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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 5203. (Contributed by NM, 26-Oct-2006.) |
Ref | Expression |
---|---|
isarep2.1 | |
isarep2.2 |
Ref | Expression |
---|---|
isarep2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resima 4847 | . . . 4 | |
2 | resopab 4858 | . . . . 5 | |
3 | 2 | imaeq1i 4873 | . . . 4 |
4 | 1, 3 | eqtr3i 2160 | . . 3 |
5 | funopab 5153 | . . . . 5 | |
6 | isarep2.2 | . . . . . . . 8 | |
7 | 6 | rspec 2482 | . . . . . . 7 |
8 | nfv 1508 | . . . . . . . 8 | |
9 | 8 | mo3 2051 | . . . . . . 7 |
10 | 7, 9 | sylibr 133 | . . . . . 6 |
11 | moanimv 2072 | . . . . . 6 | |
12 | 10, 11 | mpbir 145 | . . . . 5 |
13 | 5, 12 | mpgbir 1429 | . . . 4 |
14 | isarep2.1 | . . . . 5 | |
15 | 14 | funimaex 5203 | . . . 4 |
16 | 13, 15 | ax-mp 5 | . . 3 |
17 | 4, 16 | eqeltri 2210 | . 2 |
18 | 17 | isseti 2689 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wal 1329 wceq 1331 wex 1468 wcel 1480 wsb 1735 wmo 1998 wral 2414 cvv 2681 copab 3983 cres 4536 cima 4537 wfun 5112 |
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 2119 ax-coll 4038 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-fun 5120 |
This theorem is referenced by: (None) |
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