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Theorem bropfvvvvlem 8079
Description: Lemma for bropfvvvv 8080. (Contributed by AV, 31-Dec-2020.) (Revised by AV, 16-Jan-2021.)
Hypotheses
Ref Expression
bropfvvvv.o 𝑂 = (𝑎𝑈 ↦ (𝑏𝑉, 𝑐𝑊 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜑}))
bropfvvvv.oo ((𝐴𝑈𝐵𝑆𝐶𝑇) → (𝐵(𝑂𝐴)𝐶) = {⟨𝑑, 𝑒⟩ ∣ 𝜃})
Assertion
Ref Expression
bropfvvvvlem ((⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑇) ∧ 𝐷(𝐵(𝑂𝐴)𝐶)𝐸) → (𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)))
Distinct variable group:   𝑈,𝑎
Allowed substitution hints:   𝜑(𝑒,𝑎,𝑏,𝑐,𝑑)   𝜃(𝑒,𝑎,𝑏,𝑐,𝑑)   𝐴(𝑒,𝑎,𝑏,𝑐,𝑑)   𝐵(𝑒,𝑎,𝑏,𝑐,𝑑)   𝐶(𝑒,𝑎,𝑏,𝑐,𝑑)   𝐷(𝑒,𝑎,𝑏,𝑐,𝑑)   𝑆(𝑒,𝑎,𝑏,𝑐,𝑑)   𝑇(𝑒,𝑎,𝑏,𝑐,𝑑)   𝑈(𝑒,𝑏,𝑐,𝑑)   𝐸(𝑒,𝑎,𝑏,𝑐,𝑑)   𝑂(𝑒,𝑎,𝑏,𝑐,𝑑)   𝑉(𝑒,𝑎,𝑏,𝑐,𝑑)   𝑊(𝑒,𝑎,𝑏,𝑐,𝑑)

Proof of Theorem bropfvvvvlem
StepHypRef Expression
1 opelxp 5712 . . 3 (⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑇) ↔ (𝐵𝑆𝐶𝑇))
2 brne0 5198 . . . . . . 7 (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (𝐵(𝑂𝐴)𝐶) ≠ ∅)
3 bropfvvvv.oo . . . . . . . . . . . . . 14 ((𝐴𝑈𝐵𝑆𝐶𝑇) → (𝐵(𝑂𝐴)𝐶) = {⟨𝑑, 𝑒⟩ ∣ 𝜃})
433expb 1120 . . . . . . . . . . . . 13 ((𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇)) → (𝐵(𝑂𝐴)𝐶) = {⟨𝑑, 𝑒⟩ ∣ 𝜃})
54breqd 5159 . . . . . . . . . . . 12 ((𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇)) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸𝐷{⟨𝑑, 𝑒⟩ ∣ 𝜃}𝐸))
6 brabv 5569 . . . . . . . . . . . . . . 15 (𝐷{⟨𝑑, 𝑒⟩ ∣ 𝜃}𝐸 → (𝐷 ∈ V ∧ 𝐸 ∈ V))
76anim2i 617 . . . . . . . . . . . . . 14 ((𝐴𝑈𝐷{⟨𝑑, 𝑒⟩ ∣ 𝜃}𝐸) → (𝐴𝑈 ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)))
87ex 413 . . . . . . . . . . . . 13 (𝐴𝑈 → (𝐷{⟨𝑑, 𝑒⟩ ∣ 𝜃}𝐸 → (𝐴𝑈 ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))
98adantr 481 . . . . . . . . . . . 12 ((𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇)) → (𝐷{⟨𝑑, 𝑒⟩ ∣ 𝜃}𝐸 → (𝐴𝑈 ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))
105, 9sylbid 239 . . . . . . . . . . 11 ((𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇)) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (𝐴𝑈 ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))
1110ex 413 . . . . . . . . . 10 (𝐴𝑈 → ((𝐵𝑆𝐶𝑇) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (𝐴𝑈 ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)))))
1211com23 86 . . . . . . . . 9 (𝐴𝑈 → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → ((𝐵𝑆𝐶𝑇) → (𝐴𝑈 ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)))))
1312a1d 25 . . . . . . . 8 (𝐴𝑈 → ((𝐵(𝑂𝐴)𝐶) ≠ ∅ → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → ((𝐵𝑆𝐶𝑇) → (𝐴𝑈 ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))))
14 bropfvvvv.o . . . . . . . . . 10 𝑂 = (𝑎𝑈 ↦ (𝑏𝑉, 𝑐𝑊 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜑}))
1514fvmptndm 7028 . . . . . . . . 9 𝐴𝑈 → (𝑂𝐴) = ∅)
16 df-ov 7414 . . . . . . . . . . 11 (𝐵(𝑂𝐴)𝐶) = ((𝑂𝐴)‘⟨𝐵, 𝐶⟩)
17 fveq1 6890 . . . . . . . . . . 11 ((𝑂𝐴) = ∅ → ((𝑂𝐴)‘⟨𝐵, 𝐶⟩) = (∅‘⟨𝐵, 𝐶⟩))
1816, 17eqtrid 2784 . . . . . . . . . 10 ((𝑂𝐴) = ∅ → (𝐵(𝑂𝐴)𝐶) = (∅‘⟨𝐵, 𝐶⟩))
19 0fv 6935 . . . . . . . . . 10 (∅‘⟨𝐵, 𝐶⟩) = ∅
2018, 19eqtrdi 2788 . . . . . . . . 9 ((𝑂𝐴) = ∅ → (𝐵(𝑂𝐴)𝐶) = ∅)
21 eqneqall 2951 . . . . . . . . 9 ((𝐵(𝑂𝐴)𝐶) = ∅ → ((𝐵(𝑂𝐴)𝐶) ≠ ∅ → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → ((𝐵𝑆𝐶𝑇) → (𝐴𝑈 ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))))
2215, 20, 213syl 18 . . . . . . . 8 𝐴𝑈 → ((𝐵(𝑂𝐴)𝐶) ≠ ∅ → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → ((𝐵𝑆𝐶𝑇) → (𝐴𝑈 ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))))
2313, 22pm2.61i 182 . . . . . . 7 ((𝐵(𝑂𝐴)𝐶) ≠ ∅ → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → ((𝐵𝑆𝐶𝑇) → (𝐴𝑈 ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)))))
242, 23mpcom 38 . . . . . 6 (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → ((𝐵𝑆𝐶𝑇) → (𝐴𝑈 ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))
2524com12 32 . . . . 5 ((𝐵𝑆𝐶𝑇) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (𝐴𝑈 ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))
2625anc2ri 557 . . . 4 ((𝐵𝑆𝐶𝑇) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → ((𝐴𝑈 ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)) ∧ (𝐵𝑆𝐶𝑇))))
27 3anan32 1097 . . . 4 ((𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)) ↔ ((𝐴𝑈 ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)) ∧ (𝐵𝑆𝐶𝑇)))
2826, 27imbitrrdi 251 . . 3 ((𝐵𝑆𝐶𝑇) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))
291, 28sylbi 216 . 2 (⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑇) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))
3029imp 407 1 ((⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑇) ∧ 𝐷(𝐵(𝑂𝐴)𝐶)𝐸) → (𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2940  Vcvv 3474  c0 4322  cop 4634   class class class wbr 5148  {copab 5210  cmpt 5231   × cxp 5674  cfv 6543  (class class class)co 7411  cmpo 7413
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5682  df-dm 5686  df-iota 6495  df-fv 6551  df-ov 7414
This theorem is referenced by:  bropfvvvv  8080
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