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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 6148 | . 2 ⊢ Fun ◡{〈0, ( I ‘𝐴)〉} | |
2 | df-s1 13612 | . . . 4 ⊢ 〈“𝐴”〉 = {〈0, ( I ‘𝐴)〉} | |
3 | 2 | cnveqi 5498 | . . 3 ⊢ ◡〈“𝐴”〉 = ◡{〈0, ( I ‘𝐴)〉} |
4 | 3 | funeqi 6120 | . 2 ⊢ (Fun ◡〈“𝐴”〉 ↔ Fun ◡{〈0, ( I ‘𝐴)〉}) |
5 | 1, 4 | mpbir 223 | 1 ⊢ Fun ◡〈“𝐴”〉 |
Colors of variables: wff setvar class |
Syntax hints: {csn 4366 〈cop 4372 I cid 5217 ◡ccnv 5309 Fun wfun 6093 ‘cfv 6099 0cc0 10222 〈“cs1 13611 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2375 ax-ext 2775 ax-sep 4973 ax-nul 4981 ax-pr 5095 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ral 3092 df-rab 3096 df-v 3385 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-sn 4367 df-pr 4369 df-op 4373 df-br 4842 df-opab 4904 df-id 5218 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-fun 6101 df-s1 13612 |
This theorem is referenced by: uhgrwkspthlem1 26998 1trld 27477 |
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