MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  bropfvvvv Structured version   Visualization version   GIF version

Theorem bropfvvvv 7774
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 7771 . 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 5108 . . . . . . . . . 10 (𝑎 = 𝐴 → {⟨𝑑, 𝑒⟩ ∣ 𝜑} = {⟨𝑑, 𝑒⟩ ∣ 𝜓})
62, 3, 5mpoeq123dv 7213 . . . . . . . . 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 6748 . . . . . . . 8 ((𝐴𝑈 ∧ (𝑏𝑆, 𝑐𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V) → (𝑂𝐴) = (𝑏𝑆, 𝑐𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}))
98dmeqd 5751 . . . . . . 7 ((𝐴𝑈 ∧ (𝑏𝑆, 𝑐𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V) → dom (𝑂𝐴) = dom (𝑏𝑆, 𝑐𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}))
109eleq2d 2899 . . . . . 6 ((𝐴𝑈 ∧ (𝑏𝑆, 𝑐𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V) → (⟨𝐵, 𝐶⟩ ∈ dom (𝑂𝐴) ↔ ⟨𝐵, 𝐶⟩ ∈ dom (𝑏𝑆, 𝑐𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓})))
11 dmoprabss 7240 . . . . . . . . 9 dom {⟨⟨𝑏, 𝑐⟩, 𝑧⟩ ∣ ((𝑏𝑆𝑐𝑇) ∧ 𝑧 = {⟨𝑑, 𝑒⟩ ∣ 𝜓})} ⊆ (𝑆 × 𝑇)
1211sseli 3938 . . . . . . . 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 7773 . . . . . . . . 9 ((⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑇) ∧ 𝐷(𝐵(𝑂𝐴)𝐶)𝐸) → (𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)))
1514ex 416 . . . . . . . 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 7145 . . . . . . . 8 (𝑏𝑆, 𝑐𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) = {⟨⟨𝑏, 𝑐⟩, 𝑧⟩ ∣ ((𝑏𝑆𝑐𝑇) ∧ 𝑧 = {⟨𝑑, 𝑒⟩ ∣ 𝜓})}
1817dmeqi 5750 . . . . . . 7 dom (𝑏𝑆, 𝑐𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) = dom {⟨⟨𝑏, 𝑐⟩, 𝑧⟩ ∣ ((𝑏𝑆𝑐𝑇) ∧ 𝑧 = {⟨𝑑, 𝑒⟩ ∣ 𝜓})}
1916, 18eleq2s 2932 . . . . . 6 (⟨𝐵, 𝐶⟩ ∈ dom (𝑏𝑆, 𝑐𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))
2010, 19syl6bi 256 . . . . 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 979 . . . . 5 (¬ (𝐴𝑈 ∧ (𝑏𝑆, 𝑐𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V) ↔ (¬ 𝐴𝑈 ∨ ¬ (𝑏𝑆, 𝑐𝑇 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜓}) ∈ V))
247fvmptndm 6780 . . . . . . . . . . 11 𝐴𝑈 → (𝑂𝐴) = ∅)
2524dmeqd 5751 . . . . . . . . . 10 𝐴𝑈 → dom (𝑂𝐴) = dom ∅)
2625eleq2d 2899 . . . . . . . . 9 𝐴𝑈 → (⟨𝐵, 𝐶⟩ ∈ dom (𝑂𝐴) ↔ ⟨𝐵, 𝐶⟩ ∈ dom ∅))
27 dm0 5767 . . . . . . . . . 10 dom ∅ = ∅
2827eleq2i 2905 . . . . . . . . 9 (⟨𝐵, 𝐶⟩ ∈ dom ∅ ↔ ⟨𝐵, 𝐶⟩ ∈ ∅)
2926, 28syl6bb 290 . . . . . . . 8 𝐴𝑈 → (⟨𝐵, 𝐶⟩ ∈ dom (𝑂𝐴) ↔ ⟨𝐵, 𝐶⟩ ∈ ∅))
30 noel 4269 . . . . . . . . 9 ¬ ⟨𝐵, 𝐶⟩ ∈ ∅
3130pm2.21i 119 . . . . . . . 8 (⟨𝐵, 𝐶⟩ ∈ ∅ → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))
3229, 31syl6bi 256 . . . . . . 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 318 . . . . . . . 8 (𝐴𝑈 ↔ ¬ ¬ 𝐴𝑈)
35 elex 3487 . . . . . . . . . . . . . 14 (𝑆𝑋𝑆 ∈ V)
36 elex 3487 . . . . . . . . . . . . . 14 (𝑇𝑌𝑇 ∈ V)
3735, 36anim12i 615 . . . . . . . . . . . . 13 ((𝑆𝑋𝑇𝑌) → (𝑆 ∈ V ∧ 𝑇 ∈ V))
3837adantl 485 . . . . . . . . . . . 12 ((𝐴𝑈 ∧ (𝑆𝑋𝑇𝑌)) → (𝑆 ∈ V ∧ 𝑇 ∈ V))
39 mpoexga 7762 . . . . . . . . . . . 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 154 . . . . . . . . . 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 416 . . . . . . . . 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 238 . . . . . . 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 410 . . . . . 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 1056 . . . . 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 220 . . . 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 185 . 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 209  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2114  Vcvv 3469  c0 4265  cop 4545   class class class wbr 5042  {copab 5104  cmpt 5122   × cxp 5530  dom cdm 5532  cfv 6334  (class class class)co 7140  {coprab 7141  cmpo 7142
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676
This theorem is referenced by:  wlkonprop  27446  wksonproplem  27492
  Copyright terms: Public domain W3C validator