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Theorem bropfvvvv 8004
Description: If a binary relation holds for the result of an operation which is a function value, the involved classes are sets. (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})
bropfvvvv.s (𝑎 = 𝐴𝑉 = 𝑆)
bropfvvvv.t (𝑎 = 𝐴𝑊 = 𝑇)
bropfvvvv.p (𝑎 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
bropfvvvv ((𝑆𝑋𝑇𝑌) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))
Distinct variable groups:   𝑈,𝑎   𝐴,𝑎,𝑏,𝑐,𝑑,𝑒   𝑆,𝑎,𝑏,𝑐   𝑇,𝑎,𝑏,𝑐   𝜓,𝑎
Allowed substitution hints:   𝜑(𝑒,𝑎,𝑏,𝑐,𝑑)   𝜓(𝑒,𝑏,𝑐,𝑑)   𝜃(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,𝑑)   𝑇(𝑒,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)   𝑋(𝑒,𝑎,𝑏,𝑐,𝑑)   𝑌(𝑒,𝑎,𝑏,𝑐,𝑑)

Proof of Theorem bropfvvvv
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 brovpreldm 8001 . 2 (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → ⟨𝐵, 𝐶⟩ ∈ dom (𝑂𝐴))
2 bropfvvvv.s . . . . . . . . . 10 (𝑎 = 𝐴𝑉 = 𝑆)
3 bropfvvvv.t . . . . . . . . . 10 (𝑎 = 𝐴𝑊 = 𝑇)
4 bropfvvvv.p . . . . . . . . . . 11 (𝑎 = 𝐴 → (𝜑𝜓))
54opabbidv 5162 . . . . . . . . . 10 (𝑎 = 𝐴 → {⟨𝑑, 𝑒⟩ ∣ 𝜑} = {⟨𝑑, 𝑒⟩ ∣ 𝜓})
62, 3, 5mpoeq123dv 7416 . . . . . . . . 9 (𝑎 = 𝐴 → (𝑏𝑉, 𝑐𝑊 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜑}) = (𝑏𝑆, 𝑐𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}))
7 bropfvvvv.o . . . . . . . . 9 𝑂 = (𝑎𝑈 ↦ (𝑏𝑉, 𝑐𝑊 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜑}))
86, 7fvmptg 6933 . . . . . . . 8 ((𝐴𝑈 ∧ (𝑏𝑆, 𝑐𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V) → (𝑂𝐴) = (𝑏𝑆, 𝑐𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}))
98dmeqd 5851 . . . . . . 7 ((𝐴𝑈 ∧ (𝑏𝑆, 𝑐𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V) → dom (𝑂𝐴) = dom (𝑏𝑆, 𝑐𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}))
109eleq2d 2823 . . . . . 6 ((𝐴𝑈 ∧ (𝑏𝑆, 𝑐𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V) → (⟨𝐵, 𝐶⟩ ∈ dom (𝑂𝐴) ↔ ⟨𝐵, 𝐶⟩ ∈ dom (𝑏𝑆, 𝑐𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓})))
11 dmoprabss 7443 . . . . . . . . 9 dom {⟨⟨𝑏, 𝑐⟩, 𝑧⟩ ∣ ((𝑏𝑆𝑐𝑇) ∧ 𝑧 = {⟨𝑑, 𝑒⟩ ∣ 𝜓})} ⊆ (𝑆 × 𝑇)
1211sseli 3931 . . . . . . . 8 (⟨𝐵, 𝐶⟩ ∈ dom {⟨⟨𝑏, 𝑐⟩, 𝑧⟩ ∣ ((𝑏𝑆𝑐𝑇) ∧ 𝑧 = {⟨𝑑, 𝑒⟩ ∣ 𝜓})} → ⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑇))
13 bropfvvvv.oo . . . . . . . . . 10 ((𝐴𝑈𝐵𝑆𝐶𝑇) → (𝐵(𝑂𝐴)𝐶) = {⟨𝑑, 𝑒⟩ ∣ 𝜃})
147, 13bropfvvvvlem 8003 . . . . . . . . 9 ((⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑇) ∧ 𝐷(𝐵(𝑂𝐴)𝐶)𝐸) → (𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)))
1514ex 414 . . . . . . . 8 (⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑇) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))
1612, 15syl 17 . . . . . . 7 (⟨𝐵, 𝐶⟩ ∈ dom {⟨⟨𝑏, 𝑐⟩, 𝑧⟩ ∣ ((𝑏𝑆𝑐𝑇) ∧ 𝑧 = {⟨𝑑, 𝑒⟩ ∣ 𝜓})} → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))
17 df-mpo 7346 . . . . . . . 8 (𝑏𝑆, 𝑐𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) = {⟨⟨𝑏, 𝑐⟩, 𝑧⟩ ∣ ((𝑏𝑆𝑐𝑇) ∧ 𝑧 = {⟨𝑑, 𝑒⟩ ∣ 𝜓})}
1817dmeqi 5850 . . . . . . 7 dom (𝑏𝑆, 𝑐𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) = dom {⟨⟨𝑏, 𝑐⟩, 𝑧⟩ ∣ ((𝑏𝑆𝑐𝑇) ∧ 𝑧 = {⟨𝑑, 𝑒⟩ ∣ 𝜓})}
1916, 18eleq2s 2856 . . . . . 6 (⟨𝐵, 𝐶⟩ ∈ dom (𝑏𝑆, 𝑐𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))
2010, 19syl6bi 253 . . . . 5 ((𝐴𝑈 ∧ (𝑏𝑆, 𝑐𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V) → (⟨𝐵, 𝐶⟩ ∈ dom (𝑂𝐴) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)))))
2120com23 86 . . . 4 ((𝐴𝑈 ∧ (𝑏𝑆, 𝑐𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (⟨𝐵, 𝐶⟩ ∈ dom (𝑂𝐴) → (𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)))))
2221a1d 25 . . 3 ((𝐴𝑈 ∧ (𝑏𝑆, 𝑐𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V) → ((𝑆𝑋𝑇𝑌) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (⟨𝐵, 𝐶⟩ ∈ dom (𝑂𝐴) → (𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))))
23 ianor 980 . . . . 5 (¬ (𝐴𝑈 ∧ (𝑏𝑆, 𝑐𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V) ↔ (¬ 𝐴𝑈 ∨ ¬ (𝑏𝑆, 𝑐𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V))
247fvmptndm 6965 . . . . . . . . . . 11 𝐴𝑈 → (𝑂𝐴) = ∅)
2524dmeqd 5851 . . . . . . . . . 10 𝐴𝑈 → dom (𝑂𝐴) = dom ∅)
2625eleq2d 2823 . . . . . . . . 9 𝐴𝑈 → (⟨𝐵, 𝐶⟩ ∈ dom (𝑂𝐴) ↔ ⟨𝐵, 𝐶⟩ ∈ dom ∅))
27 dm0 5866 . . . . . . . . . 10 dom ∅ = ∅
2827eleq2i 2829 . . . . . . . . 9 (⟨𝐵, 𝐶⟩ ∈ dom ∅ ↔ ⟨𝐵, 𝐶⟩ ∈ ∅)
2926, 28bitrdi 287 . . . . . . . 8 𝐴𝑈 → (⟨𝐵, 𝐶⟩ ∈ dom (𝑂𝐴) ↔ ⟨𝐵, 𝐶⟩ ∈ ∅))
30 noel 4281 . . . . . . . . 9 ¬ ⟨𝐵, 𝐶⟩ ∈ ∅
3130pm2.21i 119 . . . . . . . 8 (⟨𝐵, 𝐶⟩ ∈ ∅ → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))
3229, 31syl6bi 253 . . . . . . 7 𝐴𝑈 → (⟨𝐵, 𝐶⟩ ∈ dom (𝑂𝐴) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)))))
3332a1d 25 . . . . . 6 𝐴𝑈 → ((𝑆𝑋𝑇𝑌) → (⟨𝐵, 𝐶⟩ ∈ dom (𝑂𝐴) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))))
34 notnotb 315 . . . . . . . 8 (𝐴𝑈 ↔ ¬ ¬ 𝐴𝑈)
35 elex 3460 . . . . . . . . . . . . . 14 (𝑆𝑋𝑆 ∈ V)
36 elex 3460 . . . . . . . . . . . . . 14 (𝑇𝑌𝑇 ∈ V)
3735, 36anim12i 614 . . . . . . . . . . . . 13 ((𝑆𝑋𝑇𝑌) → (𝑆 ∈ V ∧ 𝑇 ∈ V))
3837adantl 483 . . . . . . . . . . . 12 ((𝐴𝑈 ∧ (𝑆𝑋𝑇𝑌)) → (𝑆 ∈ V ∧ 𝑇 ∈ V))
39 mpoexga 7990 . . . . . . . . . . . 12 ((𝑆 ∈ V ∧ 𝑇 ∈ V) → (𝑏𝑆, 𝑐𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V)
4038, 39syl 17 . . . . . . . . . . 11 ((𝐴𝑈 ∧ (𝑆𝑋𝑇𝑌)) → (𝑏𝑆, 𝑐𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V)
4140pm2.24d 151 . . . . . . . . . 10 ((𝐴𝑈 ∧ (𝑆𝑋𝑇𝑌)) → (¬ (𝑏𝑆, 𝑐𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V → (⟨𝐵, 𝐶⟩ ∈ dom (𝑂𝐴) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))))
4241ex 414 . . . . . . . . 9 (𝐴𝑈 → ((𝑆𝑋𝑇𝑌) → (¬ (𝑏𝑆, 𝑐𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V → (⟨𝐵, 𝐶⟩ ∈ dom (𝑂𝐴) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)))))))
4342com23 86 . . . . . . . 8 (𝐴𝑈 → (¬ (𝑏𝑆, 𝑐𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V → ((𝑆𝑋𝑇𝑌) → (⟨𝐵, 𝐶⟩ ∈ dom (𝑂𝐴) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)))))))
4434, 43sylbir 234 . . . . . . 7 (¬ ¬ 𝐴𝑈 → (¬ (𝑏𝑆, 𝑐𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V → ((𝑆𝑋𝑇𝑌) → (⟨𝐵, 𝐶⟩ ∈ dom (𝑂𝐴) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)))))))
4544imp 408 . . . . . 6 ((¬ ¬ 𝐴𝑈 ∧ ¬ (𝑏𝑆, 𝑐𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V) → ((𝑆𝑋𝑇𝑌) → (⟨𝐵, 𝐶⟩ ∈ dom (𝑂𝐴) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))))
4633, 45jaoi3 1059 . . . . 5 ((¬ 𝐴𝑈 ∨ ¬ (𝑏𝑆, 𝑐𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V) → ((𝑆𝑋𝑇𝑌) → (⟨𝐵, 𝐶⟩ ∈ dom (𝑂𝐴) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))))
4723, 46sylbi 216 . . . 4 (¬ (𝐴𝑈 ∧ (𝑏𝑆, 𝑐𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V) → ((𝑆𝑋𝑇𝑌) → (⟨𝐵, 𝐶⟩ ∈ dom (𝑂𝐴) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))))
4847com34 91 . . 3 (¬ (𝐴𝑈 ∧ (𝑏𝑆, 𝑐𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V) → ((𝑆𝑋𝑇𝑌) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (⟨𝐵, 𝐶⟩ ∈ dom (𝑂𝐴) → (𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))))
4922, 48pm2.61i 182 . 2 ((𝑆𝑋𝑇𝑌) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (⟨𝐵, 𝐶⟩ ∈ dom (𝑂𝐴) → (𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)))))
501, 49mpdi 45 1 ((𝑆𝑋𝑇𝑌) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wo 845  w3a 1087   = wceq 1541  wcel 2106  Vcvv 3442  c0 4273  cop 4583   class class class wbr 5096  {copab 5158  cmpt 5179   × cxp 5622  dom cdm 5624  cfv 6483  (class class class)co 7341  {coprab 7342  cmpo 7343
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 2708  ax-rep 5233  ax-sep 5247  ax-nul 5254  ax-pow 5312  ax-pr 5376  ax-un 7654
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3444  df-sbc 3731  df-csb 3847  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4274  df-if 4478  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4857  df-iun 4947  df-br 5097  df-opab 5159  df-mpt 5180  df-id 5522  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6435  df-fun 6485  df-fn 6486  df-f 6487  df-f1 6488  df-fo 6489  df-f1o 6490  df-fv 6491  df-ov 7344  df-oprab 7345  df-mpo 7346  df-1st 7903  df-2nd 7904
This theorem is referenced by:  wlkonprop  28313  wksonproplem  28359  wksonproplemOLD  28360
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