Theorem ssref 22117

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 2805	. . . 4 ⊢ (𝑋 = 𝑌 ↔ 𝑌 = 𝑋)
2	1	biimpi 219	. . 3 ⊢ (𝑋 = 𝑌 → 𝑌 = 𝑋)
3	2	3ad2ant3 1132	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐵 ∧ 𝑋 = 𝑌) → 𝑌 = 𝑋)
4		ssel2 3910	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
5	4	3ad2antl2 1183	. . . 4 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐵 ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
6		ssid 3937	. . . 4 ⊢ 𝑥 ⊆ 𝑥
7		sseq2 3941	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝑥))
8	7	rspcev 3571	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 ⊆ 𝑥) → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
9	5, 6, 8	sylancl 589	. . 3 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐵 ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
10	9	ralrimiva 3149	. 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 22114	. . 3 ⊢ (𝐴 ∈ 𝐶 → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)))
14	13	3ad2ant1 1130	. 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)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   ⊆ wss 3881  ∪ cuni 4800   class class class wbr 5030  Refcref 22107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-xp 5525  df-rel 5526  df-ref 22110

This theorem is referenced by:  cmpcref  31203  refssfne  33819