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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 6655. (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 6033 | . . . 4 ⊢ (({〈𝑥, 𝑦〉 ∣ 𝜑} ↾ 𝐴) “ 𝐴) = ({〈𝑥, 𝑦〉 ∣ 𝜑} “ 𝐴) | |
| 2 | resopab 6052 | . . . . 5 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
| 3 | 2 | imaeq1i 6075 | . . . 4 ⊢ (({〈𝑥, 𝑦〉 ∣ 𝜑} ↾ 𝐴) “ 𝐴) = ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} “ 𝐴) | 
| 4 | 1, 3 | eqtr3i 2767 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} “ 𝐴) = ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} “ 𝐴) | 
| 5 | funopab 6601 | . . . . 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 3250 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦∀𝑧((𝜑 ∧ [𝑧 / 𝑦]𝜑) → 𝑦 = 𝑧)) | 
| 8 | nfv 1914 | . . . . . . . 8 ⊢ Ⅎ𝑧𝜑 | |
| 9 | 8 | mo3 2564 | . . . . . . 7 ⊢ (∃*𝑦𝜑 ↔ ∀𝑦∀𝑧((𝜑 ∧ [𝑧 / 𝑦]𝜑) → 𝑦 = 𝑧)) | 
| 10 | 7, 9 | sylibr 234 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ∃*𝑦𝜑) | 
| 11 | moanimv 2619 | . . . . . 6 ⊢ (∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 → ∃*𝑦𝜑)) | |
| 12 | 10, 11 | mpbir 231 | . . . . 5 ⊢ ∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) | 
| 13 | 5, 12 | mpgbir 1799 | . . . 4 ⊢ Fun {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | 
| 14 | isarep2.1 | . . . . 5 ⊢ 𝐴 ∈ V | |
| 15 | 14 | funimaex 6655 | . . . 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 2837 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} “ 𝐴) ∈ V | 
| 18 | 17 | isseti 3498 | 1 ⊢ ∃𝑤 𝑤 = ({〈𝑥, 𝑦〉 ∣ 𝜑} “ 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1538 = wceq 1540 ∃wex 1779 [wsb 2064 ∈ wcel 2108 ∃*wmo 2538 ∀wral 3061 Vcvv 3480 {copab 5205 ↾ cres 5687 “ cima 5688 Fun wfun 6555 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-fun 6563 | 
| This theorem is referenced by: (None) | 
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