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 5563 | . 2 ⊢ ((𝐴 ∖ ◡◡𝐴) “ 𝐵) = ran ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) | |
2 | resnonrel 39945 | . . 3 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) = ∅ | |
3 | 2 | rneqi 5802 | . 2 ⊢ ran ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) = ran ∅ |
4 | rn0 5791 | . 2 ⊢ ran ∅ = ∅ | |
5 | 1, 3, 4 | 3eqtri 2848 | 1 ⊢ ((𝐴 ∖ ◡◡𝐴) “ 𝐵) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∖ cdif 3933 ∅c0 4291 ◡ccnv 5549 ran crn 5551 ↾ cres 5552 “ cima 5553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-br 5060 df-opab 5122 df-xp 5556 df-rel 5557 df-cnv 5558 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |