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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 6435. (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 5881 | . . . 4 ⊢ (({〈𝑥, 𝑦〉 ∣ 𝜑} ↾ 𝐴) “ 𝐴) = ({〈𝑥, 𝑦〉 ∣ 𝜑} “ 𝐴) | |
2 | resopab 5896 | . . . . 5 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
3 | 2 | imaeq1i 5920 | . . . 4 ⊢ (({〈𝑥, 𝑦〉 ∣ 𝜑} ↾ 𝐴) “ 𝐴) = ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} “ 𝐴) |
4 | 1, 3 | eqtr3i 2846 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} “ 𝐴) = ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} “ 𝐴) |
5 | funopab 6384 | . . . . 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 3207 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦∀𝑧((𝜑 ∧ [𝑧 / 𝑦]𝜑) → 𝑦 = 𝑧)) |
8 | nfv 1911 | . . . . . . . 8 ⊢ Ⅎ𝑧𝜑 | |
9 | 8 | mo3 2644 | . . . . . . 7 ⊢ (∃*𝑦𝜑 ↔ ∀𝑦∀𝑧((𝜑 ∧ [𝑧 / 𝑦]𝜑) → 𝑦 = 𝑧)) |
10 | 7, 9 | sylibr 236 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ∃*𝑦𝜑) |
11 | moanimv 2700 | . . . . . 6 ⊢ (∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 → ∃*𝑦𝜑)) | |
12 | 10, 11 | mpbir 233 | . . . . 5 ⊢ ∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) |
13 | 5, 12 | mpgbir 1796 | . . . 4 ⊢ Fun {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} |
14 | isarep2.1 | . . . . 5 ⊢ 𝐴 ∈ V | |
15 | 14 | funimaex 6435 | . . . 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 2909 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} “ 𝐴) ∈ V |
18 | 17 | isseti 3508 | 1 ⊢ ∃𝑤 𝑤 = ({〈𝑥, 𝑦〉 ∣ 𝜑} “ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∀wal 1531 = wceq 1533 ∃wex 1776 [wsb 2065 ∈ wcel 2110 ∃*wmo 2616 ∀wral 3138 Vcvv 3494 {copab 5120 ↾ cres 5551 “ cima 5552 Fun wfun 6343 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-fun 6351 |
This theorem is referenced by: (None) |
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