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Theorem genipv 6605
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 5519 . . . 4 (𝑓 = 𝐴 → (𝑓𝐹𝑔) = (𝐴𝐹𝑔))
2 fveq2 5178 . . . . . . 7 (𝑓 = 𝐴 → (1st𝑓) = (1st𝐴))
32rexeqdv 2512 . . . . . 6 (𝑓 = 𝐴 → (∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)))
43rabbidv 2549 . . . . 5 (𝑓 = 𝐴 → {𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)} = {𝑥Q ∣ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)})
5 fveq2 5178 . . . . . . 7 (𝑓 = 𝐴 → (2nd𝑓) = (2nd𝐴))
65rexeqdv 2512 . . . . . 6 (𝑓 = 𝐴 → (∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)))
76rabbidv 2549 . . . . 5 (𝑓 = 𝐴 → {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)} = {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)})
84, 7opeq12d 3557 . . . 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 2054 . . 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 5520 . . . 4 (𝑔 = 𝐵 → (𝐴𝐹𝑔) = (𝐴𝐹𝐵))
11 fveq2 5178 . . . . . . . 8 (𝑔 = 𝐵 → (1st𝑔) = (1st𝐵))
1211rexeqdv 2512 . . . . . . 7 (𝑔 = 𝐵 → (∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦𝐺𝑧)))
1312rexbidv 2327 . . . . . 6 (𝑔 = 𝐵 → (∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦𝐺𝑧)))
1413rabbidv 2549 . . . . 5 (𝑔 = 𝐵 → {𝑥Q ∣ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)} = {𝑥Q ∣ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦𝐺𝑧)})
15 fveq2 5178 . . . . . . . 8 (𝑔 = 𝐵 → (2nd𝑔) = (2nd𝐵))
1615rexeqdv 2512 . . . . . . 7 (𝑔 = 𝐵 → (∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦𝐺𝑧)))
1716rexbidv 2327 . . . . . 6 (𝑔 = 𝐵 → (∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦𝐺𝑧)))
1817rabbidv 2549 . . . . 5 (𝑔 = 𝐵 → {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)} = {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦𝐺𝑧)})
1914, 18opeq12d 3557 . . . 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 2054 . . 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 6459 . . . . . . 7 Q ∈ V
2221a1i 9 . . . . . 6 ((𝑓P𝑔P) → Q ∈ V)
23 rabssab 3027 . . . . . . 7 {𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)} ⊆ {𝑥 ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)}
24 prop 6571 . . . . . . . . . . . 12 (𝑓P → ⟨(1st𝑓), (2nd𝑓)⟩ ∈ P)
25 elprnql 6577 . . . . . . . . . . . 12 ((⟨(1st𝑓), (2nd𝑓)⟩ ∈ P𝑦 ∈ (1st𝑓)) → 𝑦Q)
2624, 25sylan 267 . . . . . . . . . . 11 ((𝑓P𝑦 ∈ (1st𝑓)) → 𝑦Q)
27 prop 6571 . . . . . . . . . . . 12 (𝑔P → ⟨(1st𝑔), (2nd𝑔)⟩ ∈ P)
28 elprnql 6577 . . . . . . . . . . . 12 ((⟨(1st𝑔), (2nd𝑔)⟩ ∈ P𝑧 ∈ (1st𝑔)) → 𝑧Q)
2927, 28sylan 267 . . . . . . . . . . 11 ((𝑔P𝑧 ∈ (1st𝑔)) → 𝑧Q)
30 genp.2 . . . . . . . . . . . 12 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
31 eleq1 2100 . . . . . . . . . . . 12 (𝑥 = (𝑦𝐺𝑧) → (𝑥Q ↔ (𝑦𝐺𝑧) ∈ Q))
3230, 31syl5ibrcom 146 . . . . . . . . . . 11 ((𝑦Q𝑧Q) → (𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
3326, 29, 32syl2an 273 . . . . . . . . . 10 (((𝑓P𝑦 ∈ (1st𝑓)) ∧ (𝑔P𝑧 ∈ (1st𝑔))) → (𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
3433an4s 522 . . . . . . . . 9 (((𝑓P𝑔P) ∧ (𝑦 ∈ (1st𝑓) ∧ 𝑧 ∈ (1st𝑔))) → (𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
3534rexlimdvva 2440 . . . . . . . 8 ((𝑓P𝑔P) → (∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
3635abssdv 3014 . . . . . . 7 ((𝑓P𝑔P) → {𝑥 ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)} ⊆ Q)
3723, 36syl5ss 2956 . . . . . 6 ((𝑓P𝑔P) → {𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)} ⊆ Q)
3822, 37ssexd 3897 . . . . 5 ((𝑓P𝑔P) → {𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)} ∈ V)
39 rabssab 3027 . . . . . . 7 {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)} ⊆ {𝑥 ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)}
40 elprnqu 6578 . . . . . . . . . . . 12 ((⟨(1st𝑓), (2nd𝑓)⟩ ∈ P𝑦 ∈ (2nd𝑓)) → 𝑦Q)
4124, 40sylan 267 . . . . . . . . . . 11 ((𝑓P𝑦 ∈ (2nd𝑓)) → 𝑦Q)
42 elprnqu 6578 . . . . . . . . . . . 12 ((⟨(1st𝑔), (2nd𝑔)⟩ ∈ P𝑧 ∈ (2nd𝑔)) → 𝑧Q)
4327, 42sylan 267 . . . . . . . . . . 11 ((𝑔P𝑧 ∈ (2nd𝑔)) → 𝑧Q)
4441, 43, 32syl2an 273 . . . . . . . . . 10 (((𝑓P𝑦 ∈ (2nd𝑓)) ∧ (𝑔P𝑧 ∈ (2nd𝑔))) → (𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
4544an4s 522 . . . . . . . . 9 (((𝑓P𝑔P) ∧ (𝑦 ∈ (2nd𝑓) ∧ 𝑧 ∈ (2nd𝑔))) → (𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
4645rexlimdvva 2440 . . . . . . . 8 ((𝑓P𝑔P) → (∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
4746abssdv 3014 . . . . . . 7 ((𝑓P𝑔P) → {𝑥 ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)} ⊆ Q)
4839, 47syl5ss 2956 . . . . . 6 ((𝑓P𝑔P) → {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)} ⊆ Q)
4922, 48ssexd 3897 . . . . 5 ((𝑓P𝑔P) → {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)} ∈ V)
50 opelxp 4374 . . . . 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 394 . . . 4 ((𝑓P𝑔P) → ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)}⟩ ∈ (V × V))
52 fveq2 5178 . . . . . . . 8 (𝑤 = 𝑓 → (1st𝑤) = (1st𝑓))
5352rexeqdv 2512 . . . . . . 7 (𝑤 = 𝑓 → (∃𝑦 ∈ (1st𝑤)∃𝑧 ∈ (1st𝑣)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑣)𝑥 = (𝑦𝐺𝑧)))
5453rabbidv 2549 . . . . . 6 (𝑤 = 𝑓 → {𝑥Q ∣ ∃𝑦 ∈ (1st𝑤)∃𝑧 ∈ (1st𝑣)𝑥 = (𝑦𝐺𝑧)} = {𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑣)𝑥 = (𝑦𝐺𝑧)})
55 fveq2 5178 . . . . . . . 8 (𝑤 = 𝑓 → (2nd𝑤) = (2nd𝑓))
5655rexeqdv 2512 . . . . . . 7 (𝑤 = 𝑓 → (∃𝑦 ∈ (2nd𝑤)∃𝑧 ∈ (2nd𝑣)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑣)𝑥 = (𝑦𝐺𝑧)))
5756rabbidv 2549 . . . . . 6 (𝑤 = 𝑓 → {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑤)∃𝑧 ∈ (2nd𝑣)𝑥 = (𝑦𝐺𝑧)} = {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑣)𝑥 = (𝑦𝐺𝑧)})
5854, 57opeq12d 3557 . . . . 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 5178 . . . . . . . . 9 (𝑣 = 𝑔 → (1st𝑣) = (1st𝑔))
6059rexeqdv 2512 . . . . . . . 8 (𝑣 = 𝑔 → (∃𝑧 ∈ (1st𝑣)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)))
6160rexbidv 2327 . . . . . . 7 (𝑣 = 𝑔 → (∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑣)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)))
6261rabbidv 2549 . . . . . 6 (𝑣 = 𝑔 → {𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑣)𝑥 = (𝑦𝐺𝑧)} = {𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)})
63 fveq2 5178 . . . . . . . . 9 (𝑣 = 𝑔 → (2nd𝑣) = (2nd𝑔))
6463rexeqdv 2512 . . . . . . . 8 (𝑣 = 𝑔 → (∃𝑧 ∈ (2nd𝑣)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)))
6564rexbidv 2327 . . . . . . 7 (𝑣 = 𝑔 → (∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑣)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)))
6665rabbidv 2549 . . . . . 6 (𝑣 = 𝑔 → {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑣)𝑥 = (𝑦𝐺𝑧)} = {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)})
6762, 66opeq12d 3557 . . . . 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 6604 . . . . 5 𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝑤)∃𝑧 ∈ (1st𝑣)𝑥 = (𝑦𝐺𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑤)∃𝑧 ∈ (2nd𝑣)𝑥 = (𝑦𝐺𝑧)}⟩)
7058, 67, 69ovmpt2g 5635 . . . 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 1233 . . 3 ((𝑓P𝑔P) → (𝑓𝐹𝑔) = ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)}⟩)
729, 20, 71vtocl2ga 2621 . 2 ((𝐴P𝐵P) → (𝐴𝐹𝐵) = ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦𝐺𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦𝐺𝑧)}⟩)
73 eqeq1 2046 . . . . . 6 (𝑥 = 𝑞 → (𝑥 = (𝑦𝐺𝑧) ↔ 𝑞 = (𝑦𝐺𝑧)))
74732rexbidv 2349 . . . . 5 (𝑥 = 𝑞 → (∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑞 = (𝑦𝐺𝑧)))
75 oveq1 5519 . . . . . . 7 (𝑦 = 𝑟 → (𝑦𝐺𝑧) = (𝑟𝐺𝑧))
7675eqeq2d 2051 . . . . . 6 (𝑦 = 𝑟 → (𝑞 = (𝑦𝐺𝑧) ↔ 𝑞 = (𝑟𝐺𝑧)))
77 oveq2 5520 . . . . . . 7 (𝑧 = 𝑠 → (𝑟𝐺𝑧) = (𝑟𝐺𝑠))
7877eqeq2d 2051 . . . . . 6 (𝑧 = 𝑠 → (𝑞 = (𝑟𝐺𝑧) ↔ 𝑞 = (𝑟𝐺𝑠)))
7976, 78cbvrex2v 2542 . . . . 5 (∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑞 = (𝑦𝐺𝑧) ↔ ∃𝑟 ∈ (1st𝐴)∃𝑠 ∈ (1st𝐵)𝑞 = (𝑟𝐺𝑠))
8074, 79syl6bb 185 . . . 4 (𝑥 = 𝑞 → (∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑟 ∈ (1st𝐴)∃𝑠 ∈ (1st𝐵)𝑞 = (𝑟𝐺𝑠)))
8180cbvrabv 2556 . . 3 {𝑥Q ∣ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦𝐺𝑧)} = {𝑞Q ∣ ∃𝑟 ∈ (1st𝐴)∃𝑠 ∈ (1st𝐵)𝑞 = (𝑟𝐺𝑠)}
82732rexbidv 2349 . . . . 5 (𝑥 = 𝑞 → (∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑞 = (𝑦𝐺𝑧)))
8376, 78cbvrex2v 2542 . . . . 5 (∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑞 = (𝑦𝐺𝑧) ↔ ∃𝑟 ∈ (2nd𝐴)∃𝑠 ∈ (2nd𝐵)𝑞 = (𝑟𝐺𝑠))
8482, 83syl6bb 185 . . . 4 (𝑥 = 𝑞 → (∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑟 ∈ (2nd𝐴)∃𝑠 ∈ (2nd𝐵)𝑞 = (𝑟𝐺𝑠)))
8584cbvrabv 2556 . . 3 {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦𝐺𝑧)} = {𝑞Q ∣ ∃𝑟 ∈ (2nd𝐴)∃𝑠 ∈ (2nd𝐵)𝑞 = (𝑟𝐺𝑠)}
8681, 85opeq12i 3554 . 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, 86syl6eq 2088 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 97  w3a 885   = wceq 1243  wcel 1393  {cab 2026  wrex 2307  {crab 2310  Vcvv 2557  cop 3378   × cxp 4343  cfv 4902  (class class class)co 5512  cmpt2 5514  1st c1st 5765  2nd c2nd 5766  Qcnq 6376  Pcnp 6387
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-qs 6112  df-ni 6400  df-nqqs 6444  df-inp 6562
This theorem is referenced by:  genpelvl  6608  genpelvu  6609  plpvlu  6634  mpvlu  6635
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