Theorem reftr 23457

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 2736	. . . 4 ⊢ ∪ 𝐵 = ∪ 𝐵
2		eqid 2736	. . . 4 ⊢ ∪ 𝐶 = ∪ 𝐶
3	1, 2	refbas 23453	. . 3 ⊢ (𝐵Ref𝐶 → ∪ 𝐶 = ∪ 𝐵)
4		eqid 2736	. . . 4 ⊢ ∪ 𝐴 = ∪ 𝐴
5	4, 1	refbas 23453	. . 3 ⊢ (𝐴Ref𝐵 → ∪ 𝐵 = ∪ 𝐴)
6	3, 5	sylan9eqr 2793	. 2 ⊢ ((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) → ∪ 𝐶 = ∪ 𝐴)
7		refssex 23454	. . . . . 6 ⊢ ((𝐴Ref𝐵 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
8	7	ex 412	. . . . 5 ⊢ (𝐴Ref𝐵 → (𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦))
9	8	adantr 480	. . . 4 ⊢ ((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) → (𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦))
10		refssex 23454	. . . . . . 7 ⊢ ((𝐵Ref𝐶 ∧ 𝑦 ∈ 𝐵) → ∃𝑧 ∈ 𝐶 𝑦 ⊆ 𝑧)
11	10	ad2ant2lr 748	. . . . . 6 ⊢ (((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ⊆ 𝑦)) → ∃𝑧 ∈ 𝐶 𝑦 ⊆ 𝑧)
12		sstr2 3970	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝑦 → (𝑦 ⊆ 𝑧 → 𝑥 ⊆ 𝑧))
13	12	reximdv 3156	. . . . . . 7 ⊢ (𝑥 ⊆ 𝑦 → (∃𝑧 ∈ 𝐶 𝑦 ⊆ 𝑧 → ∃𝑧 ∈ 𝐶 𝑥 ⊆ 𝑧))
14	13	ad2antll 729	. . . . . 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 3143	. . . 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 3132	. 2 ⊢ ((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) → ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐶 𝑥 ⊆ 𝑧)
19		refrel 23451	. . . . 5 ⊢ Rel Ref
20	19	brrelex1i 5715	. . . 4 ⊢ (𝐴Ref𝐵 → 𝐴 ∈ V)
21	20	adantr 480	. . 3 ⊢ ((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) → 𝐴 ∈ V)
22	4, 2	isref 23452	. . 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 713	1 ⊢ ((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) → 𝐴Ref𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3052  ∃wrex 3061  Vcvv 3464   ⊆ wss 3931  ∪ cuni 4888   class class class wbr 5124  Refcref 23445

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 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

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 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-xp 5665  df-rel 5666  df-ref 23448

This theorem is referenced by:  refssfne  36381