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Theorem mptrcllem 40310
Description: Show two versions of a closure with reflexive properties are equal. (Contributed by RP, 19-Oct-2020.)
Hypotheses
Ref Expression
mptrcllem.ex1 (𝑥𝑉 {𝑦 ∣ (𝑥𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} ∈ V)
mptrcllem.ex2 (𝑥𝑉 {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧𝜓)} ∈ V)
mptrcllem.hyp1 (𝑥𝑉𝜒)
mptrcllem.hyp2 (𝑥𝑉𝜃)
mptrcllem.hyp3 (𝑥𝑉𝜏)
mptrcllem.sub1 (𝑦 = {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧𝜓)} → (𝜑𝜒))
mptrcllem.sub2 (𝑦 = {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧𝜓)} → (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦𝜃))
mptrcllem.sub3 (𝑧 = {𝑦 ∣ (𝑥𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} → (𝜓𝜏))
Assertion
Ref Expression
mptrcllem (𝑥𝑉 {𝑦 ∣ (𝑥𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))}) = (𝑥𝑉 {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧𝜓)})
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
StepHypRef Expression
1 mptrcllem.ex2 . . . 4 (𝑥𝑉 {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧𝜓)} ∈ V)
2 mptrcllem.sub1 . . . . 5 (𝑦 = {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧𝜓)} → (𝜑𝜒))
3 mptrcllem.sub2 . . . . 5 (𝑦 = {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧𝜓)} → (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦𝜃))
42, 3anbi12d 633 . . . 4 (𝑦 = {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧𝜓)} → ((𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦) ↔ (𝜒𝜃)))
5 id 22 . . . . . . . . 9 ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 → (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧)
65unssad 4117 . . . . . . . 8 ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧𝑥𝑧)
76adantr 484 . . . . . . 7 (((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧𝜓) → 𝑥𝑧)
87a1i 11 . . . . . 6 (𝑥𝑉 → (((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧𝜓) → 𝑥𝑧))
98alrimiv 1928 . . . . 5 (𝑥𝑉 → ∀𝑧(((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧𝜓) → 𝑥𝑧))
10 ssintab 4858 . . . . 5 (𝑥 {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧𝜓)} ↔ ∀𝑧(((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧𝜓) → 𝑥𝑧))
119, 10sylibr 237 . . . 4 (𝑥𝑉𝑥 {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧𝜓)})
12 mptrcllem.hyp1 . . . . 5 (𝑥𝑉𝜒)
13 mptrcllem.hyp2 . . . . 5 (𝑥𝑉𝜃)
1412, 13jca 515 . . . 4 (𝑥𝑉 → (𝜒𝜃))
151, 4, 11, 14clublem 40307 . . 3 (𝑥𝑉 {𝑦 ∣ (𝑥𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} ⊆ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧𝜓)})
16 mptrcllem.ex1 . . . 4 (𝑥𝑉 {𝑦 ∣ (𝑥𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} ∈ V)
17 mptrcllem.sub3 . . . 4 (𝑧 = {𝑦 ∣ (𝑥𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} → (𝜓𝜏))
18 simpl 486 . . . . . . . . 9 ((𝑥𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)) → 𝑥𝑦)
19 dmss 5739 . . . . . . . . . . . . 13 (𝑥𝑦 → dom 𝑥 ⊆ dom 𝑦)
20 rnss 5777 . . . . . . . . . . . . 13 (𝑥𝑦 → ran 𝑥 ⊆ ran 𝑦)
2119, 20jca 515 . . . . . . . . . . . 12 (𝑥𝑦 → (dom 𝑥 ⊆ dom 𝑦 ∧ ran 𝑥 ⊆ ran 𝑦))
22 unss12 4112 . . . . . . . . . . . 12 ((dom 𝑥 ⊆ dom 𝑦 ∧ ran 𝑥 ⊆ ran 𝑦) → (dom 𝑥 ∪ ran 𝑥) ⊆ (dom 𝑦 ∪ ran 𝑦))
23 ssres2 5850 . . . . . . . . . . . 12 ((dom 𝑥 ∪ ran 𝑥) ⊆ (dom 𝑦 ∪ ran 𝑦) → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ ( I ↾ (dom 𝑦 ∪ ran 𝑦)))
2421, 22, 233syl 18 . . . . . . . . . . 11 (𝑥𝑦 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ ( I ↾ (dom 𝑦 ∪ ran 𝑦)))
2524adantr 484 . . . . . . . . . 10 ((𝑥𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)) → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ ( I ↾ (dom 𝑦 ∪ ran 𝑦)))
26 simprr 772 . . . . . . . . . 10 ((𝑥𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)) → ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)
2725, 26sstrd 3928 . . . . . . . . 9 ((𝑥𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)) → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑦)
2818, 27jca 515 . . . . . . . 8 ((𝑥𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)) → (𝑥𝑦 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑦))
2928a1i 11 . . . . . . 7 (𝑥𝑉 → ((𝑥𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)) → (𝑥𝑦 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑦)))
30 unss 4114 . . . . . . 7 ((𝑥𝑦 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑦) ↔ (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑦)
3129, 30syl6ib 254 . . . . . 6 (𝑥𝑉 → ((𝑥𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)) → (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑦))
3231alrimiv 1928 . . . . 5 (𝑥𝑉 → ∀𝑦((𝑥𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)) → (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑦))
33 ssintab 4858 . . . . 5 ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ {𝑦 ∣ (𝑥𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} ↔ ∀𝑦((𝑥𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)) → (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑦))
3432, 33sylibr 237 . . . 4 (𝑥𝑉 → (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ {𝑦 ∣ (𝑥𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))})
35 mptrcllem.hyp3 . . . 4 (𝑥𝑉𝜏)
3616, 17, 34, 35clublem 40307 . . 3 (𝑥𝑉 {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧𝜓)} ⊆ {𝑦 ∣ (𝑥𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))})
3715, 36eqssd 3935 . 2 (𝑥𝑉 {𝑦 ∣ (𝑥𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} = {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧𝜓)})
3837mpteq2ia 5124 1 (𝑥𝑉 {𝑦 ∣ (𝑥𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))}) = (𝑥𝑉 {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧𝜓)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536   = wceq 1538  wcel 2112  {cab 2779  Vcvv 3444  cun 3882  wss 3884   cint 4841  cmpt 5113   I cid 5427  dom cdm 5523  ran crn 5524  cres 5525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rab 3118  df-v 3446  df-un 3889  df-in 3891  df-ss 3901  df-sn 4529  df-pr 4531  df-op 4535  df-int 4842  df-br 5034  df-opab 5096  df-mpt 5114  df-xp 5529  df-cnv 5531  df-dm 5533  df-rn 5534  df-res 5535
This theorem is referenced by:  dfrtrcl5  40326
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