Theorem mptrcllem 44379

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 644	. . . 4 ⊢ (𝑦 = ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ 𝜓)} → ((𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦) ↔ (𝜒 ∧ 𝜃)))
5		id 23	. . . . . . . . 9 ⊢ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 → (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧)
6	5	unssad 4149	. . . . . . . 8 ⊢ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 → 𝑥 ⊆ 𝑧)
7	6	adantr 486	. . . . . . 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 1960	. . . . 5 ⊢ (𝑥 ∈ 𝑉 → ∀𝑧(((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ 𝜓) → 𝑥 ⊆ 𝑧))
10		ssintab 4935	. . . . 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 521	. . . 4 ⊢ (𝑥 ∈ 𝑉 → (𝜒 ∧ 𝜃))
15	1, 4, 11, 14	clublem 44376	. . 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 488	. . . . . . . . 9 ⊢ ((𝑥 ⊆ 𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)) → 𝑥 ⊆ 𝑦)
19		dmss 5897	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊆ 𝑦 → dom 𝑥 ⊆ dom 𝑦)
20		rnss 5934	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊆ 𝑦 → ran 𝑥 ⊆ ran 𝑦)
21	19, 20	jca 521	. . . . . . . . . . . 12 ⊢ (𝑥 ⊆ 𝑦 → (dom 𝑥 ⊆ dom 𝑦 ∧ ran 𝑥 ⊆ ran 𝑦))
22		unss12 4144	. . . . . . . . . . . 12 ⊢ ((dom 𝑥 ⊆ dom 𝑦 ∧ ran 𝑥 ⊆ ran 𝑦) → (dom 𝑥 ∪ ran 𝑥) ⊆ (dom 𝑦 ∪ ran 𝑦))
23		ssres2 6008	. . . . . . . . . . . 12 ⊢ ((dom 𝑥 ∪ ran 𝑥) ⊆ (dom 𝑦 ∪ ran 𝑦) → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ ( I ↾ (dom 𝑦 ∪ ran 𝑦)))
24	21, 22, 23	3syl 19	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ 𝑦 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ ( I ↾ (dom 𝑦 ∪ ran 𝑦)))
25	24	adantr 486	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ 𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)) → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ ( I ↾ (dom 𝑦 ∪ ran 𝑦)))
26		simprr 785	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ 𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)) → ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)
27	25, 26	sstrd 3950	. . . . . . . . 9 ⊢ ((𝑥 ⊆ 𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)) → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑦)
28	18, 27	jca 521	. . . . . . . 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 4146	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝑦 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑦) ↔ (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑦)
31	29, 30	imbitrdi 254	. . . . . 6 ⊢ (𝑥 ∈ 𝑉 → ((𝑥 ⊆ 𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)) → (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑦))
32	31	alrimiv 1960	. . . . 5 ⊢ (𝑥 ∈ 𝑉 → ∀𝑦((𝑥 ⊆ 𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)) → (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑦))
33		ssintab 4935	. . . . 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 44376	. . 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 3957	. 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 5211	1 ⊢ (𝑥 ∈ 𝑉 ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))}) = (𝑥 ∈ 𝑉 ↦ ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ 𝜓)})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2146  {cab 2744  Vcvv 3458   ∪ cun 3906   ⊆ wss 3908  ∩ cint 4917   ↦ cmpt 5197   I cid 5560  dom cdm 5666  ran crn 5667   ↾ cres 5668

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-int 4918  df-br 5115  df-opab 5179  df-mpt 5198  df-xp 5672  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678

This theorem is used by:  dfrtrcl5  44395