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Mirrors > Home > ILE Home > Th. List > fnexALT | Unicode version |
Description: If the domain of a function is a set, the function is a set. Theorem 6.16(1) of [TakeutiZaring] p. 28. This theorem is derived using the Axiom of Replacement in the form of funimaexg 5177. This version of fnex 5610 uses ax-pow 4068 and ax-un 4325, whereas fnex 5610 does not. (Contributed by NM, 14-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
fnexALT |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnrel 5191 | . . . 4 | |
2 | relssdmrn 5029 | . . . 4 | |
3 | 1, 2 | syl 14 | . . 3 |
4 | 3 | adantr 274 | . 2 |
5 | fndm 5192 | . . . . 5 | |
6 | 5 | eleq1d 2186 | . . . 4 |
7 | 6 | biimpar 295 | . . 3 |
8 | fnfun 5190 | . . . . 5 | |
9 | funimaexg 5177 | . . . . 5 | |
10 | 8, 9 | sylan 281 | . . . 4 |
11 | imadmrn 4861 | . . . . . . 7 | |
12 | 5 | imaeq2d 4851 | . . . . . . 7 |
13 | 11, 12 | syl5eqr 2164 | . . . . . 6 |
14 | 13 | eleq1d 2186 | . . . . 5 |
15 | 14 | biimpar 295 | . . . 4 |
16 | 10, 15 | syldan 280 | . . 3 |
17 | xpexg 4623 | . . 3 | |
18 | 7, 16, 17 | syl2anc 408 | . 2 |
19 | ssexg 4037 | . 2 | |
20 | 4, 18, 19 | syl2anc 408 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 1465 cvv 2660 wss 3041 cxp 4507 cdm 4509 crn 4510 cima 4512 wrel 4514 wfun 5087 wfn 5088 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-fun 5095 df-fn 5096 |
This theorem is referenced by: (None) |
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