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Mirrors > Home > NFE Home > Th. List > fsn2 | Unicode version |
Description: A function that maps a singleton to a class is the singleton of an ordered pair. (Contributed by set.mm contributors, 19-May-2004.) |
Ref | Expression |
---|---|
fsn2.1 |
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Ref | Expression |
---|---|
fsn2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsn2.1 |
. . . . . 6
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2 | 1 | snid 3761 |
. . . . 5
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3 | ffvelrn 5416 |
. . . . 5
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4 | 2, 3 | mpan2 652 |
. . . 4
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5 | ffn 5224 |
. . . . 5
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6 | dffn3 5230 |
. . . . . . 7
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7 | 6 | biimpi 186 |
. . . . . 6
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8 | imadmrn 5009 |
. . . . . . . . 9
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9 | fndm 5183 |
. . . . . . . . . 10
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10 | 9 | imaeq2d 4943 |
. . . . . . . . 9
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11 | 8, 10 | syl5eqr 2399 |
. . . . . . . 8
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12 | fnsnfv 5374 |
. . . . . . . . 9
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13 | 2, 12 | mpan2 652 |
. . . . . . . 8
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14 | 11, 13 | eqtr4d 2388 |
. . . . . . 7
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15 | feq3 5213 |
. . . . . . 7
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16 | 14, 15 | syl 15 |
. . . . . 6
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17 | 7, 16 | mpbid 201 |
. . . . 5
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18 | 5, 17 | syl 15 |
. . . 4
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19 | 4, 18 | jca 518 |
. . 3
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20 | snssi 3853 |
. . . 4
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21 | fss 5231 |
. . . . 5
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22 | 21 | ancoms 439 |
. . . 4
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23 | 20, 22 | sylan 457 |
. . 3
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24 | 19, 23 | impbii 180 |
. 2
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25 | fvex 5340 |
. . . 4
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26 | 1, 25 | fsn 5433 |
. . 3
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27 | 26 | anbi2i 675 |
. 2
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28 | 24, 27 | bitri 240 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4079 ax-xp 4080 ax-cnv 4081 ax-1c 4082 ax-sset 4083 ax-si 4084 ax-ins2 4085 ax-ins3 4086 ax-typlower 4087 ax-sn 4088 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2479 df-ne 2519 df-ral 2620 df-rex 2621 df-reu 2622 df-rmo 2623 df-rab 2624 df-v 2862 df-sbc 3048 df-nin 3212 df-compl 3213 df-in 3214 df-un 3215 df-dif 3216 df-symdif 3217 df-ss 3260 df-pss 3262 df-nul 3552 df-if 3664 df-pw 3725 df-sn 3742 df-pr 3743 df-uni 3893 df-int 3928 df-opk 4059 df-1c 4137 df-pw1 4138 df-uni1 4139 df-xpk 4186 df-cnvk 4187 df-ins2k 4188 df-ins3k 4189 df-imak 4190 df-cok 4191 df-p6 4192 df-sik 4193 df-ssetk 4194 df-imagek 4195 df-idk 4196 df-iota 4340 df-0c 4378 df-addc 4379 df-nnc 4380 df-fin 4381 df-lefin 4441 df-ltfin 4442 df-ncfin 4443 df-tfin 4444 df-evenfin 4445 df-oddfin 4446 df-sfin 4447 df-spfin 4448 df-phi 4566 df-op 4567 df-proj1 4568 df-proj2 4569 df-opab 4624 df-br 4641 df-co 4727 df-ima 4728 df-id 4768 df-xp 4785 df-cnv 4786 df-rn 4787 df-dm 4788 df-res 4789 df-fun 4790 df-fn 4791 df-f 4792 df-f1 4793 df-fo 4794 df-f1o 4795 df-fv 4796 |
This theorem is referenced by: fnressn 5439 fressnfv 5440 1cnc 6140 |
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