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Theorem bropfvvvvlem 7301
Description: Lemma for bropfvvvv 7302. (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 5180 . . 3 (⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑇) ↔ (𝐵𝑆𝐶𝑇))
2 brne0 4735 . . . . . . 7 (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (𝐵(𝑂𝐴)𝐶) ≠ ∅)
3 bropfvvvv.oo . . . . . . . . . . . . . 14 ((𝐴𝑈𝐵𝑆𝐶𝑇) → (𝐵(𝑂𝐴)𝐶) = {⟨𝑑, 𝑒⟩ ∣ 𝜃})
433expb 1285 . . . . . . . . . . . . 13 ((𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇)) → (𝐵(𝑂𝐴)𝐶) = {⟨𝑑, 𝑒⟩ ∣ 𝜃})
54breqd 4696 . . . . . . . . . . . 12 ((𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇)) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸𝐷{⟨𝑑, 𝑒⟩ ∣ 𝜃}𝐸))
6 brabv 6741 . . . . . . . . . . . . . . 15 (𝐷{⟨𝑑, 𝑒⟩ ∣ 𝜃}𝐸 → (𝐷 ∈ V ∧ 𝐸 ∈ V))
76anim2i 592 . . . . . . . . . . . . . 14 ((𝐴𝑈𝐷{⟨𝑑, 𝑒⟩ ∣ 𝜃}𝐸) → (𝐴𝑈 ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)))
87ex 449 . . . . . . . . . . . . 13 (𝐴𝑈 → (𝐷{⟨𝑑, 𝑒⟩ ∣ 𝜃}𝐸 → (𝐴𝑈 ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))
98adantr 480 . . . . . . . . . . . 12 ((𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇)) → (𝐷{⟨𝑑, 𝑒⟩ ∣ 𝜃}𝐸 → (𝐴𝑈 ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))
105, 9sylbid 230 . . . . . . . . . . 11 ((𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇)) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (𝐴𝑈 ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))
1110ex 449 . . . . . . . . . 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 6347 . . . . . . . . 9 𝐴𝑈 → (𝑂𝐴) = ∅)
16 df-ov 6693 . . . . . . . . . . 11 (𝐵(𝑂𝐴)𝐶) = ((𝑂𝐴)‘⟨𝐵, 𝐶⟩)
17 fveq1 6228 . . . . . . . . . . 11 ((𝑂𝐴) = ∅ → ((𝑂𝐴)‘⟨𝐵, 𝐶⟩) = (∅‘⟨𝐵, 𝐶⟩))
1816, 17syl5eq 2697 . . . . . . . . . 10 ((𝑂𝐴) = ∅ → (𝐵(𝑂𝐴)𝐶) = (∅‘⟨𝐵, 𝐶⟩))
19 0fv 6265 . . . . . . . . . 10 (∅‘⟨𝐵, 𝐶⟩) = ∅
2018, 19syl6eq 2701 . . . . . . . . 9 ((𝑂𝐴) = ∅ → (𝐵(𝑂𝐴)𝐶) = ∅)
21 eqneqall 2834 . . . . . . . . 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 176 . . . . . . 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 580 . . . 4 ((𝐵𝑆𝐶𝑇) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → ((𝐴𝑈 ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)) ∧ (𝐵𝑆𝐶𝑇))))
27 3anan32 1068 . . . 4 ((𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)) ↔ ((𝐴𝑈 ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)) ∧ (𝐵𝑆𝐶𝑇)))
2826, 27syl6ibr 242 . . 3 ((𝐵𝑆𝐶𝑇) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))
291, 28sylbi 207 . 2 (⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑇) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))
3029imp 444 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 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  Vcvv 3231  c0 3948  cop 4216   class class class wbr 4685  {copab 4745  cmpt 4762   × cxp 5141  cfv 5926  (class class class)co 6690  cmpt2 6692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-xp 5149  df-dm 5153  df-iota 5889  df-fv 5934  df-ov 6693
This theorem is referenced by:  bropfvvvv  7302
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