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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 7694 | . 2 ⊢ (2nd ‘∅) = ∪ ran {∅} | |
2 | dmsn0 6068 | . . . 4 ⊢ dom {∅} = ∅ | |
3 | dm0rn0 5797 | . . . 4 ⊢ (dom {∅} = ∅ ↔ ran {∅} = ∅) | |
4 | 2, 3 | mpbi 232 | . . 3 ⊢ ran {∅} = ∅ |
5 | 4 | unieqi 4853 | . 2 ⊢ ∪ ran {∅} = ∪ ∅ |
6 | uni0 4868 | . 2 ⊢ ∪ ∅ = ∅ | |
7 | 1, 5, 6 | 3eqtri 2850 | 1 ⊢ (2nd ‘∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∅c0 4293 {csn 4569 ∪ cuni 4840 dom cdm 5557 ran crn 5558 ‘cfv 6357 2nd c2nd 7690 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fv 6365 df-2nd 7692 |
This theorem is referenced by: smfval 28384 |
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