Theorem reftr 23498

Description: Refinement is transitive. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)

Assertion

Ref	Expression
reftr	⊢ ((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) → 𝐴Ref𝐶)

Proof of Theorem reftr

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

Step	Hyp	Ref	Expression
1		eqid 2739	. . . 4 ⊢ ∪ 𝐵 = ∪ 𝐵
2		eqid 2739	. . . 4 ⊢ ∪ 𝐶 = ∪ 𝐶
3	1, 2	refbas 23494	. . 3 ⊢ (𝐵Ref𝐶 → ∪ 𝐶 = ∪ 𝐵)
4		eqid 2739	. . . 4 ⊢ ∪ 𝐴 = ∪ 𝐴
5	4, 1	refbas 23494	. . 3 ⊢ (𝐴Ref𝐵 → ∪ 𝐵 = ∪ 𝐴)
6	3, 5	sylan9eqr 2796	. 2 ⊢ ((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) → ∪ 𝐶 = ∪ 𝐴)
7		refssex 23495	. . . . . 6 ⊢ ((𝐴Ref𝐵 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
8	7	ex 413	. . . . 5 ⊢ (𝐴Ref𝐵 → (𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦))
9	8	adantr 481	. . . 4 ⊢ ((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) → (𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦))
10		refssex 23495	. . . . . . 7 ⊢ ((𝐵Ref𝐶 ∧ 𝑦 ∈ 𝐵) → ∃𝑧 ∈ 𝐶 𝑦 ⊆ 𝑧)
11	10	ad2ant2lr 754	. . . . . 6 ⊢ (((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ⊆ 𝑦)) → ∃𝑧 ∈ 𝐶 𝑦 ⊆ 𝑧)
12		sstr2 3922	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝑦 → (𝑦 ⊆ 𝑧 → 𝑥 ⊆ 𝑧))
13	12	reximdv 3154	. . . . . . 7 ⊢ (𝑥 ⊆ 𝑦 → (∃𝑧 ∈ 𝐶 𝑦 ⊆ 𝑧 → ∃𝑧 ∈ 𝐶 𝑥 ⊆ 𝑧))
14	13	ad2antll 735	. . . . . 6 ⊢ (((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ⊆ 𝑦)) → (∃𝑧 ∈ 𝐶 𝑦 ⊆ 𝑧 → ∃𝑧 ∈ 𝐶 𝑥 ⊆ 𝑧))
15	11, 14	mpd 15	. . . . 5 ⊢ (((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ⊆ 𝑦)) → ∃𝑧 ∈ 𝐶 𝑥 ⊆ 𝑧)
16	15	rexlimdvaa 3141	. . . 4 ⊢ ((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) → (∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦 → ∃𝑧 ∈ 𝐶 𝑥 ⊆ 𝑧))
17	9, 16	syld 47	. . 3 ⊢ ((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) → (𝑥 ∈ 𝐴 → ∃𝑧 ∈ 𝐶 𝑥 ⊆ 𝑧))
18	17	ralrimiv 3130	. 2 ⊢ ((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) → ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐶 𝑥 ⊆ 𝑧)
19		refrel 23492	. . . . 5 ⊢ Rel Ref
20	19	brrelex1i 5675	. . . 4 ⊢ (𝐴Ref𝐵 → 𝐴 ∈ V)
21	20	adantr 481	. . 3 ⊢ ((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) → 𝐴 ∈ V)
22	4, 2	isref 23493	. . 3 ⊢ (𝐴 ∈ V → (𝐴Ref𝐶 ↔ (∪ 𝐶 = ∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐶 𝑥 ⊆ 𝑧)))
23	21, 22	syl 17	. 2 ⊢ ((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) → (𝐴Ref𝐶 ↔ (∪ 𝐶 = ∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐶 𝑥 ⊆ 𝑧)))
24	6, 18, 23	mpbir2and 719	1 ⊢ ((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) → 𝐴Ref𝐶)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053  ∃wrex 3063  Vcvv 3431   ⊆ wss 3883  ∪ cuni 4839   class class class wbr 5073  Refcref 23486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5219  ax-pow 5295  ax-pr 5363  ax-un 7679

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-br 5074  df-opab 5136  df-xp 5625  df-rel 5626  df-ref 23489

This theorem is referenced by:  refssfne  36595