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Theorem bropfvvvvlem 7493
Description: Lemma for bropfvvvv 7494. (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 5348 . . 3 (⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑇) ↔ (𝐵𝑆𝐶𝑇))
2 brne0 4893 . . . . . . 7 (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (𝐵(𝑂𝐴)𝐶) ≠ ∅)
3 bropfvvvv.oo . . . . . . . . . . . . . 14 ((𝐴𝑈𝐵𝑆𝐶𝑇) → (𝐵(𝑂𝐴)𝐶) = {⟨𝑑, 𝑒⟩ ∣ 𝜃})
433expb 1150 . . . . . . . . . . . . 13 ((𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇)) → (𝐵(𝑂𝐴)𝐶) = {⟨𝑑, 𝑒⟩ ∣ 𝜃})
54breqd 4854 . . . . . . . . . . . 12 ((𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇)) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸𝐷{⟨𝑑, 𝑒⟩ ∣ 𝜃}𝐸))
6 brabv 6933 . . . . . . . . . . . . . . 15 (𝐷{⟨𝑑, 𝑒⟩ ∣ 𝜃}𝐸 → (𝐷 ∈ V ∧ 𝐸 ∈ V))
76anim2i 611 . . . . . . . . . . . . . 14 ((𝐴𝑈𝐷{⟨𝑑, 𝑒⟩ ∣ 𝜃}𝐸) → (𝐴𝑈 ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)))
87ex 402 . . . . . . . . . . . . 13 (𝐴𝑈 → (𝐷{⟨𝑑, 𝑒⟩ ∣ 𝜃}𝐸 → (𝐴𝑈 ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))
98adantr 473 . . . . . . . . . . . 12 ((𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇)) → (𝐷{⟨𝑑, 𝑒⟩ ∣ 𝜃}𝐸 → (𝐴𝑈 ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))
105, 9sylbid 232 . . . . . . . . . . 11 ((𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇)) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (𝐴𝑈 ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))
1110ex 402 . . . . . . . . . 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 6533 . . . . . . . . 9 𝐴𝑈 → (𝑂𝐴) = ∅)
16 df-ov 6881 . . . . . . . . . . 11 (𝐵(𝑂𝐴)𝐶) = ((𝑂𝐴)‘⟨𝐵, 𝐶⟩)
17 fveq1 6410 . . . . . . . . . . 11 ((𝑂𝐴) = ∅ → ((𝑂𝐴)‘⟨𝐵, 𝐶⟩) = (∅‘⟨𝐵, 𝐶⟩))
1816, 17syl5eq 2845 . . . . . . . . . 10 ((𝑂𝐴) = ∅ → (𝐵(𝑂𝐴)𝐶) = (∅‘⟨𝐵, 𝐶⟩))
19 0fv 6451 . . . . . . . . . 10 (∅‘⟨𝐵, 𝐶⟩) = ∅
2018, 19syl6eq 2849 . . . . . . . . 9 ((𝑂𝐴) = ∅ → (𝐵(𝑂𝐴)𝐶) = ∅)
21 eqneqall 2982 . . . . . . . . 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 177 . . . . . . 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 553 . . . 4 ((𝐵𝑆𝐶𝑇) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → ((𝐴𝑈 ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)) ∧ (𝐵𝑆𝐶𝑇))))
27 3anan32 1119 . . . 4 ((𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)) ↔ ((𝐴𝑈 ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)) ∧ (𝐵𝑆𝐶𝑇)))
2826, 27syl6ibr 244 . . 3 ((𝐵𝑆𝐶𝑇) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))
291, 28sylbi 209 . 2 (⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑇) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))
3029imp 396 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 385  w3a 1108   = wceq 1653  wcel 2157  wne 2971  Vcvv 3385  c0 4115  cop 4374   class class class wbr 4843  {copab 4905  cmpt 4922   × cxp 5310  cfv 6101  (class class class)co 6878  cmpt2 6880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-xp 5318  df-dm 5322  df-iota 6064  df-fv 6109  df-ov 6881
This theorem is referenced by:  bropfvvvv  7494
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