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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 3484 | . . 3 ⊢ 𝑏 ∈ V | |
| 2 | 1 | elima 6083 | . 2 ⊢ (𝑏 ∈ ({〈𝑥, 𝑦〉 ∣ 𝜑} “ 𝐴) ↔ ∃𝑧 ∈ 𝐴 𝑧{〈𝑥, 𝑦〉 ∣ 𝜑}𝑏) | 
| 3 | df-br 5144 | . . . 4 ⊢ (𝑧{〈𝑥, 𝑦〉 ∣ 𝜑}𝑏 ↔ 〈𝑧, 𝑏〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
| 4 | vopelopabsb 5534 | . . . 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 2156 | . . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑥][𝑏 / 𝑦]𝜑 | |
| 8 | nfv 1914 | . . 3 ⊢ Ⅎ𝑧[𝑏 / 𝑦]𝜑 | |
| 9 | sbequ12r 2252 | . . 3 ⊢ (𝑧 = 𝑥 → ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑)) | |
| 10 | 7, 8, 9 | cbvrexw 3307 | . 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 206 [wsb 2064 ∈ wcel 2108 ∃wrex 3070 〈cop 4632 class class class wbr 5143 {copab 5205 “ cima 5688 | 
| 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-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-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-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 | 
| This theorem is referenced by: (None) | 
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