Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 2nd0 | Structured version Visualization version GIF version |
Description: The value of the second-member function at the empty set. (Contributed by NM, 23-Apr-2007.) |
Ref | Expression |
---|---|
2nd0 | ⊢ (2nd ‘∅) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2ndval 7761 | . 2 ⊢ (2nd ‘∅) = ∪ ran {∅} | |
2 | dmsn0 6069 | . . . 4 ⊢ dom {∅} = ∅ | |
3 | dm0rn0 5791 | . . . 4 ⊢ (dom {∅} = ∅ ↔ ran {∅} = ∅) | |
4 | 2, 3 | mpbi 233 | . . 3 ⊢ ran {∅} = ∅ |
5 | 4 | unieqi 4829 | . 2 ⊢ ∪ ran {∅} = ∪ ∅ |
6 | uni0 4846 | . 2 ⊢ ∪ ∅ = ∅ | |
7 | 1, 5, 6 | 3eqtri 2769 | 1 ⊢ (2nd ‘∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∅c0 4234 {csn 4538 ∪ cuni 4816 dom cdm 5548 ran crn 5549 ‘cfv 6377 2nd c2nd 7757 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5189 ax-nul 5196 ax-pr 5319 ax-un 7520 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-ral 3063 df-rex 3064 df-rab 3067 df-v 3407 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-nul 4235 df-if 4437 df-sn 4539 df-pr 4541 df-op 4545 df-uni 4817 df-br 5051 df-opab 5113 df-mpt 5133 df-id 5452 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6335 df-fun 6379 df-fv 6385 df-2nd 7759 |
This theorem is referenced by: smfval 28683 |
Copyright terms: Public domain | W3C validator |