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Mirrors > Home > MPE Home > Th. List > isarep2 | Structured version Visualization version 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 6223. (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 5682 | . . . 4 ⊢ (({〈𝑥, 𝑦〉 ∣ 𝜑} ↾ 𝐴) “ 𝐴) = ({〈𝑥, 𝑦〉 ∣ 𝜑} “ 𝐴) | |
2 | resopab 5698 | . . . . 5 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
3 | 2 | imaeq1i 5719 | . . . 4 ⊢ (({〈𝑥, 𝑦〉 ∣ 𝜑} ↾ 𝐴) “ 𝐴) = ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} “ 𝐴) |
4 | 1, 3 | eqtr3i 2804 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} “ 𝐴) = ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} “ 𝐴) |
5 | funopab 6172 | . . . . 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 3113 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦∀𝑧((𝜑 ∧ [𝑧 / 𝑦]𝜑) → 𝑦 = 𝑧)) |
8 | nfv 1957 | . . . . . . . 8 ⊢ Ⅎ𝑧𝜑 | |
9 | 8 | mo3 2580 | . . . . . . 7 ⊢ (∃*𝑦𝜑 ↔ ∀𝑦∀𝑧((𝜑 ∧ [𝑧 / 𝑦]𝜑) → 𝑦 = 𝑧)) |
10 | 7, 9 | sylibr 226 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ∃*𝑦𝜑) |
11 | moanimv 2654 | . . . . . 6 ⊢ (∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 → ∃*𝑦𝜑)) | |
12 | 10, 11 | mpbir 223 | . . . . 5 ⊢ ∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) |
13 | 5, 12 | mpgbir 1843 | . . . 4 ⊢ Fun {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} |
14 | isarep2.1 | . . . . 5 ⊢ 𝐴 ∈ V | |
15 | 14 | funimaex 6223 | . . . 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 2855 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} “ 𝐴) ∈ V |
18 | 17 | isseti 3411 | 1 ⊢ ∃𝑤 𝑤 = ({〈𝑥, 𝑦〉 ∣ 𝜑} “ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∀wal 1599 = wceq 1601 ∃wex 1823 [wsb 2011 ∈ wcel 2107 ∃*wmo 2549 ∀wral 3090 Vcvv 3398 {copab 4950 ↾ cres 5359 “ cima 5360 Fun wfun 6131 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pr 5140 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4889 df-opab 4951 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-fun 6139 |
This theorem is referenced by: (None) |
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