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Mirrors > Home > ILE Home > Th. List > funfvima3 | Unicode version |
Description: A class including a function contains the function's value in the image of the singleton of the argument. (Contributed by NM, 23-Mar-2004.) |
Ref | Expression |
---|---|
funfvima3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funfvop 5608 | . . . . . 6 | |
2 | ssel 3141 | . . . . . 6 | |
3 | 1, 2 | syl5 32 | . . . . 5 |
4 | 3 | imp 123 | . . . 4 |
5 | simpr 109 | . . . . . 6 | |
6 | sneq 3594 | . . . . . . . . . 10 | |
7 | 6 | imaeq2d 4953 | . . . . . . . . 9 |
8 | 7 | eleq2d 2240 | . . . . . . . 8 |
9 | opeq1 3765 | . . . . . . . . 9 | |
10 | 9 | eleq1d 2239 | . . . . . . . 8 |
11 | 8, 10 | bibi12d 234 | . . . . . . 7 |
12 | 11 | adantl 275 | . . . . . 6 |
13 | vex 2733 | . . . . . . 7 | |
14 | funfvex 5513 | . . . . . . 7 | |
15 | elimasng 4979 | . . . . . . 7 | |
16 | 13, 14, 15 | sylancr 412 | . . . . . 6 |
17 | 5, 12, 16 | vtocld 2782 | . . . . 5 |
18 | 17 | adantl 275 | . . . 4 |
19 | 4, 18 | mpbird 166 | . . 3 |
20 | 19 | exp32 363 | . 2 |
21 | 20 | impcom 124 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1348 wcel 2141 cvv 2730 wss 3121 csn 3583 cop 3586 cdm 4611 cima 4614 wfun 5192 cfv 5198 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-fv 5206 |
This theorem is referenced by: (None) |
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