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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > crefdf | Structured version Visualization version GIF version |
Description: A formulation of crefi 31200 easier to use for definitions. (Contributed by Thierry Arnoux, 7-Jan-2020.) |
Ref | Expression |
---|---|
crefi.x | ⊢ 𝑋 = ∪ 𝐽 |
crefdf.b | ⊢ 𝐵 = CovHasRef𝐴 |
crefdf.p | ⊢ (𝑧 ∈ 𝐴 → 𝜑) |
Ref | Expression |
---|---|
crefdf | ⊢ ((𝐽 ∈ 𝐵 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → ∃𝑧 ∈ 𝒫 𝐽(𝜑 ∧ 𝑧Ref𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crefdf.b | . . . 4 ⊢ 𝐵 = CovHasRef𝐴 | |
2 | 1 | eleq2i 2881 | . . 3 ⊢ (𝐽 ∈ 𝐵 ↔ 𝐽 ∈ CovHasRef𝐴) |
3 | crefi.x | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
4 | 3 | crefi 31200 | . . 3 ⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝐶) |
5 | 2, 4 | syl3an1b 1400 | . 2 ⊢ ((𝐽 ∈ 𝐵 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝐶) |
6 | elin 3897 | . . . . . 6 ⊢ (𝑧 ∈ (𝒫 𝐽 ∩ 𝐴) ↔ (𝑧 ∈ 𝒫 𝐽 ∧ 𝑧 ∈ 𝐴)) | |
7 | crefdf.p | . . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → 𝜑) | |
8 | 7 | anim2i 619 | . . . . . 6 ⊢ ((𝑧 ∈ 𝒫 𝐽 ∧ 𝑧 ∈ 𝐴) → (𝑧 ∈ 𝒫 𝐽 ∧ 𝜑)) |
9 | 6, 8 | sylbi 220 | . . . . 5 ⊢ (𝑧 ∈ (𝒫 𝐽 ∩ 𝐴) → (𝑧 ∈ 𝒫 𝐽 ∧ 𝜑)) |
10 | 9 | anim1i 617 | . . . 4 ⊢ ((𝑧 ∈ (𝒫 𝐽 ∩ 𝐴) ∧ 𝑧Ref𝐶) → ((𝑧 ∈ 𝒫 𝐽 ∧ 𝜑) ∧ 𝑧Ref𝐶)) |
11 | anass 472 | . . . 4 ⊢ (((𝑧 ∈ 𝒫 𝐽 ∧ 𝜑) ∧ 𝑧Ref𝐶) ↔ (𝑧 ∈ 𝒫 𝐽 ∧ (𝜑 ∧ 𝑧Ref𝐶))) | |
12 | 10, 11 | sylib 221 | . . 3 ⊢ ((𝑧 ∈ (𝒫 𝐽 ∩ 𝐴) ∧ 𝑧Ref𝐶) → (𝑧 ∈ 𝒫 𝐽 ∧ (𝜑 ∧ 𝑧Ref𝐶))) |
13 | 12 | reximi2 3207 | . 2 ⊢ (∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝐶 → ∃𝑧 ∈ 𝒫 𝐽(𝜑 ∧ 𝑧Ref𝐶)) |
14 | 5, 13 | syl 17 | 1 ⊢ ((𝐽 ∈ 𝐵 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → ∃𝑧 ∈ 𝒫 𝐽(𝜑 ∧ 𝑧Ref𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 ∩ cin 3880 ⊆ wss 3881 𝒫 cpw 4497 ∪ cuni 4800 class class class wbr 5030 Refcref 22107 CovHasRefccref 31195 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-cref 31196 |
This theorem is referenced by: cmpfiref 31204 ldlfcntref 31207 |
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