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Theorem bropfvvvvlem 7201
Description: Lemma for bropfvvvv 7202. (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 5106 . . 3 (⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑇) ↔ (𝐵𝑆𝐶𝑇))
2 brne0 4662 . . . . . . 7 (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (𝐵(𝑂𝐴)𝐶) ≠ ∅)
3 bropfvvvv.oo . . . . . . . . . . . . . 14 ((𝐴𝑈𝐵𝑆𝐶𝑇) → (𝐵(𝑂𝐴)𝐶) = {⟨𝑑, 𝑒⟩ ∣ 𝜃})
433expb 1263 . . . . . . . . . . . . 13 ((𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇)) → (𝐵(𝑂𝐴)𝐶) = {⟨𝑑, 𝑒⟩ ∣ 𝜃})
54breqd 4624 . . . . . . . . . . . 12 ((𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇)) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸𝐷{⟨𝑑, 𝑒⟩ ∣ 𝜃}𝐸))
6 brabv 6652 . . . . . . . . . . . . . . 15 (𝐷{⟨𝑑, 𝑒⟩ ∣ 𝜃}𝐸 → (𝐷 ∈ V ∧ 𝐸 ∈ V))
76anim2i 592 . . . . . . . . . . . . . 14 ((𝐴𝑈𝐷{⟨𝑑, 𝑒⟩ ∣ 𝜃}𝐸) → (𝐴𝑈 ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)))
87ex 450 . . . . . . . . . . . . 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 230 . . . . . . . . . . 11 ((𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇)) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (𝐴𝑈 ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))
1110ex 450 . . . . . . . . . 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 6264 . . . . . . . . 9 𝐴𝑈 → (𝑂𝐴) = ∅)
16 df-ov 6607 . . . . . . . . . . 11 (𝐵(𝑂𝐴)𝐶) = ((𝑂𝐴)‘⟨𝐵, 𝐶⟩)
17 fveq1 6147 . . . . . . . . . . 11 ((𝑂𝐴) = ∅ → ((𝑂𝐴)‘⟨𝐵, 𝐶⟩) = (∅‘⟨𝐵, 𝐶⟩))
1816, 17syl5eq 2667 . . . . . . . . . 10 ((𝑂𝐴) = ∅ → (𝐵(𝑂𝐴)𝐶) = (∅‘⟨𝐵, 𝐶⟩))
19 0fv 6184 . . . . . . . . . 10 (∅‘⟨𝐵, 𝐶⟩) = ∅
2018, 19syl6eq 2671 . . . . . . . . 9 ((𝑂𝐴) = ∅ → (𝐵(𝑂𝐴)𝐶) = ∅)
21 eqneqall 2801 . . . . . . . . 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 1048 . . . 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 445 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 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  Vcvv 3186  c0 3891  cop 4154   class class class wbr 4613  {copab 4672  cmpt 4673   × cxp 5072  cfv 5847  (class class class)co 6604  cmpt2 6606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-xp 5080  df-dm 5084  df-iota 5810  df-fv 5855  df-ov 6607
This theorem is referenced by:  bropfvvvv  7202
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