![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ssref | Structured version Visualization version GIF version |
Description: A subcover is a refinement of the original cover. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.) |
Ref | Expression |
---|---|
ssref.1 | ⊢ 𝑋 = ∪ 𝐴 |
ssref.2 | ⊢ 𝑌 = ∪ 𝐵 |
Ref | Expression |
---|---|
ssref | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐵 ∧ 𝑋 = 𝑌) → 𝐴Ref𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcom 2740 | . . . 4 ⊢ (𝑋 = 𝑌 ↔ 𝑌 = 𝑋) | |
2 | 1 | biimpi 215 | . . 3 ⊢ (𝑋 = 𝑌 → 𝑌 = 𝑋) |
3 | 2 | 3ad2ant3 1136 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐵 ∧ 𝑋 = 𝑌) → 𝑌 = 𝑋) |
4 | ssel2 3943 | . . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵) | |
5 | 4 | 3ad2antl2 1187 | . . . 4 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐵 ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵) |
6 | ssid 3970 | . . . 4 ⊢ 𝑥 ⊆ 𝑥 | |
7 | sseq2 3974 | . . . . 5 ⊢ (𝑦 = 𝑥 → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝑥)) | |
8 | 7 | rspcev 3583 | . . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 ⊆ 𝑥) → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦) |
9 | 5, 6, 8 | sylancl 587 | . . 3 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐵 ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦) |
10 | 9 | ralrimiva 3140 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐵 ∧ 𝑋 = 𝑌) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦) |
11 | ssref.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐴 | |
12 | ssref.2 | . . . 4 ⊢ 𝑌 = ∪ 𝐵 | |
13 | 11, 12 | isref 22883 | . . 3 ⊢ (𝐴 ∈ 𝐶 → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦))) |
14 | 13 | 3ad2ant1 1134 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐵 ∧ 𝑋 = 𝑌) → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦))) |
15 | 3, 10, 14 | mpbir2and 712 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐵 ∧ 𝑋 = 𝑌) → 𝐴Ref𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3061 ∃wrex 3070 ⊆ wss 3914 ∪ cuni 4869 class class class wbr 5109 Refcref 22876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-xp 5643 df-rel 5644 df-ref 22879 |
This theorem is referenced by: cmpcref 32495 refssfne 34883 |
Copyright terms: Public domain | W3C validator |