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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 4984. (Contributed by NM, 26-Oct-2006.) |
Ref | Expression |
---|---|
isarep2.1 | |
isarep2.2 |
Ref | Expression |
---|---|
isarep2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resima 4643 | . . . 4 | |
2 | resopab 4652 | . . . . 5 | |
3 | 2 | imaeq1i 4665 | . . . 4 |
4 | 1, 3 | eqtr3i 2062 | . . 3 |
5 | funopab 4935 | . . . . 5 | |
6 | isarep2.2 | . . . . . . . 8 | |
7 | 6 | rspec 2373 | . . . . . . 7 |
8 | nfv 1421 | . . . . . . . 8 | |
9 | 8 | mo3 1954 | . . . . . . 7 |
10 | 7, 9 | sylibr 137 | . . . . . 6 |
11 | moanimv 1975 | . . . . . 6 | |
12 | 10, 11 | mpbir 134 | . . . . 5 |
13 | 5, 12 | mpgbir 1342 | . . . 4 |
14 | isarep2.1 | . . . . 5 | |
15 | 14 | funimaex 4984 | . . . 4 |
16 | 13, 15 | ax-mp 7 | . . 3 |
17 | 4, 16 | eqeltri 2110 | . 2 |
18 | 17 | isseti 2563 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wal 1241 wceq 1243 wex 1381 wcel 1393 wsb 1645 wmo 1901 wral 2306 cvv 2557 copab 3817 cres 4347 cima 4348 wfun 4896 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-fun 4904 |
This theorem is referenced by: (None) |
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