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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 3497 | . . 3 ⊢ 𝑏 ∈ V | |
2 | 1 | elima 5934 | . 2 ⊢ (𝑏 ∈ ({〈𝑥, 𝑦〉 ∣ 𝜑} “ 𝐴) ↔ ∃𝑧 ∈ 𝐴 𝑧{〈𝑥, 𝑦〉 ∣ 𝜑}𝑏) |
3 | df-br 5067 | . . . 4 ⊢ (𝑧{〈𝑥, 𝑦〉 ∣ 𝜑}𝑏 ↔ 〈𝑧, 𝑏〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
4 | opelopabsb 5417 | . . . 4 ⊢ (〈𝑧, 𝑏〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑) | |
5 | sbsbc 3776 | . . . . . 6 ⊢ ([𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑) | |
6 | 5 | sbbii 2081 | . . . . 5 ⊢ ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑) |
7 | sbsbc 3776 | . . . . 5 ⊢ ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑) | |
8 | 6, 7 | bitr2i 278 | . . . 4 ⊢ ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑) |
9 | 3, 4, 8 | 3bitri 299 | . . 3 ⊢ (𝑧{〈𝑥, 𝑦〉 ∣ 𝜑}𝑏 ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑) |
10 | 9 | rexbii 3247 | . 2 ⊢ (∃𝑧 ∈ 𝐴 𝑧{〈𝑥, 𝑦〉 ∣ 𝜑}𝑏 ↔ ∃𝑧 ∈ 𝐴 [𝑧 / 𝑥][𝑏 / 𝑦]𝜑) |
11 | nfs1v 2160 | . . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑥][𝑏 / 𝑦]𝜑 | |
12 | nfv 1915 | . . 3 ⊢ Ⅎ𝑧[𝑏 / 𝑦]𝜑 | |
13 | sbequ12r 2254 | . . 3 ⊢ (𝑧 = 𝑥 → ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑)) | |
14 | 11, 12, 13 | cbvrexw 3442 | . 2 ⊢ (∃𝑧 ∈ 𝐴 [𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ ∃𝑥 ∈ 𝐴 [𝑏 / 𝑦]𝜑) |
15 | 2, 10, 14 | 3bitri 299 | 1 ⊢ (𝑏 ∈ ({〈𝑥, 𝑦〉 ∣ 𝜑} “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 [𝑏 / 𝑦]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 [wsb 2069 ∈ wcel 2114 ∃wrex 3139 [wsbc 3772 〈cop 4573 class class class wbr 5066 {copab 5128 “ cima 5558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-cnv 5563 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 |
This theorem is referenced by: (None) |
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