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Mirrors > Home > ILE Home > Th. List > abrexex | Unicode version |
Description: Existence of a class abstraction of existentially restricted sets. is normally a free-variable parameter in the class expression substituted for , which can be thought of as . This simple-looking theorem is actually quite powerful and appears to involve the Axiom of Replacement in an intrinsic way, as can be seen by tracing back through the path mptexg 5719, funex 5717, fnex 5716, resfunexg 5715, and funimaexg 5280. See also abrexex2 6101. (Contributed by NM, 16-Oct-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
abrexex.1 |
Ref | Expression |
---|---|
abrexex |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2170 | . . 3 | |
2 | 1 | rnmpt 4857 | . 2 |
3 | abrexex.1 | . . . 4 | |
4 | 3 | mptex 5720 | . . 3 |
5 | 4 | rnex 4876 | . 2 |
6 | 2, 5 | eqeltrri 2244 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1348 wcel 2141 cab 2156 wrex 2449 cvv 2730 cmpt 4048 crn 4610 |
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-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 |
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-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 |
This theorem is referenced by: ab2rexex 6108 shftfval 10774 |
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