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| Mirrors > Home > ILE Home > Th. List > resfunexgALT | Unicode version | ||
| Description: The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28. This version has a shorter proof than resfunexg 5805 but requires ax-pow 4218 and ax-un 4480. (Contributed by NM, 7-Apr-1995.) (Proof modification is discouraged.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| resfunexgALT |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dmresexg 4982 |
. . . 4
| |
| 2 | 1 | adantl 277 |
. . 3
|
| 3 | df-ima 4688 |
. . . 4
| |
| 4 | funimaexg 5358 |
. . . 4
| |
| 5 | 3, 4 | eqeltrrid 2293 |
. . 3
|
| 6 | 2, 5 | jca 306 |
. 2
|
| 7 | xpexg 4789 |
. 2
| |
| 8 | relres 4987 |
. . . 4
| |
| 9 | relssdmrn 5203 |
. . . 4
| |
| 10 | 8, 9 | ax-mp 5 |
. . 3
|
| 11 | ssexg 4183 |
. . 3
| |
| 12 | 10, 11 | mpan 424 |
. 2
|
| 13 | 6, 7, 12 | 3syl 17 |
1
|
| 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-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-coll 4159 ax-sep 4162 ax-pow 4218 ax-pr 4253 ax-un 4480 |
| This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ral 2489 df-rex 2490 df-v 2774 df-un 3170 df-in 3172 df-ss 3179 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-br 4045 df-opab 4106 df-id 4340 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-rn 4686 df-res 4687 df-ima 4688 df-fun 5273 |
| This theorem is referenced by: (None) |
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