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Mirrors > Home > ILE Home > Th. List > 2nd0 | 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 | 0ex 4050 | . . 3 ⊢ ∅ ∈ V | |
2 | 2ndvalg 6034 | . . 3 ⊢ (∅ ∈ V → (2nd ‘∅) = ∪ ran {∅}) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (2nd ‘∅) = ∪ ran {∅} |
4 | dmsn0 5001 | . . . 4 ⊢ dom {∅} = ∅ | |
5 | dm0rn0 4751 | . . . 4 ⊢ (dom {∅} = ∅ ↔ ran {∅} = ∅) | |
6 | 4, 5 | mpbi 144 | . . 3 ⊢ ran {∅} = ∅ |
7 | 6 | unieqi 3741 | . 2 ⊢ ∪ ran {∅} = ∪ ∅ |
8 | uni0 3758 | . 2 ⊢ ∪ ∅ = ∅ | |
9 | 3, 7, 8 | 3eqtri 2162 | 1 ⊢ (2nd ‘∅) = ∅ |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 Vcvv 2681 ∅c0 3358 {csn 3522 ∪ cuni 3731 dom cdm 4534 ran crn 4535 ‘cfv 5118 2nd c2nd 6030 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-iota 5083 df-fun 5120 df-fv 5126 df-2nd 6032 |
This theorem is referenced by: (None) |
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