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Theorem genipv 7324
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 5781 . . . 4 (𝑓 = 𝐴 → (𝑓𝐹𝑔) = (𝐴𝐹𝑔))
2 fveq2 5421 . . . . . . 7 (𝑓 = 𝐴 → (1st𝑓) = (1st𝐴))
32rexeqdv 2633 . . . . . 6 (𝑓 = 𝐴 → (∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)))
43rabbidv 2675 . . . . 5 (𝑓 = 𝐴 → {𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)} = {𝑥Q ∣ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)})
5 fveq2 5421 . . . . . . 7 (𝑓 = 𝐴 → (2nd𝑓) = (2nd𝐴))
65rexeqdv 2633 . . . . . 6 (𝑓 = 𝐴 → (∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)))
76rabbidv 2675 . . . . 5 (𝑓 = 𝐴 → {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)} = {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)})
84, 7opeq12d 3713 . . . 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 2154 . . 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 5782 . . . 4 (𝑔 = 𝐵 → (𝐴𝐹𝑔) = (𝐴𝐹𝐵))
11 fveq2 5421 . . . . . . . 8 (𝑔 = 𝐵 → (1st𝑔) = (1st𝐵))
1211rexeqdv 2633 . . . . . . 7 (𝑔 = 𝐵 → (∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦𝐺𝑧)))
1312rexbidv 2438 . . . . . 6 (𝑔 = 𝐵 → (∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦𝐺𝑧)))
1413rabbidv 2675 . . . . 5 (𝑔 = 𝐵 → {𝑥Q ∣ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)} = {𝑥Q ∣ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦𝐺𝑧)})
15 fveq2 5421 . . . . . . . 8 (𝑔 = 𝐵 → (2nd𝑔) = (2nd𝐵))
1615rexeqdv 2633 . . . . . . 7 (𝑔 = 𝐵 → (∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦𝐺𝑧)))
1716rexbidv 2438 . . . . . 6 (𝑔 = 𝐵 → (∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦𝐺𝑧)))
1817rabbidv 2675 . . . . 5 (𝑔 = 𝐵 → {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)} = {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦𝐺𝑧)})
1914, 18opeq12d 3713 . . . 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 2154 . . 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 7178 . . . . . . 7 Q ∈ V
2221a1i 9 . . . . . 6 ((𝑓P𝑔P) → Q ∈ V)
23 rabssab 3184 . . . . . . 7 {𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)} ⊆ {𝑥 ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)}
24 prop 7290 . . . . . . . . . . . 12 (𝑓P → ⟨(1st𝑓), (2nd𝑓)⟩ ∈ P)
25 elprnql 7296 . . . . . . . . . . . 12 ((⟨(1st𝑓), (2nd𝑓)⟩ ∈ P𝑦 ∈ (1st𝑓)) → 𝑦Q)
2624, 25sylan 281 . . . . . . . . . . 11 ((𝑓P𝑦 ∈ (1st𝑓)) → 𝑦Q)
27 prop 7290 . . . . . . . . . . . 12 (𝑔P → ⟨(1st𝑔), (2nd𝑔)⟩ ∈ P)
28 elprnql 7296 . . . . . . . . . . . 12 ((⟨(1st𝑔), (2nd𝑔)⟩ ∈ P𝑧 ∈ (1st𝑔)) → 𝑧Q)
2927, 28sylan 281 . . . . . . . . . . 11 ((𝑔P𝑧 ∈ (1st𝑔)) → 𝑧Q)
30 genp.2 . . . . . . . . . . . 12 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
31 eleq1 2202 . . . . . . . . . . . 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 577 . . . . . . . . 9 (((𝑓P𝑔P) ∧ (𝑦 ∈ (1st𝑓) ∧ 𝑧 ∈ (1st𝑔))) → (𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
3534rexlimdvva 2557 . . . . . . . 8 ((𝑓P𝑔P) → (∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
3635abssdv 3171 . . . . . . 7 ((𝑓P𝑔P) → {𝑥 ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)} ⊆ Q)
3723, 36sstrid 3108 . . . . . 6 ((𝑓P𝑔P) → {𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)} ⊆ Q)
3822, 37ssexd 4068 . . . . 5 ((𝑓P𝑔P) → {𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)} ∈ V)
39 rabssab 3184 . . . . . . 7 {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)} ⊆ {𝑥 ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)}
40 elprnqu 7297 . . . . . . . . . . . 12 ((⟨(1st𝑓), (2nd𝑓)⟩ ∈ P𝑦 ∈ (2nd𝑓)) → 𝑦Q)
4124, 40sylan 281 . . . . . . . . . . 11 ((𝑓P𝑦 ∈ (2nd𝑓)) → 𝑦Q)
42 elprnqu 7297 . . . . . . . . . . . 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 577 . . . . . . . . 9 (((𝑓P𝑔P) ∧ (𝑦 ∈ (2nd𝑓) ∧ 𝑧 ∈ (2nd𝑔))) → (𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
4645rexlimdvva 2557 . . . . . . . 8 ((𝑓P𝑔P) → (∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
4746abssdv 3171 . . . . . . 7 ((𝑓P𝑔P) → {𝑥 ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)} ⊆ Q)
4839, 47sstrid 3108 . . . . . 6 ((𝑓P𝑔P) → {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)} ⊆ Q)
4922, 48ssexd 4068 . . . . 5 ((𝑓P𝑔P) → {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)} ∈ V)
50 opelxp 4569 . . . . 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 413 . . . 4 ((𝑓P𝑔P) → ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)}⟩ ∈ (V × V))
52 fveq2 5421 . . . . . . . 8 (𝑤 = 𝑓 → (1st𝑤) = (1st𝑓))
5352rexeqdv 2633 . . . . . . 7 (𝑤 = 𝑓 → (∃𝑦 ∈ (1st𝑤)∃𝑧 ∈ (1st𝑣)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑣)𝑥 = (𝑦𝐺𝑧)))
5453rabbidv 2675 . . . . . 6 (𝑤 = 𝑓 → {𝑥Q ∣ ∃𝑦 ∈ (1st𝑤)∃𝑧 ∈ (1st𝑣)𝑥 = (𝑦𝐺𝑧)} = {𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑣)𝑥 = (𝑦𝐺𝑧)})
55 fveq2 5421 . . . . . . . 8 (𝑤 = 𝑓 → (2nd𝑤) = (2nd𝑓))
5655rexeqdv 2633 . . . . . . 7 (𝑤 = 𝑓 → (∃𝑦 ∈ (2nd𝑤)∃𝑧 ∈ (2nd𝑣)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑣)𝑥 = (𝑦𝐺𝑧)))
5756rabbidv 2675 . . . . . 6 (𝑤 = 𝑓 → {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑤)∃𝑧 ∈ (2nd𝑣)𝑥 = (𝑦𝐺𝑧)} = {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑣)𝑥 = (𝑦𝐺𝑧)})
5854, 57opeq12d 3713 . . . . 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 5421 . . . . . . . . 9 (𝑣 = 𝑔 → (1st𝑣) = (1st𝑔))
6059rexeqdv 2633 . . . . . . . 8 (𝑣 = 𝑔 → (∃𝑧 ∈ (1st𝑣)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)))
6160rexbidv 2438 . . . . . . 7 (𝑣 = 𝑔 → (∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑣)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)))
6261rabbidv 2675 . . . . . 6 (𝑣 = 𝑔 → {𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑣)𝑥 = (𝑦𝐺𝑧)} = {𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)})
63 fveq2 5421 . . . . . . . . 9 (𝑣 = 𝑔 → (2nd𝑣) = (2nd𝑔))
6463rexeqdv 2633 . . . . . . . 8 (𝑣 = 𝑔 → (∃𝑧 ∈ (2nd𝑣)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)))
6564rexbidv 2438 . . . . . . 7 (𝑣 = 𝑔 → (∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑣)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)))
6665rabbidv 2675 . . . . . 6 (𝑣 = 𝑔 → {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑣)𝑥 = (𝑦𝐺𝑧)} = {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)})
6762, 66opeq12d 3713 . . . . 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 7323 . . . . 5 𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝑤)∃𝑧 ∈ (1st𝑣)𝑥 = (𝑦𝐺𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑤)∃𝑧 ∈ (2nd𝑣)𝑥 = (𝑦𝐺𝑧)}⟩)
7058, 67, 69ovmpog 5905 . . . 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 1316 . . 3 ((𝑓P𝑔P) → (𝑓𝐹𝑔) = ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)}⟩)
729, 20, 71vtocl2ga 2754 . 2 ((𝐴P𝐵P) → (𝐴𝐹𝐵) = ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦𝐺𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦𝐺𝑧)}⟩)
73 eqeq1 2146 . . . . . 6 (𝑥 = 𝑞 → (𝑥 = (𝑦𝐺𝑧) ↔ 𝑞 = (𝑦𝐺𝑧)))
74732rexbidv 2460 . . . . 5 (𝑥 = 𝑞 → (∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑞 = (𝑦𝐺𝑧)))
75 oveq1 5781 . . . . . . 7 (𝑦 = 𝑟 → (𝑦𝐺𝑧) = (𝑟𝐺𝑧))
7675eqeq2d 2151 . . . . . 6 (𝑦 = 𝑟 → (𝑞 = (𝑦𝐺𝑧) ↔ 𝑞 = (𝑟𝐺𝑧)))
77 oveq2 5782 . . . . . . 7 (𝑧 = 𝑠 → (𝑟𝐺𝑧) = (𝑟𝐺𝑠))
7877eqeq2d 2151 . . . . . 6 (𝑧 = 𝑠 → (𝑞 = (𝑟𝐺𝑧) ↔ 𝑞 = (𝑟𝐺𝑠)))
7976, 78cbvrex2v 2666 . . . . 5 (∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑞 = (𝑦𝐺𝑧) ↔ ∃𝑟 ∈ (1st𝐴)∃𝑠 ∈ (1st𝐵)𝑞 = (𝑟𝐺𝑠))
8074, 79syl6bb 195 . . . 4 (𝑥 = 𝑞 → (∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑟 ∈ (1st𝐴)∃𝑠 ∈ (1st𝐵)𝑞 = (𝑟𝐺𝑠)))
8180cbvrabv 2685 . . 3 {𝑥Q ∣ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦𝐺𝑧)} = {𝑞Q ∣ ∃𝑟 ∈ (1st𝐴)∃𝑠 ∈ (1st𝐵)𝑞 = (𝑟𝐺𝑠)}
82732rexbidv 2460 . . . . 5 (𝑥 = 𝑞 → (∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑞 = (𝑦𝐺𝑧)))
8376, 78cbvrex2v 2666 . . . . 5 (∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑞 = (𝑦𝐺𝑧) ↔ ∃𝑟 ∈ (2nd𝐴)∃𝑠 ∈ (2nd𝐵)𝑞 = (𝑟𝐺𝑠))
8482, 83syl6bb 195 . . . 4 (𝑥 = 𝑞 → (∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑟 ∈ (2nd𝐴)∃𝑠 ∈ (2nd𝐵)𝑞 = (𝑟𝐺𝑠)))
8584cbvrabv 2685 . . 3 {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦𝐺𝑧)} = {𝑞Q ∣ ∃𝑟 ∈ (2nd𝐴)∃𝑠 ∈ (2nd𝐵)𝑞 = (𝑟𝐺𝑠)}
8681, 85opeq12i 3710 . 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 2188 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 962   = wceq 1331  wcel 1480  {cab 2125  wrex 2417  {crab 2420  Vcvv 2686  cop 3530   × cxp 4537  cfv 5123  (class class class)co 5774  cmpo 5776  1st c1st 6036  2nd c2nd 6037  Qcnq 7095  Pcnp 7106
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-qs 6435  df-ni 7119  df-nqqs 7163  df-inp 7281
This theorem is referenced by:  genpelvl  7327  genpelvu  7328  plpvlu  7353  mpvlu  7354
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