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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 6374 | . 2 ⊢ Fun ◡{〈0, ( I ‘𝐴)〉} | |
2 | df-s1 13941 | . . . 4 ⊢ 〈“𝐴”〉 = {〈0, ( I ‘𝐴)〉} | |
3 | 2 | cnveqi 5709 | . . 3 ⊢ ◡〈“𝐴”〉 = ◡{〈0, ( I ‘𝐴)〉} |
4 | 3 | funeqi 6345 | . 2 ⊢ (Fun ◡〈“𝐴”〉 ↔ Fun ◡{〈0, ( I ‘𝐴)〉}) |
5 | 1, 4 | mpbir 234 | 1 ⊢ Fun ◡〈“𝐴”〉 |
Colors of variables: wff setvar class |
Syntax hints: {csn 4525 〈cop 4531 I cid 5424 ◡ccnv 5518 Fun wfun 6318 ‘cfv 6324 0cc0 10526 〈“cs1 13940 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-fun 6326 df-s1 13941 |
This theorem is referenced by: uhgrwkspthlem1 27542 1trld 27927 |
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