Theorem ssref 22114

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 2828	. . . 4 ⊢ (𝑋 = 𝑌 ↔ 𝑌 = 𝑋)
2	1	biimpi 218	. . 3 ⊢ (𝑋 = 𝑌 → 𝑌 = 𝑋)
3	2	3ad2ant3 1131	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐵 ∧ 𝑋 = 𝑌) → 𝑌 = 𝑋)
4		ssel2 3962	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
5	4	3ad2antl2 1182	. . . 4 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐵 ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
6		ssid 3989	. . . 4 ⊢ 𝑥 ⊆ 𝑥
7		sseq2 3993	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝑥))
8	7	rspcev 3623	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 ⊆ 𝑥) → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
9	5, 6, 8	sylancl 588	. . 3 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐵 ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
10	9	ralrimiva 3182	. 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 22111	. . 3 ⊢ (𝐴 ∈ 𝐶 → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)))
14	13	3ad2ant1 1129	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐵 ∧ 𝑋 = 𝑌) → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)))
15	3, 10, 14	mpbir2and 711	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐵 ∧ 𝑋 = 𝑌) → 𝐴Ref𝐵)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139   ⊆ wss 3936  ∪ cuni 4832   class class class wbr 5059  Refcref 22104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-xp 5556  df-rel 5557  df-ref 22107

This theorem is referenced by:  cmpcref  31109  refssfne  33701