴 ∖ ◡◡𝐴)‘𝑋) ∈ {∅} ↔ ((𝐴 ∖ ◡◡𝐴)‘𝑋) = ∅) |
10 | 7, 9 | mpbi 229 | 1 ⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 ∖ cdif 3946 ∪ cun 3947 ⊆ wss 3949 ∅c0 4326 {csn 4632 ◡ccnv 5681 ran crn 5683 ‘cfv 6553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-xp 5688 df-rel 5689 df-cnv 5690 df-dm 5692 df-rn 5693 df-iota 6505 df-fv 6561 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |