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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 6609 | . 2 ⊢ Fun ◡{〈0, ( I ‘𝐴)〉} | |
2 | df-s1 14604 | . . . 4 ⊢ 〈“𝐴”〉 = {〈0, ( I ‘𝐴)〉} | |
3 | 2 | cnveqi 5881 | . . 3 ⊢ ◡〈“𝐴”〉 = ◡{〈0, ( I ‘𝐴)〉} |
4 | 3 | funeqi 6580 | . 2 ⊢ (Fun ◡〈“𝐴”〉 ↔ Fun ◡{〈0, ( I ‘𝐴)〉}) |
5 | 1, 4 | mpbir 230 | 1 ⊢ Fun ◡〈“𝐴”〉 |
Colors of variables: wff setvar class |
Syntax hints: {csn 4633 〈cop 4639 I cid 5579 ◡ccnv 5681 Fun wfun 6548 ‘cfv 6554 0cc0 11158 〈“cs1 14603 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-mo 2529 df-clab 2704 df-cleq 2718 df-clel 2803 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-br 5154 df-opab 5216 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-fun 6556 df-s1 14604 |
This theorem is referenced by: uhgrwkspthlem1 29690 1trld 30075 |
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