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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 4103 | . . 3 ⊢ ∅ ∈ V | |
2 | 2ndvalg 6103 | . . 3 ⊢ (∅ ∈ V → (2nd ‘∅) = ∪ ran {∅}) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (2nd ‘∅) = ∪ ran {∅} |
4 | dmsn0 5065 | . . . 4 ⊢ dom {∅} = ∅ | |
5 | dm0rn0 4815 | . . . 4 ⊢ (dom {∅} = ∅ ↔ ran {∅} = ∅) | |
6 | 4, 5 | mpbi 144 | . . 3 ⊢ ran {∅} = ∅ |
7 | 6 | unieqi 3793 | . 2 ⊢ ∪ ran {∅} = ∪ ∅ |
8 | uni0 3810 | . 2 ⊢ ∪ ∅ = ∅ | |
9 | 3, 7, 8 | 3eqtri 2189 | 1 ⊢ (2nd ‘∅) = ∅ |
Colors of variables: wff set class |
Syntax hints: = wceq 1342 ∈ wcel 2135 Vcvv 2721 ∅c0 3404 {csn 3570 ∪ cuni 3783 dom cdm 4598 ran crn 4599 ‘cfv 5182 2nd c2nd 6099 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-iota 5147 df-fun 5184 df-fv 5190 df-2nd 6101 |
This theorem is referenced by: (None) |
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