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Theorem genipv 7431
Description: Value of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingon, 3-Oct-2019.)
Hypotheses
Ref Expression
genp.1 𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
genp.2 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
Assertion
Ref Expression
genipv ((𝐴P𝐵P) → (𝐴𝐹𝐵) = ⟨{𝑞Q ∣ ∃𝑟 ∈ (1st𝐴)∃𝑠 ∈ (1st𝐵)𝑞 = (𝑟𝐺𝑠)}, {𝑞Q ∣ ∃𝑟 ∈ (2nd𝐴)∃𝑠 ∈ (2nd𝐵)𝑞 = (𝑟𝐺𝑠)}⟩)
Distinct variable groups:   𝑥,𝑦,𝑧,𝑞,𝑟,𝑠,𝐴   𝑥,𝐵,𝑦,𝑧,𝑞,𝑟,𝑠   𝑥,𝑤,𝑣,𝐺,𝑦,𝑧,𝑞,𝑟,𝑠
Allowed substitution hints:   𝐴(𝑤,𝑣)   𝐵(𝑤,𝑣)   𝐹(𝑥,𝑦,𝑧,𝑤,𝑣,𝑠,𝑟,𝑞)

Proof of Theorem genipv
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 5833 . . . 4 (𝑓 = 𝐴 → (𝑓𝐹𝑔) = (𝐴𝐹𝑔))
2 fveq2 5470 . . . . . . 7 (𝑓 = 𝐴 → (1st𝑓) = (1st𝐴))
32rexeqdv 2659 . . . . . 6 (𝑓 = 𝐴 → (∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)))
43rabbidv 2701 . . . . 5 (𝑓 = 𝐴 → {𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)} = {𝑥Q ∣ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)})
5 fveq2 5470 . . . . . . 7 (𝑓 = 𝐴 → (2nd𝑓) = (2nd𝐴))
65rexeqdv 2659 . . . . . 6 (𝑓 = 𝐴 → (∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)))
76rabbidv 2701 . . . . 5 (𝑓 = 𝐴 → {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)} = {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)})
84, 7opeq12d 3751 . . . 4 (𝑓 = 𝐴 → ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)}⟩ = ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)}⟩)
91, 8eqeq12d 2172 . . 3 (𝑓 = 𝐴 → ((𝑓𝐹𝑔) = ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)}⟩ ↔ (𝐴𝐹𝑔) = ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)}⟩))
10 oveq2 5834 . . . 4 (𝑔 = 𝐵 → (𝐴𝐹𝑔) = (𝐴𝐹𝐵))
11 fveq2 5470 . . . . . . . 8 (𝑔 = 𝐵 → (1st𝑔) = (1st𝐵))
1211rexeqdv 2659 . . . . . . 7 (𝑔 = 𝐵 → (∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦𝐺𝑧)))
1312rexbidv 2458 . . . . . 6 (𝑔 = 𝐵 → (∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦𝐺𝑧)))
1413rabbidv 2701 . . . . 5 (𝑔 = 𝐵 → {𝑥Q ∣ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)} = {𝑥Q ∣ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦𝐺𝑧)})
15 fveq2 5470 . . . . . . . 8 (𝑔 = 𝐵 → (2nd𝑔) = (2nd𝐵))
1615rexeqdv 2659 . . . . . . 7 (𝑔 = 𝐵 → (∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦𝐺𝑧)))
1716rexbidv 2458 . . . . . 6 (𝑔 = 𝐵 → (∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦𝐺𝑧)))
1817rabbidv 2701 . . . . 5 (𝑔 = 𝐵 → {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)} = {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦𝐺𝑧)})
1914, 18opeq12d 3751 . . . 4 (𝑔 = 𝐵 → ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)}⟩ = ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦𝐺𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦𝐺𝑧)}⟩)
2010, 19eqeq12d 2172 . . 3 (𝑔 = 𝐵 → ((𝐴𝐹𝑔) = ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)}⟩ ↔ (𝐴𝐹𝐵) = ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦𝐺𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦𝐺𝑧)}⟩))
21 nqex 7285 . . . . . . 7 Q ∈ V
2221a1i 9 . . . . . 6 ((𝑓P𝑔P) → Q ∈ V)
23 rabssab 3216 . . . . . . 7 {𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)} ⊆ {𝑥 ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)}
24 prop 7397 . . . . . . . . . . . 12 (𝑓P → ⟨(1st𝑓), (2nd𝑓)⟩ ∈ P)
25 elprnql 7403 . . . . . . . . . . . 12 ((⟨(1st𝑓), (2nd𝑓)⟩ ∈ P𝑦 ∈ (1st𝑓)) → 𝑦Q)
2624, 25sylan 281 . . . . . . . . . . 11 ((𝑓P𝑦 ∈ (1st𝑓)) → 𝑦Q)
27 prop 7397 . . . . . . . . . . . 12 (𝑔P → ⟨(1st𝑔), (2nd𝑔)⟩ ∈ P)
28 elprnql 7403 . . . . . . . . . . . 12 ((⟨(1st𝑔), (2nd𝑔)⟩ ∈ P𝑧 ∈ (1st𝑔)) → 𝑧Q)
2927, 28sylan 281 . . . . . . . . . . 11 ((𝑔P𝑧 ∈ (1st𝑔)) → 𝑧Q)
30 genp.2 . . . . . . . . . . . 12 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
31 eleq1 2220 . . . . . . . . . . . 12 (𝑥 = (𝑦𝐺𝑧) → (𝑥Q ↔ (𝑦𝐺𝑧) ∈ Q))
3230, 31syl5ibrcom 156 . . . . . . . . . . 11 ((𝑦Q𝑧Q) → (𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
3326, 29, 32syl2an 287 . . . . . . . . . 10 (((𝑓P𝑦 ∈ (1st𝑓)) ∧ (𝑔P𝑧 ∈ (1st𝑔))) → (𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
3433an4s 578 . . . . . . . . 9 (((𝑓P𝑔P) ∧ (𝑦 ∈ (1st𝑓) ∧ 𝑧 ∈ (1st𝑔))) → (𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
3534rexlimdvva 2582 . . . . . . . 8 ((𝑓P𝑔P) → (∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
3635abssdv 3202 . . . . . . 7 ((𝑓P𝑔P) → {𝑥 ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)} ⊆ Q)
3723, 36sstrid 3139 . . . . . 6 ((𝑓P𝑔P) → {𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)} ⊆ Q)
3822, 37ssexd 4106 . . . . 5 ((𝑓P𝑔P) → {𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)} ∈ V)
39 rabssab 3216 . . . . . . 7 {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)} ⊆ {𝑥 ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)}
40 elprnqu 7404 . . . . . . . . . . . 12 ((⟨(1st𝑓), (2nd𝑓)⟩ ∈ P𝑦 ∈ (2nd𝑓)) → 𝑦Q)
4124, 40sylan 281 . . . . . . . . . . 11 ((𝑓P𝑦 ∈ (2nd𝑓)) → 𝑦Q)
42 elprnqu 7404 . . . . . . . . . . . 12 ((⟨(1st𝑔), (2nd𝑔)⟩ ∈ P𝑧 ∈ (2nd𝑔)) → 𝑧Q)
4327, 42sylan 281 . . . . . . . . . . 11 ((𝑔P𝑧 ∈ (2nd𝑔)) → 𝑧Q)
4441, 43, 32syl2an 287 . . . . . . . . . 10 (((𝑓P𝑦 ∈ (2nd𝑓)) ∧ (𝑔P𝑧 ∈ (2nd𝑔))) → (𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
4544an4s 578 . . . . . . . . 9 (((𝑓P𝑔P) ∧ (𝑦 ∈ (2nd𝑓) ∧ 𝑧 ∈ (2nd𝑔))) → (𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
4645rexlimdvva 2582 . . . . . . . 8 ((𝑓P𝑔P) → (∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
4746abssdv 3202 . . . . . . 7 ((𝑓P𝑔P) → {𝑥 ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)} ⊆ Q)
4839, 47sstrid 3139 . . . . . 6 ((𝑓P𝑔P) → {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)} ⊆ Q)
4922, 48ssexd 4106 . . . . 5 ((𝑓P𝑔P) → {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)} ∈ V)
50 opelxp 4618 . . . . 5 (⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)}⟩ ∈ (V × V) ↔ ({𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)} ∈ V ∧ {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)} ∈ V))
5138, 49, 50sylanbrc 414 . . . 4 ((𝑓P𝑔P) → ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)}⟩ ∈ (V × V))
52 fveq2 5470 . . . . . . . 8 (𝑤 = 𝑓 → (1st𝑤) = (1st𝑓))
5352rexeqdv 2659 . . . . . . 7 (𝑤 = 𝑓 → (∃𝑦 ∈ (1st𝑤)∃𝑧 ∈ (1st𝑣)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑣)𝑥 = (𝑦𝐺𝑧)))
5453rabbidv 2701 . . . . . 6 (𝑤 = 𝑓 → {𝑥Q ∣ ∃𝑦 ∈ (1st𝑤)∃𝑧 ∈ (1st𝑣)𝑥 = (𝑦𝐺𝑧)} = {𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑣)𝑥 = (𝑦𝐺𝑧)})
55 fveq2 5470 . . . . . . . 8 (𝑤 = 𝑓 → (2nd𝑤) = (2nd𝑓))
5655rexeqdv 2659 . . . . . . 7 (𝑤 = 𝑓 → (∃𝑦 ∈ (2nd𝑤)∃𝑧 ∈ (2nd𝑣)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑣)𝑥 = (𝑦𝐺𝑧)))
5756rabbidv 2701 . . . . . 6 (𝑤 = 𝑓 → {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑤)∃𝑧 ∈ (2nd𝑣)𝑥 = (𝑦𝐺𝑧)} = {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑣)𝑥 = (𝑦𝐺𝑧)})
5854, 57opeq12d 3751 . . . . 5 (𝑤 = 𝑓 → ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝑤)∃𝑧 ∈ (1st𝑣)𝑥 = (𝑦𝐺𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑤)∃𝑧 ∈ (2nd𝑣)𝑥 = (𝑦𝐺𝑧)}⟩ = ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑣)𝑥 = (𝑦𝐺𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑣)𝑥 = (𝑦𝐺𝑧)}⟩)
59 fveq2 5470 . . . . . . . . 9 (𝑣 = 𝑔 → (1st𝑣) = (1st𝑔))
6059rexeqdv 2659 . . . . . . . 8 (𝑣 = 𝑔 → (∃𝑧 ∈ (1st𝑣)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)))
6160rexbidv 2458 . . . . . . 7 (𝑣 = 𝑔 → (∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑣)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)))
6261rabbidv 2701 . . . . . 6 (𝑣 = 𝑔 → {𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑣)𝑥 = (𝑦𝐺𝑧)} = {𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)})
63 fveq2 5470 . . . . . . . . 9 (𝑣 = 𝑔 → (2nd𝑣) = (2nd𝑔))
6463rexeqdv 2659 . . . . . . . 8 (𝑣 = 𝑔 → (∃𝑧 ∈ (2nd𝑣)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)))
6564rexbidv 2458 . . . . . . 7 (𝑣 = 𝑔 → (∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑣)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)))
6665rabbidv 2701 . . . . . 6 (𝑣 = 𝑔 → {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑣)𝑥 = (𝑦𝐺𝑧)} = {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)})
6762, 66opeq12d 3751 . . . . 5 (𝑣 = 𝑔 → ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑣)𝑥 = (𝑦𝐺𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑣)𝑥 = (𝑦𝐺𝑧)}⟩ = ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)}⟩)
68 genp.1 . . . . . 6 𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
6968genpdf 7430 . . . . 5 𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝑤)∃𝑧 ∈ (1st𝑣)𝑥 = (𝑦𝐺𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑤)∃𝑧 ∈ (2nd𝑣)𝑥 = (𝑦𝐺𝑧)}⟩)
7058, 67, 69ovmpog 5957 . . . 4 ((𝑓P𝑔P ∧ ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)}⟩ ∈ (V × V)) → (𝑓𝐹𝑔) = ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)}⟩)
7151, 70mpd3an3 1320 . . 3 ((𝑓P𝑔P) → (𝑓𝐹𝑔) = ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)}⟩)
729, 20, 71vtocl2ga 2780 . 2 ((𝐴P𝐵P) → (𝐴𝐹𝐵) = ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦𝐺𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦𝐺𝑧)}⟩)
73 eqeq1 2164 . . . . . 6 (𝑥 = 𝑞 → (𝑥 = (𝑦𝐺𝑧) ↔ 𝑞 = (𝑦𝐺𝑧)))
74732rexbidv 2482 . . . . 5 (𝑥 = 𝑞 → (∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑞 = (𝑦𝐺𝑧)))
75 oveq1 5833 . . . . . . 7 (𝑦 = 𝑟 → (𝑦𝐺𝑧) = (𝑟𝐺𝑧))
7675eqeq2d 2169 . . . . . 6 (𝑦 = 𝑟 → (𝑞 = (𝑦𝐺𝑧) ↔ 𝑞 = (𝑟𝐺𝑧)))
77 oveq2 5834 . . . . . . 7 (𝑧 = 𝑠 → (𝑟𝐺𝑧) = (𝑟𝐺𝑠))
7877eqeq2d 2169 . . . . . 6 (𝑧 = 𝑠 → (𝑞 = (𝑟𝐺𝑧) ↔ 𝑞 = (𝑟𝐺𝑠)))
7976, 78cbvrex2v 2692 . . . . 5 (∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑞 = (𝑦𝐺𝑧) ↔ ∃𝑟 ∈ (1st𝐴)∃𝑠 ∈ (1st𝐵)𝑞 = (𝑟𝐺𝑠))
8074, 79bitrdi 195 . . . 4 (𝑥 = 𝑞 → (∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑟 ∈ (1st𝐴)∃𝑠 ∈ (1st𝐵)𝑞 = (𝑟𝐺𝑠)))
8180cbvrabv 2711 . . 3 {𝑥Q ∣ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦𝐺𝑧)} = {𝑞Q ∣ ∃𝑟 ∈ (1st𝐴)∃𝑠 ∈ (1st𝐵)𝑞 = (𝑟𝐺𝑠)}
82732rexbidv 2482 . . . . 5 (𝑥 = 𝑞 → (∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑞 = (𝑦𝐺𝑧)))
8376, 78cbvrex2v 2692 . . . . 5 (∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑞 = (𝑦𝐺𝑧) ↔ ∃𝑟 ∈ (2nd𝐴)∃𝑠 ∈ (2nd𝐵)𝑞 = (𝑟𝐺𝑠))
8482, 83bitrdi 195 . . . 4 (𝑥 = 𝑞 → (∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑟 ∈ (2nd𝐴)∃𝑠 ∈ (2nd𝐵)𝑞 = (𝑟𝐺𝑠)))
8584cbvrabv 2711 . . 3 {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦𝐺𝑧)} = {𝑞Q ∣ ∃𝑟 ∈ (2nd𝐴)∃𝑠 ∈ (2nd𝐵)𝑞 = (𝑟𝐺𝑠)}
8681, 85opeq12i 3748 . 2 ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦𝐺𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦𝐺𝑧)}⟩ = ⟨{𝑞Q ∣ ∃𝑟 ∈ (1st𝐴)∃𝑠 ∈ (1st𝐵)𝑞 = (𝑟𝐺𝑠)}, {𝑞Q ∣ ∃𝑟 ∈ (2nd𝐴)∃𝑠 ∈ (2nd𝐵)𝑞 = (𝑟𝐺𝑠)}⟩
8772, 86eqtrdi 2206 1 ((𝐴P𝐵P) → (𝐴𝐹𝐵) = ⟨{𝑞Q ∣ ∃𝑟 ∈ (1st𝐴)∃𝑠 ∈ (1st𝐵)𝑞 = (𝑟𝐺𝑠)}, {𝑞Q ∣ ∃𝑟 ∈ (2nd𝐴)∃𝑠 ∈ (2nd𝐵)𝑞 = (𝑟𝐺𝑠)}⟩)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 963   = wceq 1335  wcel 2128  {cab 2143  wrex 2436  {crab 2439  Vcvv 2712  cop 3564   × cxp 4586  cfv 5172  (class class class)co 5826  cmpo 5828  1st c1st 6088  2nd c2nd 6089  Qcnq 7202  Pcnp 7213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4081  ax-sep 4084  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-setind 4498  ax-iinf 4549
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4028  df-mpt 4029  df-id 4255  df-iom 4552  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-f1 5177  df-fo 5178  df-f1o 5179  df-fv 5180  df-ov 5829  df-oprab 5830  df-mpo 5831  df-1st 6090  df-2nd 6091  df-qs 6488  df-ni 7226  df-nqqs 7270  df-inp 7388
This theorem is referenced by:  genpelvl  7434  genpelvu  7435  plpvlu  7460  mpvlu  7461
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