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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 5976. (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 5431 | . . . 4 ⊢ (({〈𝑥, 𝑦〉 ∣ 𝜑} ↾ 𝐴) “ 𝐴) = ({〈𝑥, 𝑦〉 ∣ 𝜑} “ 𝐴) | |
2 | resopab 5446 | . . . . 5 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
3 | 2 | imaeq1i 5463 | . . . 4 ⊢ (({〈𝑥, 𝑦〉 ∣ 𝜑} ↾ 𝐴) “ 𝐴) = ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} “ 𝐴) |
4 | 1, 3 | eqtr3i 2646 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} “ 𝐴) = ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} “ 𝐴) |
5 | funopab 5923 | . . . . 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 2931 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦∀𝑧((𝜑 ∧ [𝑧 / 𝑦]𝜑) → 𝑦 = 𝑧)) |
8 | nfv 1843 | . . . . . . . 8 ⊢ Ⅎ𝑧𝜑 | |
9 | 8 | mo3 2507 | . . . . . . 7 ⊢ (∃*𝑦𝜑 ↔ ∀𝑦∀𝑧((𝜑 ∧ [𝑧 / 𝑦]𝜑) → 𝑦 = 𝑧)) |
10 | 7, 9 | sylibr 224 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ∃*𝑦𝜑) |
11 | moanimv 2531 | . . . . . 6 ⊢ (∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 → ∃*𝑦𝜑)) | |
12 | 10, 11 | mpbir 221 | . . . . 5 ⊢ ∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) |
13 | 5, 12 | mpgbir 1726 | . . . 4 ⊢ Fun {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} |
14 | isarep2.1 | . . . . 5 ⊢ 𝐴 ∈ V | |
15 | 14 | funimaex 5976 | . . . 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 2697 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} “ 𝐴) ∈ V |
18 | 17 | isseti 3209 | 1 ⊢ ∃𝑤 𝑤 = ({〈𝑥, 𝑦〉 ∣ 𝜑} “ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∀wal 1481 = wceq 1483 ∃wex 1704 [wsb 1880 ∈ wcel 1990 ∃*wmo 2471 ∀wral 2912 Vcvv 3200 {copab 4712 ↾ cres 5116 “ cima 5117 Fun wfun 5882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-fun 5890 |
This theorem is referenced by: (None) |
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