![]() |
Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > imanonrel | Structured version Visualization version GIF version |
Description: An image under the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.) |
Ref | Expression |
---|---|
imanonrel | ⊢ ((𝐴 ∖ ◡◡𝐴) “ 𝐵) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ima 5712 | . 2 ⊢ ((𝐴 ∖ ◡◡𝐴) “ 𝐵) = ran ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) | |
2 | resnonrel 43495 | . . 3 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) = ∅ | |
3 | 2 | rneqi 5961 | . 2 ⊢ ran ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) = ran ∅ |
4 | rn0 5949 | . 2 ⊢ ran ∅ = ∅ | |
5 | 1, 3, 4 | 3eqtri 2766 | 1 ⊢ ((𝐴 ∖ ◡◡𝐴) “ 𝐵) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∖ cdif 3967 ∅c0 4347 ◡ccnv 5698 ran crn 5700 ↾ cres 5701 “ cima 5702 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pr 5450 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3439 df-v 3484 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-br 5170 df-opab 5232 df-xp 5705 df-rel 5706 df-cnv 5707 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |