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Mirrors > Home > MPE Home > Th. List > funcnvs1 | Structured version Visualization version GIF version |
Description: The converse of a singleton word is a function. (Contributed by AV, 22-Jan-2021.) |
Ref | Expression |
---|---|
funcnvs1 | ⊢ Fun ◡〈“𝐴”〉 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funcnvsn 5850 | . 2 ⊢ Fun ◡{〈0, ( I ‘𝐴)〉} | |
2 | df-s1 13157 | . . . 4 ⊢ 〈“𝐴”〉 = {〈0, ( I ‘𝐴)〉} | |
3 | 2 | cnveqi 5219 | . . 3 ⊢ ◡〈“𝐴”〉 = ◡{〈0, ( I ‘𝐴)〉} |
4 | 3 | funeqi 5824 | . 2 ⊢ (Fun ◡〈“𝐴”〉 ↔ Fun ◡{〈0, ( I ‘𝐴)〉}) |
5 | 1, 4 | mpbir 220 | 1 ⊢ Fun ◡〈“𝐴”〉 |
Colors of variables: wff setvar class |
Syntax hints: {csn 4125 〈cop 4131 I cid 4948 ◡ccnv 5037 Fun wfun 5798 ‘cfv 5804 0cc0 9815 〈“cs1 13149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-fun 5806 df-s1 13157 |
This theorem is referenced by: uhgr1wlkspthlem1 40959 1trld 41309 |
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