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Mirrors > Home > MPE Home > Th. List > isarep1 | Structured version Visualization version GIF version |
Description: Part of a study of the Axiom of Replacement used by the Isabelle prover. The object PrimReplace is apparently the image of the function encoded by 𝜑(𝑥, 𝑦) i.e. the class ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴). If so, we can prove Isabelle's "Axiom of Replacement" conclusion without using the Axiom of Replacement, for which I (N. Megill) currently have no explanation. (Contributed by NM, 26-Oct-2006.) (Proof shortened by Mario Carneiro, 4-Dec-2016.) (Proof shortened by SN, 19-Dec-2024.) |
Ref | Expression |
---|---|
isarep1 | ⊢ (𝑏 ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 [𝑏 / 𝑦]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3448 | . . 3 ⊢ 𝑏 ∈ V | |
2 | 1 | elima 6019 | . 2 ⊢ (𝑏 ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) ↔ ∃𝑧 ∈ 𝐴 𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑏) |
3 | df-br 5107 | . . . 4 ⊢ (𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑏 ↔ ⟨𝑧, 𝑏⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) | |
4 | vopelopabsb 5487 | . . . 4 ⊢ (⟨𝑧, 𝑏⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑) | |
5 | 3, 4 | bitri 275 | . . 3 ⊢ (𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑏 ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑) |
6 | 5 | rexbii 3094 | . 2 ⊢ (∃𝑧 ∈ 𝐴 𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑏 ↔ ∃𝑧 ∈ 𝐴 [𝑧 / 𝑥][𝑏 / 𝑦]𝜑) |
7 | nfs1v 2154 | . . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑥][𝑏 / 𝑦]𝜑 | |
8 | nfv 1918 | . . 3 ⊢ Ⅎ𝑧[𝑏 / 𝑦]𝜑 | |
9 | sbequ12r 2245 | . . 3 ⊢ (𝑧 = 𝑥 → ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑)) | |
10 | 7, 8, 9 | cbvrexw 3289 | . 2 ⊢ (∃𝑧 ∈ 𝐴 [𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ ∃𝑥 ∈ 𝐴 [𝑏 / 𝑦]𝜑) |
11 | 2, 6, 10 | 3bitri 297 | 1 ⊢ (𝑏 ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 [𝑏 / 𝑦]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 [wsb 2068 ∈ wcel 2107 ∃wrex 3070 ⟨cop 4593 class class class wbr 5106 {copab 5168 “ cima 5637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-xp 5640 df-cnv 5642 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 |
This theorem is referenced by: (None) |
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