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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 30755 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 2851 | . . 3 ⊢ (𝐽 ∈ 𝐵 ↔ 𝐽 ∈ CovHasRef𝐴) |
3 | crefi.x | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
4 | 3 | crefi 30755 | . . 3 ⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝐶) |
5 | 2, 4 | syl3an1b 1383 | . 2 ⊢ ((𝐽 ∈ 𝐵 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝐶) |
6 | elin 4051 | . . . . . 6 ⊢ (𝑧 ∈ (𝒫 𝐽 ∩ 𝐴) ↔ (𝑧 ∈ 𝒫 𝐽 ∧ 𝑧 ∈ 𝐴)) | |
7 | crefdf.p | . . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → 𝜑) | |
8 | 7 | anim2i 607 | . . . . . 6 ⊢ ((𝑧 ∈ 𝒫 𝐽 ∧ 𝑧 ∈ 𝐴) → (𝑧 ∈ 𝒫 𝐽 ∧ 𝜑)) |
9 | 6, 8 | sylbi 209 | . . . . 5 ⊢ (𝑧 ∈ (𝒫 𝐽 ∩ 𝐴) → (𝑧 ∈ 𝒫 𝐽 ∧ 𝜑)) |
10 | 9 | anim1i 605 | . . . 4 ⊢ ((𝑧 ∈ (𝒫 𝐽 ∩ 𝐴) ∧ 𝑧Ref𝐶) → ((𝑧 ∈ 𝒫 𝐽 ∧ 𝜑) ∧ 𝑧Ref𝐶)) |
11 | anass 461 | . . . 4 ⊢ (((𝑧 ∈ 𝒫 𝐽 ∧ 𝜑) ∧ 𝑧Ref𝐶) ↔ (𝑧 ∈ 𝒫 𝐽 ∧ (𝜑 ∧ 𝑧Ref𝐶))) | |
12 | 10, 11 | sylib 210 | . . 3 ⊢ ((𝑧 ∈ (𝒫 𝐽 ∩ 𝐴) ∧ 𝑧Ref𝐶) → (𝑧 ∈ 𝒫 𝐽 ∧ (𝜑 ∧ 𝑧Ref𝐶))) |
13 | 12 | reximi2 3185 | . 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 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 ∃wrex 3083 ∩ cin 3822 ⊆ wss 3823 𝒫 cpw 4416 ∪ cuni 4706 class class class wbr 4923 Refcref 21808 CovHasRefccref 30750 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-ext 2744 ax-sep 5054 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-br 4924 df-cref 30751 |
This theorem is referenced by: cmpfiref 30759 ldlfcntref 30762 |
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