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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 6404 | . 2 ⊢ Fun ◡{〈0, ( I ‘𝐴)〉} | |
2 | df-s1 13950 | . . . 4 ⊢ 〈“𝐴”〉 = {〈0, ( I ‘𝐴)〉} | |
3 | 2 | cnveqi 5745 | . . 3 ⊢ ◡〈“𝐴”〉 = ◡{〈0, ( I ‘𝐴)〉} |
4 | 3 | funeqi 6376 | . 2 ⊢ (Fun ◡〈“𝐴”〉 ↔ Fun ◡{〈0, ( I ‘𝐴)〉}) |
5 | 1, 4 | mpbir 233 | 1 ⊢ Fun ◡〈“𝐴”〉 |
Colors of variables: wff setvar class |
Syntax hints: {csn 4567 〈cop 4573 I cid 5459 ◡ccnv 5554 Fun wfun 6349 ‘cfv 6355 0cc0 10537 〈“cs1 13949 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-fun 6357 df-s1 13950 |
This theorem is referenced by: uhgrwkspthlem1 27534 1trld 27921 |
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