Theorem mptrcllem 40849

Description: Show two versions of a closure with reflexive properties are equal. (Contributed by RP, 19-Oct-2020.)

Hypotheses

Ref	Expression
mptrcllem.ex1	⊢ (𝑥 ∈ 𝑉 → ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} ∈ V)
mptrcllem.ex2	⊢ (𝑥 ∈ 𝑉 → ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ 𝜓)} ∈ V)
mptrcllem.hyp1	⊢ (𝑥 ∈ 𝑉 → 𝜒)
mptrcllem.hyp2	⊢ (𝑥 ∈ 𝑉 → 𝜃)
mptrcllem.hyp3	⊢ (𝑥 ∈ 𝑉 → 𝜏)
mptrcllem.sub1	⊢ (𝑦 = ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ 𝜓)} → (𝜑 ↔ 𝜒))
mptrcllem.sub2	⊢ (𝑦 = ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ 𝜓)} → (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ↔ 𝜃))
mptrcllem.sub3	⊢ (𝑧 = ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} → (𝜓 ↔ 𝜏))

Assertion

Ref	Expression
mptrcllem	⊢ (𝑥 ∈ 𝑉 ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))}) = (𝑥 ∈ 𝑉 ↦ ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ 𝜓)})

Distinct variable groups:   𝑥,𝑦,𝑧,𝑉   𝜑,𝑥,𝑧   𝜓,𝑥,𝑦   𝜒,𝑦   𝜃,𝑦   𝜏,𝑧

Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑧)   𝜒(𝑥,𝑧)   𝜃(𝑥,𝑧)   𝜏(𝑥,𝑦)

Proof of Theorem mptrcllem

Step	Hyp	Ref	Expression
1		mptrcllem.ex2	. . . 4 ⊢ (𝑥 ∈ 𝑉 → ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ 𝜓)} ∈ V)
2		mptrcllem.sub1	. . . . 5 ⊢ (𝑦 = ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ 𝜓)} → (𝜑 ↔ 𝜒))
3		mptrcllem.sub2	. . . . 5 ⊢ (𝑦 = ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ 𝜓)} → (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ↔ 𝜃))
4	2, 3	anbi12d 634	. . . 4 ⊢ (𝑦 = ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ 𝜓)} → ((𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦) ↔ (𝜒 ∧ 𝜃)))
5		id 22	. . . . . . . . 9 ⊢ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 → (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧)
6	5	unssad 4091	. . . . . . . 8 ⊢ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 → 𝑥 ⊆ 𝑧)
7	6	adantr 484	. . . . . . 7 ⊢ (((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ 𝜓) → 𝑥 ⊆ 𝑧)
8	7	a1i 11	. . . . . 6 ⊢ (𝑥 ∈ 𝑉 → (((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ 𝜓) → 𝑥 ⊆ 𝑧))
9	8	alrimiv 1935	. . . . 5 ⊢ (𝑥 ∈ 𝑉 → ∀𝑧(((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ 𝜓) → 𝑥 ⊆ 𝑧))
10		ssintab 4866	. . . . 5 ⊢ (𝑥 ⊆ ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ 𝜓)} ↔ ∀𝑧(((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ 𝜓) → 𝑥 ⊆ 𝑧))
11	9, 10	sylibr 237	. . . 4 ⊢ (𝑥 ∈ 𝑉 → 𝑥 ⊆ ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ 𝜓)})
12		mptrcllem.hyp1	. . . . 5 ⊢ (𝑥 ∈ 𝑉 → 𝜒)
13		mptrcllem.hyp2	. . . . 5 ⊢ (𝑥 ∈ 𝑉 → 𝜃)
14	12, 13	jca 515	. . . 4 ⊢ (𝑥 ∈ 𝑉 → (𝜒 ∧ 𝜃))
15	1, 4, 11, 14	clublem 40846	. . 3 ⊢ (𝑥 ∈ 𝑉 → ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} ⊆ ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ 𝜓)})
16		mptrcllem.ex1	. . . 4 ⊢ (𝑥 ∈ 𝑉 → ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} ∈ V)
17		mptrcllem.sub3	. . . 4 ⊢ (𝑧 = ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} → (𝜓 ↔ 𝜏))
18		simpl 486	. . . . . . . . 9 ⊢ ((𝑥 ⊆ 𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)) → 𝑥 ⊆ 𝑦)
19		dmss 5760	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊆ 𝑦 → dom 𝑥 ⊆ dom 𝑦)
20		rnss 5797	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊆ 𝑦 → ran 𝑥 ⊆ ran 𝑦)
21	19, 20	jca 515	. . . . . . . . . . . 12 ⊢ (𝑥 ⊆ 𝑦 → (dom 𝑥 ⊆ dom 𝑦 ∧ ran 𝑥 ⊆ ran 𝑦))
22		unss12 4086	. . . . . . . . . . . 12 ⊢ ((dom 𝑥 ⊆ dom 𝑦 ∧ ran 𝑥 ⊆ ran 𝑦) → (dom 𝑥 ∪ ran 𝑥) ⊆ (dom 𝑦 ∪ ran 𝑦))
23		ssres2 5868	. . . . . . . . . . . 12 ⊢ ((dom 𝑥 ∪ ran 𝑥) ⊆ (dom 𝑦 ∪ ran 𝑦) → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ ( I ↾ (dom 𝑦 ∪ ran 𝑦)))
24	21, 22, 23	3syl 18	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ 𝑦 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ ( I ↾ (dom 𝑦 ∪ ran 𝑦)))
25	24	adantr 484	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ 𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)) → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ ( I ↾ (dom 𝑦 ∪ ran 𝑦)))
26		simprr 773	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ 𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)) → ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)
27	25, 26	sstrd 3901	. . . . . . . . 9 ⊢ ((𝑥 ⊆ 𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)) → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑦)
28	18, 27	jca 515	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)) → (𝑥 ⊆ 𝑦 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑦))
29	28	a1i 11	. . . . . . 7 ⊢ (𝑥 ∈ 𝑉 → ((𝑥 ⊆ 𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)) → (𝑥 ⊆ 𝑦 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑦)))
30		unss 4088	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝑦 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑦) ↔ (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑦)
31	29, 30	syl6ib 254	. . . . . 6 ⊢ (𝑥 ∈ 𝑉 → ((𝑥 ⊆ 𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)) → (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑦))
32	31	alrimiv 1935	. . . . 5 ⊢ (𝑥 ∈ 𝑉 → ∀𝑦((𝑥 ⊆ 𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)) → (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑦))
33		ssintab 4866	. . . . 5 ⊢ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} ↔ ∀𝑦((𝑥 ⊆ 𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)) → (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑦))
34	32, 33	sylibr 237	. . . 4 ⊢ (𝑥 ∈ 𝑉 → (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))})
35		mptrcllem.hyp3	. . . 4 ⊢ (𝑥 ∈ 𝑉 → 𝜏)
36	16, 17, 34, 35	clublem 40846	. . 3 ⊢ (𝑥 ∈ 𝑉 → ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ 𝜓)} ⊆ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))})
37	15, 36	eqssd 3908	. 2 ⊢ (𝑥 ∈ 𝑉 → ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} = ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ 𝜓)})
38	37	mpteq2ia 5135	1 ⊢ (𝑥 ∈ 𝑉 ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))}) = (𝑥 ∈ 𝑉 ↦ ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ 𝜓)})

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1541   = wceq 1543   ∈ wcel 2110  {cab 2712  Vcvv 3401   ∪ cun 3855   ⊆ wss 3857  ∩ cint 4849   ↦ cmpt 5124   I cid 5443  dom cdm 5540  ran crn 5541   ↾ cres 5542

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-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706

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-nf 1792  df-sb 2071  df-clab 2713  df-cleq 2726  df-clel 2812  df-ral 3059  df-rab 3063  df-v 3403  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-nul 4228  df-if 4430  df-sn 4532  df-pr 4534  df-op 4538  df-int 4850  df-br 5044  df-opab 5106  df-mpt 5125  df-xp 5546  df-cnv 5548  df-dm 5550  df-rn 5551  df-res 5552

This theorem is referenced by:  dfrtrcl5  40865