Theorem ssref 23399

Description: A subcover is a refinement of the original cover. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)

Hypotheses

Ref	Expression
ssref.1	⊢ 𝑋 = ∪ 𝐴
ssref.2	⊢ 𝑌 = ∪ 𝐵

Assertion

Ref	Expression
ssref	⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐵 ∧ 𝑋 = 𝑌) → 𝐴Ref𝐵)

Proof of Theorem ssref

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqcom 2736	. . . 4 ⊢ (𝑋 = 𝑌 ↔ 𝑌 = 𝑋)
2	1	biimpi 216	. . 3 ⊢ (𝑋 = 𝑌 → 𝑌 = 𝑋)
3	2	3ad2ant3 1135	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐵 ∧ 𝑋 = 𝑌) → 𝑌 = 𝑋)
4		ssel2 3941	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
5	4	3ad2antl2 1187	. . . 4 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐵 ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
6		ssid 3969	. . . 4 ⊢ 𝑥 ⊆ 𝑥
7		sseq2 3973	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝑥))
8	7	rspcev 3588	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 ⊆ 𝑥) → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
9	5, 6, 8	sylancl 586	. . 3 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐵 ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
10	9	ralrimiva 3125	. 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 23396	. . 3 ⊢ (𝐴 ∈ 𝐶 → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)))
14	13	3ad2ant1 1133	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐵 ∧ 𝑋 = 𝑌) → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)))
15	3, 10, 14	mpbir2and 713	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐵 ∧ 𝑋 = 𝑌) → 𝐴Ref𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ⊆ wss 3914  ∪ cuni 4871   class class class wbr 5107  Refcref 23389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-xp 5644  df-rel 5645  df-ref 23392

This theorem is referenced by:  cmpcref  33840  refssfne  36346