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Mirrors > Home > ILE Home > Th. List > isarep1 | 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.) |
Ref | Expression |
---|---|
isarep1 | ⊢ (𝑏 ∈ ({〈𝑥, 𝑦〉 ∣ 𝜑} “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 [𝑏 / 𝑦]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2692 | . . 3 ⊢ 𝑏 ∈ V | |
2 | 1 | elima 4894 | . 2 ⊢ (𝑏 ∈ ({〈𝑥, 𝑦〉 ∣ 𝜑} “ 𝐴) ↔ ∃𝑧 ∈ 𝐴 𝑧{〈𝑥, 𝑦〉 ∣ 𝜑}𝑏) |
3 | df-br 3938 | . . . 4 ⊢ (𝑧{〈𝑥, 𝑦〉 ∣ 𝜑}𝑏 ↔ 〈𝑧, 𝑏〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
4 | opelopabsb 4190 | . . . 4 ⊢ (〈𝑧, 𝑏〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑) | |
5 | sbsbc 2917 | . . . . . 6 ⊢ ([𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑) | |
6 | 5 | sbbii 1739 | . . . . 5 ⊢ ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑) |
7 | sbsbc 2917 | . . . . 5 ⊢ ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑) | |
8 | 6, 7 | bitr2i 184 | . . . 4 ⊢ ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑) |
9 | 3, 4, 8 | 3bitri 205 | . . 3 ⊢ (𝑧{〈𝑥, 𝑦〉 ∣ 𝜑}𝑏 ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑) |
10 | 9 | rexbii 2445 | . 2 ⊢ (∃𝑧 ∈ 𝐴 𝑧{〈𝑥, 𝑦〉 ∣ 𝜑}𝑏 ↔ ∃𝑧 ∈ 𝐴 [𝑧 / 𝑥][𝑏 / 𝑦]𝜑) |
11 | nfs1v 1913 | . . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑥][𝑏 / 𝑦]𝜑 | |
12 | nfv 1509 | . . 3 ⊢ Ⅎ𝑧[𝑏 / 𝑦]𝜑 | |
13 | sbequ12r 1746 | . . 3 ⊢ (𝑧 = 𝑥 → ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑)) | |
14 | 11, 12, 13 | cbvrex 2654 | . 2 ⊢ (∃𝑧 ∈ 𝐴 [𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ ∃𝑥 ∈ 𝐴 [𝑏 / 𝑦]𝜑) |
15 | 2, 10, 14 | 3bitri 205 | 1 ⊢ (𝑏 ∈ ({〈𝑥, 𝑦〉 ∣ 𝜑} “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 [𝑏 / 𝑦]𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∈ wcel 1481 [wsb 1736 ∃wrex 2418 [wsbc 2913 〈cop 3535 class class class wbr 3937 {copab 3996 “ cima 4550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-opab 3998 df-xp 4553 df-cnv 4555 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 |
This theorem is referenced by: (None) |
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