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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.) |
Ref | Expression |
---|---|
isarep1 | ⊢ (𝑏 ∈ ({〈𝑥, 𝑦〉 ∣ 𝜑} “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 [𝑏 / 𝑦]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3440 | . . 3 ⊢ 𝑏 ∈ V | |
2 | 1 | elima 5811 | . 2 ⊢ (𝑏 ∈ ({〈𝑥, 𝑦〉 ∣ 𝜑} “ 𝐴) ↔ ∃𝑧 ∈ 𝐴 𝑧{〈𝑥, 𝑦〉 ∣ 𝜑}𝑏) |
3 | df-br 4963 | . . . 4 ⊢ (𝑧{〈𝑥, 𝑦〉 ∣ 𝜑}𝑏 ↔ 〈𝑧, 𝑏〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
4 | opelopabsb 5307 | . . . 4 ⊢ (〈𝑧, 𝑏〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑) | |
5 | sbsbc 3710 | . . . . . 6 ⊢ ([𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑) | |
6 | 5 | sbbii 2054 | . . . . 5 ⊢ ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑) |
7 | sbsbc 3710 | . . . . 5 ⊢ ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑) | |
8 | 6, 7 | bitr2i 277 | . . . 4 ⊢ ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑) |
9 | 3, 4, 8 | 3bitri 298 | . . 3 ⊢ (𝑧{〈𝑥, 𝑦〉 ∣ 𝜑}𝑏 ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑) |
10 | 9 | rexbii 3211 | . 2 ⊢ (∃𝑧 ∈ 𝐴 𝑧{〈𝑥, 𝑦〉 ∣ 𝜑}𝑏 ↔ ∃𝑧 ∈ 𝐴 [𝑧 / 𝑥][𝑏 / 𝑦]𝜑) |
11 | nfs1v 2237 | . . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑥][𝑏 / 𝑦]𝜑 | |
12 | nfv 1892 | . . 3 ⊢ Ⅎ𝑧[𝑏 / 𝑦]𝜑 | |
13 | sbequ12r 2217 | . . 3 ⊢ (𝑧 = 𝑥 → ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑)) | |
14 | 11, 12, 13 | cbvrex 3400 | . 2 ⊢ (∃𝑧 ∈ 𝐴 [𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ ∃𝑥 ∈ 𝐴 [𝑏 / 𝑦]𝜑) |
15 | 2, 10, 14 | 3bitri 298 | 1 ⊢ (𝑏 ∈ ({〈𝑥, 𝑦〉 ∣ 𝜑} “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 [𝑏 / 𝑦]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 [wsb 2042 ∈ wcel 2081 ∃wrex 3106 [wsbc 3706 〈cop 4478 class class class wbr 4962 {copab 5024 “ cima 5446 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pr 5221 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-sbc 3707 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-op 4479 df-br 4963 df-opab 5025 df-xp 5449 df-cnv 5451 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 |
This theorem is referenced by: (None) |
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