Theorem reftr 22411

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 2737	. . . 4 ⊢ ∪ 𝐵 = ∪ 𝐵
2		eqid 2737	. . . 4 ⊢ ∪ 𝐶 = ∪ 𝐶
3	1, 2	refbas 22407	. . 3 ⊢ (𝐵Ref𝐶 → ∪ 𝐶 = ∪ 𝐵)
4		eqid 2737	. . . 4 ⊢ ∪ 𝐴 = ∪ 𝐴
5	4, 1	refbas 22407	. . 3 ⊢ (𝐴Ref𝐵 → ∪ 𝐵 = ∪ 𝐴)
6	3, 5	sylan9eqr 2800	. 2 ⊢ ((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) → ∪ 𝐶 = ∪ 𝐴)
7		refssex 22408	. . . . . 6 ⊢ ((𝐴Ref𝐵 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
8	7	ex 416	. . . . 5 ⊢ (𝐴Ref𝐵 → (𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦))
9	8	adantr 484	. . . 4 ⊢ ((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) → (𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦))
10		refssex 22408	. . . . . . 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 3908	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝑦 → (𝑦 ⊆ 𝑧 → 𝑥 ⊆ 𝑧))
13	12	reximdv 3192	. . . . . . 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 3204	. . . 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 3104	. 2 ⊢ ((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) → ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐶 𝑥 ⊆ 𝑧)
19		refrel 22405	. . . . 5 ⊢ Rel Ref
20	19	brrelex1i 5605	. . . 4 ⊢ (𝐴Ref𝐵 → 𝐴 ∈ V)
21	20	adantr 484	. . 3 ⊢ ((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) → 𝐴 ∈ V)
22	4, 2	isref 22406	. . 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 209   ∧ wa 399   = wceq 1543   ∈ wcel 2110  ∀wral 3061  ∃wrex 3062  Vcvv 3408   ⊆ wss 3866  ∪ cuni 4819   class class class wbr 5053  Refcref 22399

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-xp 5557  df-rel 5558  df-ref 22402

This theorem is referenced by:  refssfne  34284