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Mirrors > Home > ILE Home > Th. List > isarep2 | Unicode version |
Description: Part of a study of the Axiom of Replacement used by the Isabelle prover. In Isabelle, the sethood of PrimReplace is apparently postulated implicitly by its type signature " i, i, i => o => i", which automatically asserts that it is a set without using any axioms. To prove that it is a set in Metamath, we need the hypotheses of Isabelle's "Axiom of Replacement" as well as the Axiom of Replacement in the form funimaex 5208. (Contributed by NM, 26-Oct-2006.) |
Ref | Expression |
---|---|
isarep2.1 | |
isarep2.2 |
Ref | Expression |
---|---|
isarep2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resima 4852 | . . . 4 | |
2 | resopab 4863 | . . . . 5 | |
3 | 2 | imaeq1i 4878 | . . . 4 |
4 | 1, 3 | eqtr3i 2162 | . . 3 |
5 | funopab 5158 | . . . . 5 | |
6 | isarep2.2 | . . . . . . . 8 | |
7 | 6 | rspec 2484 | . . . . . . 7 |
8 | nfv 1508 | . . . . . . . 8 | |
9 | 8 | mo3 2053 | . . . . . . 7 |
10 | 7, 9 | sylibr 133 | . . . . . 6 |
11 | moanimv 2074 | . . . . . 6 | |
12 | 10, 11 | mpbir 145 | . . . . 5 |
13 | 5, 12 | mpgbir 1429 | . . . 4 |
14 | isarep2.1 | . . . . 5 | |
15 | 14 | funimaex 5208 | . . . 4 |
16 | 13, 15 | ax-mp 5 | . . 3 |
17 | 4, 16 | eqeltri 2212 | . 2 |
18 | 17 | isseti 2694 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wal 1329 wceq 1331 wex 1468 wcel 1480 wsb 1735 wmo 2000 wral 2416 cvv 2686 copab 3988 cres 4541 cima 4542 wfun 5117 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-fun 5125 |
This theorem is referenced by: (None) |
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