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Mirrors > Home > ILE Home > Th. List > mpteq12f | Unicode version |
Description: An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) |
Ref | Expression |
---|---|
mpteq12f |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfa1 1541 |
. . . 4
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2 | nfra1 2508 |
. . . 4
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3 | 1, 2 | nfan 1565 |
. . 3
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4 | nfv 1528 |
. . 3
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5 | rsp 2524 |
. . . . . . 7
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6 | 5 | imp 124 |
. . . . . 6
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7 | 6 | eqeq2d 2189 |
. . . . 5
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8 | 7 | pm5.32da 452 |
. . . 4
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9 | sp 1511 |
. . . . . 6
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10 | 9 | eleq2d 2247 |
. . . . 5
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11 | 10 | anbi1d 465 |
. . . 4
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12 | 8, 11 | sylan9bbr 463 |
. . 3
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13 | 3, 4, 12 | opabbid 4065 |
. 2
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14 | df-mpt 4063 |
. 2
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15 | df-mpt 4063 |
. 2
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16 | 13, 14, 15 | 3eqtr4g 2235 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-11 1506 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-ral 2460 df-opab 4062 df-mpt 4063 |
This theorem is referenced by: mpteq12dva 4081 mpteq12 4083 mpteq2ia 4086 mpteq2da 4089 |
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