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Mirrors > Home > ILE Home > Th. List > isarep2 | GIF 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 5283. (Contributed by NM, 26-Oct-2006.) |
Ref | Expression |
---|---|
isarep2.1 | ⊢ 𝐴 ∈ V |
isarep2.2 | ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦∀𝑧((𝜑 ∧ [𝑧 / 𝑦]𝜑) → 𝑦 = 𝑧) |
Ref | Expression |
---|---|
isarep2 | ⊢ ∃𝑤 𝑤 = ({〈𝑥, 𝑦〉 ∣ 𝜑} “ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resima 4924 | . . . 4 ⊢ (({〈𝑥, 𝑦〉 ∣ 𝜑} ↾ 𝐴) “ 𝐴) = ({〈𝑥, 𝑦〉 ∣ 𝜑} “ 𝐴) | |
2 | resopab 4935 | . . . . 5 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
3 | 2 | imaeq1i 4950 | . . . 4 ⊢ (({〈𝑥, 𝑦〉 ∣ 𝜑} ↾ 𝐴) “ 𝐴) = ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} “ 𝐴) |
4 | 1, 3 | eqtr3i 2193 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} “ 𝐴) = ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} “ 𝐴) |
5 | funopab 5233 | . . . . 5 ⊢ (Fun {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ ∀𝑥∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
6 | isarep2.2 | . . . . . . . 8 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦∀𝑧((𝜑 ∧ [𝑧 / 𝑦]𝜑) → 𝑦 = 𝑧) | |
7 | 6 | rspec 2522 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦∀𝑧((𝜑 ∧ [𝑧 / 𝑦]𝜑) → 𝑦 = 𝑧)) |
8 | nfv 1521 | . . . . . . . 8 ⊢ Ⅎ𝑧𝜑 | |
9 | 8 | mo3 2073 | . . . . . . 7 ⊢ (∃*𝑦𝜑 ↔ ∀𝑦∀𝑧((𝜑 ∧ [𝑧 / 𝑦]𝜑) → 𝑦 = 𝑧)) |
10 | 7, 9 | sylibr 133 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ∃*𝑦𝜑) |
11 | moanimv 2094 | . . . . . 6 ⊢ (∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 → ∃*𝑦𝜑)) | |
12 | 10, 11 | mpbir 145 | . . . . 5 ⊢ ∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) |
13 | 5, 12 | mpgbir 1446 | . . . 4 ⊢ Fun {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} |
14 | isarep2.1 | . . . . 5 ⊢ 𝐴 ∈ V | |
15 | 14 | funimaex 5283 | . . . 4 ⊢ (Fun {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} “ 𝐴) ∈ V) |
16 | 13, 15 | ax-mp 5 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} “ 𝐴) ∈ V |
17 | 4, 16 | eqeltri 2243 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} “ 𝐴) ∈ V |
18 | 17 | isseti 2738 | 1 ⊢ ∃𝑤 𝑤 = ({〈𝑥, 𝑦〉 ∣ 𝜑} “ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1346 = wceq 1348 ∃wex 1485 [wsb 1755 ∃*wmo 2020 ∈ wcel 2141 ∀wral 2448 Vcvv 2730 {copab 4049 ↾ cres 4613 “ cima 4614 Fun wfun 5192 |
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-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
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-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-fun 5200 |
This theorem is referenced by: (None) |
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