Theorem reftr 23018

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 2733	. . . 4 ⊢ ∪ 𝐵 = ∪ 𝐵
2		eqid 2733	. . . 4 ⊢ ∪ 𝐶 = ∪ 𝐶
3	1, 2	refbas 23014	. . 3 ⊢ (𝐵Ref𝐶 → ∪ 𝐶 = ∪ 𝐵)
4		eqid 2733	. . . 4 ⊢ ∪ 𝐴 = ∪ 𝐴
5	4, 1	refbas 23014	. . 3 ⊢ (𝐴Ref𝐵 → ∪ 𝐵 = ∪ 𝐴)
6	3, 5	sylan9eqr 2795	. 2 ⊢ ((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) → ∪ 𝐶 = ∪ 𝐴)
7		refssex 23015	. . . . . 6 ⊢ ((𝐴Ref𝐵 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
8	7	ex 414	. . . . 5 ⊢ (𝐴Ref𝐵 → (𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦))
9	8	adantr 482	. . . 4 ⊢ ((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) → (𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦))
10		refssex 23015	. . . . . . 7 ⊢ ((𝐵Ref𝐶 ∧ 𝑦 ∈ 𝐵) → ∃𝑧 ∈ 𝐶 𝑦 ⊆ 𝑧)
11	10	ad2ant2lr 747	. . . . . 6 ⊢ (((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ⊆ 𝑦)) → ∃𝑧 ∈ 𝐶 𝑦 ⊆ 𝑧)
12		sstr2 3990	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝑦 → (𝑦 ⊆ 𝑧 → 𝑥 ⊆ 𝑧))
13	12	reximdv 3171	. . . . . . 7 ⊢ (𝑥 ⊆ 𝑦 → (∃𝑧 ∈ 𝐶 𝑦 ⊆ 𝑧 → ∃𝑧 ∈ 𝐶 𝑥 ⊆ 𝑧))
14	13	ad2antll 728	. . . . . 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 3157	. . . 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 3146	. 2 ⊢ ((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) → ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐶 𝑥 ⊆ 𝑧)
19		refrel 23012	. . . . 5 ⊢ Rel Ref
20	19	brrelex1i 5733	. . . 4 ⊢ (𝐴Ref𝐵 → 𝐴 ∈ V)
21	20	adantr 482	. . 3 ⊢ ((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) → 𝐴 ∈ V)
22	4, 2	isref 23013	. . 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 712	1 ⊢ ((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) → 𝐴Ref𝐶)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071  Vcvv 3475   ⊆ wss 3949  ∪ cuni 4909   class class class wbr 5149  Refcref 23006

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-ref 23009

This theorem is referenced by:  refssfne  35243