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Theorem mptrcllem 38446
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 616 . . . 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 3941 . . . . . . . 8 ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧𝑥𝑧)
76adantr 466 . . . . . . 7 (((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧𝜓) → 𝑥𝑧)
87a1i 11 . . . . . 6 (𝑥𝑉 → (((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧𝜓) → 𝑥𝑧))
98alrimiv 2007 . . . . 5 (𝑥𝑉 → ∀𝑧(((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧𝜓) → 𝑥𝑧))
10 ssintab 4629 . . . . 5 (𝑥 {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧𝜓)} ↔ ∀𝑧(((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧𝜓) → 𝑥𝑧))
119, 10sylibr 224 . . . 4 (𝑥𝑉𝑥 {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧𝜓)})
12 mptrcllem.hyp1 . . . . 5 (𝑥𝑉𝜒)
13 mptrcllem.hyp2 . . . . 5 (𝑥𝑉𝜃)
1412, 13jca 501 . . . 4 (𝑥𝑉 → (𝜒𝜃))
151, 4, 11, 14clublem 38443 . . 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 468 . . . . . . . . 9 ((𝑥𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)) → 𝑥𝑦)
19 dmss 5460 . . . . . . . . . . . . 13 (𝑥𝑦 → dom 𝑥 ⊆ dom 𝑦)
20 rnss 5491 . . . . . . . . . . . . 13 (𝑥𝑦 → ran 𝑥 ⊆ ran 𝑦)
2119, 20jca 501 . . . . . . . . . . . 12 (𝑥𝑦 → (dom 𝑥 ⊆ dom 𝑦 ∧ ran 𝑥 ⊆ ran 𝑦))
22 unss12 3936 . . . . . . . . . . . 12 ((dom 𝑥 ⊆ dom 𝑦 ∧ ran 𝑥 ⊆ ran 𝑦) → (dom 𝑥 ∪ ran 𝑥) ⊆ (dom 𝑦 ∪ ran 𝑦))
23 ssres2 5565 . . . . . . . . . . . 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 466 . . . . . . . . . 10 ((𝑥𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)) → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ ( I ↾ (dom 𝑦 ∪ ran 𝑦)))
26 simprr 756 . . . . . . . . . 10 ((𝑥𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)) → ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)
2725, 26sstrd 3762 . . . . . . . . 9 ((𝑥𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)) → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑦)
2818, 27jca 501 . . . . . . . 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 3938 . . . . . . 7 ((𝑥𝑦 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑦) ↔ (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑦)
3129, 30syl6ib 241 . . . . . 6 (𝑥𝑉 → ((𝑥𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)) → (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑦))
3231alrimiv 2007 . . . . 5 (𝑥𝑉 → ∀𝑦((𝑥𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)) → (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑦))
33 ssintab 4629 . . . . 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 224 . . . 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 38443 . . 3 (𝑥𝑉 {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧𝜓)} ⊆ {𝑦 ∣ (𝑥𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))})
3715, 36eqssd 3769 . 2 (𝑥𝑉 {𝑦 ∣ (𝑥𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} = {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧𝜓)})
3837mpteq2ia 4875 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 196  wa 382  wal 1629   = wceq 1631  wcel 2145  {cab 2757  Vcvv 3351  cun 3721  wss 3723   cint 4612  cmpt 4864   I cid 5157  dom cdm 5250  ran crn 5251  cres 5252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-int 4613  df-br 4788  df-opab 4848  df-mpt 4865  df-xp 5256  df-cnv 5258  df-dm 5260  df-rn 5261  df-res 5262
This theorem is referenced by:  dfrtrcl5  38462
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