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Theorem genipv 6969
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 5596 . . . 4 (𝑓 = 𝐴 → (𝑓𝐹𝑔) = (𝐴𝐹𝑔))
2 fveq2 5251 . . . . . . 7 (𝑓 = 𝐴 → (1st𝑓) = (1st𝐴))
32rexeqdv 2562 . . . . . 6 (𝑓 = 𝐴 → (∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)))
43rabbidv 2601 . . . . 5 (𝑓 = 𝐴 → {𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)} = {𝑥Q ∣ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)})
5 fveq2 5251 . . . . . . 7 (𝑓 = 𝐴 → (2nd𝑓) = (2nd𝐴))
65rexeqdv 2562 . . . . . 6 (𝑓 = 𝐴 → (∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)))
76rabbidv 2601 . . . . 5 (𝑓 = 𝐴 → {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)} = {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)})
84, 7opeq12d 3604 . . . 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 2097 . . 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 5597 . . . 4 (𝑔 = 𝐵 → (𝐴𝐹𝑔) = (𝐴𝐹𝐵))
11 fveq2 5251 . . . . . . . 8 (𝑔 = 𝐵 → (1st𝑔) = (1st𝐵))
1211rexeqdv 2562 . . . . . . 7 (𝑔 = 𝐵 → (∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦𝐺𝑧)))
1312rexbidv 2375 . . . . . 6 (𝑔 = 𝐵 → (∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦𝐺𝑧)))
1413rabbidv 2601 . . . . 5 (𝑔 = 𝐵 → {𝑥Q ∣ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)} = {𝑥Q ∣ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦𝐺𝑧)})
15 fveq2 5251 . . . . . . . 8 (𝑔 = 𝐵 → (2nd𝑔) = (2nd𝐵))
1615rexeqdv 2562 . . . . . . 7 (𝑔 = 𝐵 → (∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦𝐺𝑧)))
1716rexbidv 2375 . . . . . 6 (𝑔 = 𝐵 → (∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦𝐺𝑧)))
1817rabbidv 2601 . . . . 5 (𝑔 = 𝐵 → {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)} = {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦𝐺𝑧)})
1914, 18opeq12d 3604 . . . 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 2097 . . 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 6823 . . . . . . 7 Q ∈ V
2221a1i 9 . . . . . 6 ((𝑓P𝑔P) → Q ∈ V)
23 rabssab 3092 . . . . . . 7 {𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)} ⊆ {𝑥 ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)}
24 prop 6935 . . . . . . . . . . . 12 (𝑓P → ⟨(1st𝑓), (2nd𝑓)⟩ ∈ P)
25 elprnql 6941 . . . . . . . . . . . 12 ((⟨(1st𝑓), (2nd𝑓)⟩ ∈ P𝑦 ∈ (1st𝑓)) → 𝑦Q)
2624, 25sylan 277 . . . . . . . . . . 11 ((𝑓P𝑦 ∈ (1st𝑓)) → 𝑦Q)
27 prop 6935 . . . . . . . . . . . 12 (𝑔P → ⟨(1st𝑔), (2nd𝑔)⟩ ∈ P)
28 elprnql 6941 . . . . . . . . . . . 12 ((⟨(1st𝑔), (2nd𝑔)⟩ ∈ P𝑧 ∈ (1st𝑔)) → 𝑧Q)
2927, 28sylan 277 . . . . . . . . . . 11 ((𝑔P𝑧 ∈ (1st𝑔)) → 𝑧Q)
30 genp.2 . . . . . . . . . . . 12 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
31 eleq1 2145 . . . . . . . . . . . 12 (𝑥 = (𝑦𝐺𝑧) → (𝑥Q ↔ (𝑦𝐺𝑧) ∈ Q))
3230, 31syl5ibrcom 155 . . . . . . . . . . 11 ((𝑦Q𝑧Q) → (𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
3326, 29, 32syl2an 283 . . . . . . . . . 10 (((𝑓P𝑦 ∈ (1st𝑓)) ∧ (𝑔P𝑧 ∈ (1st𝑔))) → (𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
3433an4s 553 . . . . . . . . 9 (((𝑓P𝑔P) ∧ (𝑦 ∈ (1st𝑓) ∧ 𝑧 ∈ (1st𝑔))) → (𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
3534rexlimdvva 2490 . . . . . . . 8 ((𝑓P𝑔P) → (∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
3635abssdv 3079 . . . . . . 7 ((𝑓P𝑔P) → {𝑥 ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)} ⊆ Q)
3723, 36syl5ss 3021 . . . . . 6 ((𝑓P𝑔P) → {𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)} ⊆ Q)
3822, 37ssexd 3944 . . . . 5 ((𝑓P𝑔P) → {𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)} ∈ V)
39 rabssab 3092 . . . . . . 7 {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)} ⊆ {𝑥 ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)}
40 elprnqu 6942 . . . . . . . . . . . 12 ((⟨(1st𝑓), (2nd𝑓)⟩ ∈ P𝑦 ∈ (2nd𝑓)) → 𝑦Q)
4124, 40sylan 277 . . . . . . . . . . 11 ((𝑓P𝑦 ∈ (2nd𝑓)) → 𝑦Q)
42 elprnqu 6942 . . . . . . . . . . . 12 ((⟨(1st𝑔), (2nd𝑔)⟩ ∈ P𝑧 ∈ (2nd𝑔)) → 𝑧Q)
4327, 42sylan 277 . . . . . . . . . . 11 ((𝑔P𝑧 ∈ (2nd𝑔)) → 𝑧Q)
4441, 43, 32syl2an 283 . . . . . . . . . 10 (((𝑓P𝑦 ∈ (2nd𝑓)) ∧ (𝑔P𝑧 ∈ (2nd𝑔))) → (𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
4544an4s 553 . . . . . . . . 9 (((𝑓P𝑔P) ∧ (𝑦 ∈ (2nd𝑓) ∧ 𝑧 ∈ (2nd𝑔))) → (𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
4645rexlimdvva 2490 . . . . . . . 8 ((𝑓P𝑔P) → (∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
4746abssdv 3079 . . . . . . 7 ((𝑓P𝑔P) → {𝑥 ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)} ⊆ Q)
4839, 47syl5ss 3021 . . . . . 6 ((𝑓P𝑔P) → {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)} ⊆ Q)
4922, 48ssexd 3944 . . . . 5 ((𝑓P𝑔P) → {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)} ∈ V)
50 opelxp 4428 . . . . 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 408 . . . 4 ((𝑓P𝑔P) → ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)}⟩ ∈ (V × V))
52 fveq2 5251 . . . . . . . 8 (𝑤 = 𝑓 → (1st𝑤) = (1st𝑓))
5352rexeqdv 2562 . . . . . . 7 (𝑤 = 𝑓 → (∃𝑦 ∈ (1st𝑤)∃𝑧 ∈ (1st𝑣)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑣)𝑥 = (𝑦𝐺𝑧)))
5453rabbidv 2601 . . . . . 6 (𝑤 = 𝑓 → {𝑥Q ∣ ∃𝑦 ∈ (1st𝑤)∃𝑧 ∈ (1st𝑣)𝑥 = (𝑦𝐺𝑧)} = {𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑣)𝑥 = (𝑦𝐺𝑧)})
55 fveq2 5251 . . . . . . . 8 (𝑤 = 𝑓 → (2nd𝑤) = (2nd𝑓))
5655rexeqdv 2562 . . . . . . 7 (𝑤 = 𝑓 → (∃𝑦 ∈ (2nd𝑤)∃𝑧 ∈ (2nd𝑣)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑣)𝑥 = (𝑦𝐺𝑧)))
5756rabbidv 2601 . . . . . 6 (𝑤 = 𝑓 → {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑤)∃𝑧 ∈ (2nd𝑣)𝑥 = (𝑦𝐺𝑧)} = {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑣)𝑥 = (𝑦𝐺𝑧)})
5854, 57opeq12d 3604 . . . . 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 5251 . . . . . . . . 9 (𝑣 = 𝑔 → (1st𝑣) = (1st𝑔))
6059rexeqdv 2562 . . . . . . . 8 (𝑣 = 𝑔 → (∃𝑧 ∈ (1st𝑣)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)))
6160rexbidv 2375 . . . . . . 7 (𝑣 = 𝑔 → (∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑣)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)))
6261rabbidv 2601 . . . . . 6 (𝑣 = 𝑔 → {𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑣)𝑥 = (𝑦𝐺𝑧)} = {𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)})
63 fveq2 5251 . . . . . . . . 9 (𝑣 = 𝑔 → (2nd𝑣) = (2nd𝑔))
6463rexeqdv 2562 . . . . . . . 8 (𝑣 = 𝑔 → (∃𝑧 ∈ (2nd𝑣)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)))
6564rexbidv 2375 . . . . . . 7 (𝑣 = 𝑔 → (∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑣)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)))
6665rabbidv 2601 . . . . . 6 (𝑣 = 𝑔 → {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑣)𝑥 = (𝑦𝐺𝑧)} = {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)})
6762, 66opeq12d 3604 . . . . 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 6968 . . . . 5 𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝑤)∃𝑧 ∈ (1st𝑣)𝑥 = (𝑦𝐺𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑤)∃𝑧 ∈ (2nd𝑣)𝑥 = (𝑦𝐺𝑧)}⟩)
7058, 67, 69ovmpt2g 5712 . . . 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 1270 . . 3 ((𝑓P𝑔P) → (𝑓𝐹𝑔) = ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝑓)∃𝑧 ∈ (1st𝑔)𝑥 = (𝑦𝐺𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝑓)∃𝑧 ∈ (2nd𝑔)𝑥 = (𝑦𝐺𝑧)}⟩)
729, 20, 71vtocl2ga 2677 . 2 ((𝐴P𝐵P) → (𝐴𝐹𝐵) = ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦𝐺𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦𝐺𝑧)}⟩)
73 eqeq1 2089 . . . . . 6 (𝑥 = 𝑞 → (𝑥 = (𝑦𝐺𝑧) ↔ 𝑞 = (𝑦𝐺𝑧)))
74732rexbidv 2397 . . . . 5 (𝑥 = 𝑞 → (∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑞 = (𝑦𝐺𝑧)))
75 oveq1 5596 . . . . . . 7 (𝑦 = 𝑟 → (𝑦𝐺𝑧) = (𝑟𝐺𝑧))
7675eqeq2d 2094 . . . . . 6 (𝑦 = 𝑟 → (𝑞 = (𝑦𝐺𝑧) ↔ 𝑞 = (𝑟𝐺𝑧)))
77 oveq2 5597 . . . . . . 7 (𝑧 = 𝑠 → (𝑟𝐺𝑧) = (𝑟𝐺𝑠))
7877eqeq2d 2094 . . . . . 6 (𝑧 = 𝑠 → (𝑞 = (𝑟𝐺𝑧) ↔ 𝑞 = (𝑟𝐺𝑠)))
7976, 78cbvrex2v 2592 . . . . 5 (∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑞 = (𝑦𝐺𝑧) ↔ ∃𝑟 ∈ (1st𝐴)∃𝑠 ∈ (1st𝐵)𝑞 = (𝑟𝐺𝑠))
8074, 79syl6bb 194 . . . 4 (𝑥 = 𝑞 → (∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑟 ∈ (1st𝐴)∃𝑠 ∈ (1st𝐵)𝑞 = (𝑟𝐺𝑠)))
8180cbvrabv 2611 . . 3 {𝑥Q ∣ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦𝐺𝑧)} = {𝑞Q ∣ ∃𝑟 ∈ (1st𝐴)∃𝑠 ∈ (1st𝐵)𝑞 = (𝑟𝐺𝑠)}
82732rexbidv 2397 . . . . 5 (𝑥 = 𝑞 → (∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑞 = (𝑦𝐺𝑧)))
8376, 78cbvrex2v 2592 . . . . 5 (∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑞 = (𝑦𝐺𝑧) ↔ ∃𝑟 ∈ (2nd𝐴)∃𝑠 ∈ (2nd𝐵)𝑞 = (𝑟𝐺𝑠))
8482, 83syl6bb 194 . . . 4 (𝑥 = 𝑞 → (∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑟 ∈ (2nd𝐴)∃𝑠 ∈ (2nd𝐵)𝑞 = (𝑟𝐺𝑠)))
8584cbvrabv 2611 . . 3 {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦𝐺𝑧)} = {𝑞Q ∣ ∃𝑟 ∈ (2nd𝐴)∃𝑠 ∈ (2nd𝐵)𝑞 = (𝑟𝐺𝑠)}
8681, 85opeq12i 3601 . 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 2131 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 102  w3a 920   = wceq 1285  wcel 1434  {cab 2069  wrex 2354  {crab 2357  Vcvv 2612  cop 3425   × cxp 4397  cfv 4967  (class class class)co 5589  cmpt2 5591  1st c1st 5842  2nd c2nd 5843  Qcnq 6740  Pcnp 6751
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-setind 4315  ax-iinf 4365
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-id 4083  df-iom 4368  df-xp 4405  df-rel 4406  df-cnv 4407  df-co 4408  df-dm 4409  df-rn 4410  df-res 4411  df-ima 4412  df-iota 4932  df-fun 4969  df-fn 4970  df-f 4971  df-f1 4972  df-fo 4973  df-f1o 4974  df-fv 4975  df-ov 5592  df-oprab 5593  df-mpt2 5594  df-1st 5844  df-2nd 5845  df-qs 6226  df-ni 6764  df-nqqs 6808  df-inp 6926
This theorem is referenced by:  genpelvl  6972  genpelvu  6973  plpvlu  6998  mpvlu  6999
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