Theorem ssref 21125

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 2617	. . . 4 ⊢ (𝑋 = 𝑌 ↔ 𝑌 = 𝑋)
2	1	biimpi 205	. . 3 ⊢ (𝑋 = 𝑌 → 𝑌 = 𝑋)
3	2	3ad2ant3 1077	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐵 ∧ 𝑋 = 𝑌) → 𝑌 = 𝑋)
4		ssel2 3563	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
5	4	3ad2antl2 1217	. . . 4 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐵 ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
6		ssid 3587	. . . 4 ⊢ 𝑥 ⊆ 𝑥
7		sseq2 3590	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝑥))
8	7	rspcev 3282	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 ⊆ 𝑥) → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
9	5, 6, 8	sylancl 693	. . 3 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐵 ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
10	9	ralrimiva 2949	. 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 21122	. . 3 ⊢ (𝐴 ∈ 𝐶 → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)))
14	13	3ad2ant1 1075	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐵 ∧ 𝑋 = 𝑌) → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)))
15	3, 10, 14	mpbir2and 959	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐵 ∧ 𝑋 = 𝑌) → 𝐴Ref𝐵)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540  ∪ cuni 4372   class class class wbr 4583  Refcref 21115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-ref 21118

This theorem is referenced by:  cmpcref  29245  refssfne  31523