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Theorem genpassl 6622
 Description: Associativity of lower cuts. Lemma for genpassg 6624. (Contributed by Jim Kingdon, 11-Dec-2019.)
Hypotheses
Ref Expression
genpelvl.1 𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
genpelvl.2 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
genpassg.4 dom 𝐹 = (P × P)
genpassg.5 ((𝑓P𝑔P) → (𝑓𝐹𝑔) ∈ P)
genpassg.6 ((𝑓Q𝑔QQ) → ((𝑓𝐺𝑔)𝐺) = (𝑓𝐺(𝑔𝐺)))
Assertion
Ref Expression
genpassl ((𝐴P𝐵P𝐶P) → (1st ‘((𝐴𝐹𝐵)𝐹𝐶)) = (1st ‘(𝐴𝐹(𝐵𝐹𝐶))))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑓,𝑔,,𝑤,𝑣,𝐴   𝑥,𝐵,𝑦,𝑧,𝑓,𝑔,,𝑤,𝑣   𝑥,𝐺,𝑦,𝑧,𝑓,𝑔,,𝑤,𝑣   𝑓,𝐹,𝑔   𝐶,𝑓,𝑔,,𝑣,𝑤,𝑥,𝑦,𝑧   ,𝐹,𝑣,𝑤,𝑥,𝑦,𝑧

Proof of Theorem genpassl
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 prop 6573 . . . . . . . . 9 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2 elprnql 6579 . . . . . . . . 9 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑓 ∈ (1st𝐴)) → 𝑓Q)
31, 2sylan 267 . . . . . . . 8 ((𝐴P𝑓 ∈ (1st𝐴)) → 𝑓Q)
4 prop 6573 . . . . . . . . . . . . . . . 16 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
5 elprnql 6579 . . . . . . . . . . . . . . . 16 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑔 ∈ (1st𝐵)) → 𝑔Q)
64, 5sylan 267 . . . . . . . . . . . . . . 15 ((𝐵P𝑔 ∈ (1st𝐵)) → 𝑔Q)
7 r19.41v 2466 . . . . . . . . . . . . . . . . 17 (∃ ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (∃ ∈ (1st𝐶)𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)))
8 prop 6573 . . . . . . . . . . . . . . . . . . . . 21 (𝐶P → ⟨(1st𝐶), (2nd𝐶)⟩ ∈ P)
9 elprnql 6579 . . . . . . . . . . . . . . . . . . . . 21 ((⟨(1st𝐶), (2nd𝐶)⟩ ∈ P ∈ (1st𝐶)) → Q)
108, 9sylan 267 . . . . . . . . . . . . . . . . . . . 20 ((𝐶P ∈ (1st𝐶)) → Q)
11 oveq2 5520 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = (𝑔𝐺) → (𝑓𝐺𝑡) = (𝑓𝐺(𝑔𝐺)))
1211adantr 261 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑡 = (𝑔𝐺) ∧ (𝑓Q𝑔QQ)) → (𝑓𝐺𝑡) = (𝑓𝐺(𝑔𝐺)))
13 genpassg.6 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓Q𝑔QQ) → ((𝑓𝐺𝑔)𝐺) = (𝑓𝐺(𝑔𝐺)))
1413adantl 262 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑡 = (𝑔𝐺) ∧ (𝑓Q𝑔QQ)) → ((𝑓𝐺𝑔)𝐺) = (𝑓𝐺(𝑔𝐺)))
1512, 14eqtr4d 2075 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑡 = (𝑔𝐺) ∧ (𝑓Q𝑔QQ)) → (𝑓𝐺𝑡) = ((𝑓𝐺𝑔)𝐺))
1615eqeq2d 2051 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑡 = (𝑔𝐺) ∧ (𝑓Q𝑔QQ)) → (𝑥 = (𝑓𝐺𝑡) ↔ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
1716expcom 109 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑓Q𝑔QQ) → (𝑡 = (𝑔𝐺) → (𝑥 = (𝑓𝐺𝑡) ↔ 𝑥 = ((𝑓𝐺𝑔)𝐺))))
1817pm5.32d 423 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓Q𝑔QQ) → ((𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺))))
19183expa 1104 . . . . . . . . . . . . . . . . . . . 20 (((𝑓Q𝑔Q) ∧ Q) → ((𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺))))
2010, 19sylan2 270 . . . . . . . . . . . . . . . . . . 19 (((𝑓Q𝑔Q) ∧ (𝐶P ∈ (1st𝐶))) → ((𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺))))
2120anassrs 380 . . . . . . . . . . . . . . . . . 18 ((((𝑓Q𝑔Q) ∧ 𝐶P) ∧ ∈ (1st𝐶)) → ((𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺))))
2221rexbidva 2323 . . . . . . . . . . . . . . . . 17 (((𝑓Q𝑔Q) ∧ 𝐶P) → (∃ ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ ∃ ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺))))
237, 22syl5rbbr 184 . . . . . . . . . . . . . . . 16 (((𝑓Q𝑔Q) ∧ 𝐶P) → (∃ ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ (∃ ∈ (1st𝐶)𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡))))
2423an32s 502 . . . . . . . . . . . . . . 15 (((𝑓Q𝐶P) ∧ 𝑔Q) → (∃ ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ (∃ ∈ (1st𝐶)𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡))))
256, 24sylan2 270 . . . . . . . . . . . . . 14 (((𝑓Q𝐶P) ∧ (𝐵P𝑔 ∈ (1st𝐵))) → (∃ ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ (∃ ∈ (1st𝐶)𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡))))
2625anassrs 380 . . . . . . . . . . . . 13 ((((𝑓Q𝐶P) ∧ 𝐵P) ∧ 𝑔 ∈ (1st𝐵)) → (∃ ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ (∃ ∈ (1st𝐶)𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡))))
2726rexbidva 2323 . . . . . . . . . . . 12 (((𝑓Q𝐶P) ∧ 𝐵P) → (∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ ∃𝑔 ∈ (1st𝐵)(∃ ∈ (1st𝐶)𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡))))
28 r19.41v 2466 . . . . . . . . . . . 12 (∃𝑔 ∈ (1st𝐵)(∃ ∈ (1st𝐶)𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)))
2927, 28syl6bb 185 . . . . . . . . . . 11 (((𝑓Q𝐶P) ∧ 𝐵P) → (∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ (∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡))))
3029an31s 504 . . . . . . . . . 10 (((𝐵P𝐶P) ∧ 𝑓Q) → (∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ (∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡))))
3130exbidv 1706 . . . . . . . . 9 (((𝐵P𝐶P) ∧ 𝑓Q) → (∃𝑡𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ ∃𝑡(∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡))))
32 genpelvl.2 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
3332caovcl 5655 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑔QQ) → (𝑔𝐺) ∈ Q)
34 elisset 2568 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑔𝐺) ∈ Q → ∃𝑡 𝑡 = (𝑔𝐺))
3533, 34syl 14 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑔QQ) → ∃𝑡 𝑡 = (𝑔𝐺))
3635biantrurd 289 . . . . . . . . . . . . . . . . . . . . 21 ((𝑔QQ) → (𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ (∃𝑡 𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺))))
37 19.41v 1782 . . . . . . . . . . . . . . . . . . . . 21 (∃𝑡(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ (∃𝑡 𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
3836, 37syl6bbr 187 . . . . . . . . . . . . . . . . . . . 20 ((𝑔QQ) → (𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺))))
3910, 38sylan2 270 . . . . . . . . . . . . . . . . . . 19 ((𝑔Q ∧ (𝐶P ∈ (1st𝐶))) → (𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺))))
4039anassrs 380 . . . . . . . . . . . . . . . . . 18 (((𝑔Q𝐶P) ∧ ∈ (1st𝐶)) → (𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺))))
4140rexbidva 2323 . . . . . . . . . . . . . . . . 17 ((𝑔Q𝐶P) → (∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃ ∈ (1st𝐶)∃𝑡(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺))))
42 rexcom4 2577 . . . . . . . . . . . . . . . . 17 (∃ ∈ (1st𝐶)∃𝑡(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ ∃𝑡 ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
4341, 42syl6bb 185 . . . . . . . . . . . . . . . 16 ((𝑔Q𝐶P) → (∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡 ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺))))
4443ancoms 255 . . . . . . . . . . . . . . 15 ((𝐶P𝑔Q) → (∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡 ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺))))
456, 44sylan2 270 . . . . . . . . . . . . . 14 ((𝐶P ∧ (𝐵P𝑔 ∈ (1st𝐵))) → (∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡 ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺))))
4645anassrs 380 . . . . . . . . . . . . 13 (((𝐶P𝐵P) ∧ 𝑔 ∈ (1st𝐵)) → (∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡 ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺))))
4746rexbidva 2323 . . . . . . . . . . . 12 ((𝐶P𝐵P) → (∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑔 ∈ (1st𝐵)∃𝑡 ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺))))
4847ancoms 255 . . . . . . . . . . 11 ((𝐵P𝐶P) → (∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑔 ∈ (1st𝐵)∃𝑡 ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺))))
49 rexcom4 2577 . . . . . . . . . . 11 (∃𝑔 ∈ (1st𝐵)∃𝑡 ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ ∃𝑡𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
5048, 49syl6bb 185 . . . . . . . . . 10 ((𝐵P𝐶P) → (∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺))))
5150adantr 261 . . . . . . . . 9 (((𝐵P𝐶P) ∧ 𝑓Q) → (∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺))))
52 df-rex 2312 . . . . . . . . . . 11 (∃𝑡 ∈ (1st ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑡(𝑡 ∈ (1st ‘(𝐵𝐹𝐶)) ∧ 𝑥 = (𝑓𝐺𝑡)))
53 genpelvl.1 . . . . . . . . . . . . . 14 𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
5453, 32genpelvl 6610 . . . . . . . . . . . . 13 ((𝐵P𝐶P) → (𝑡 ∈ (1st ‘(𝐵𝐹𝐶)) ↔ ∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑡 = (𝑔𝐺)))
5554anbi1d 438 . . . . . . . . . . . 12 ((𝐵P𝐶P) → ((𝑡 ∈ (1st ‘(𝐵𝐹𝐶)) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡))))
5655exbidv 1706 . . . . . . . . . . 11 ((𝐵P𝐶P) → (∃𝑡(𝑡 ∈ (1st ‘(𝐵𝐹𝐶)) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ ∃𝑡(∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡))))
5752, 56syl5bb 181 . . . . . . . . . 10 ((𝐵P𝐶P) → (∃𝑡 ∈ (1st ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑡(∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡))))
5857adantr 261 . . . . . . . . 9 (((𝐵P𝐶P) ∧ 𝑓Q) → (∃𝑡 ∈ (1st ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑡(∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡))))
5931, 51, 583bitr4rd 210 . . . . . . . 8 (((𝐵P𝐶P) ∧ 𝑓Q) → (∃𝑡 ∈ (1st ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺)))
603, 59sylan2 270 . . . . . . 7 (((𝐵P𝐶P) ∧ (𝐴P𝑓 ∈ (1st𝐴))) → (∃𝑡 ∈ (1st ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺)))
6160anassrs 380 . . . . . 6 ((((𝐵P𝐶P) ∧ 𝐴P) ∧ 𝑓 ∈ (1st𝐴)) → (∃𝑡 ∈ (1st ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺)))
6261rexbidva 2323 . . . . 5 (((𝐵P𝐶P) ∧ 𝐴P) → (∃𝑓 ∈ (1st𝐴)∃𝑡 ∈ (1st ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺)))
6362ancoms 255 . . . 4 ((𝐴P ∧ (𝐵P𝐶P)) → (∃𝑓 ∈ (1st𝐴)∃𝑡 ∈ (1st ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺)))
64633impb 1100 . . 3 ((𝐴P𝐵P𝐶P) → (∃𝑓 ∈ (1st𝐴)∃𝑡 ∈ (1st ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺)))
65 genpassg.5 . . . . . 6 ((𝑓P𝑔P) → (𝑓𝐹𝑔) ∈ P)
6665caovcl 5655 . . . . 5 ((𝐵P𝐶P) → (𝐵𝐹𝐶) ∈ P)
6753, 32genpelvl 6610 . . . . 5 ((𝐴P ∧ (𝐵𝐹𝐶) ∈ P) → (𝑥 ∈ (1st ‘(𝐴𝐹(𝐵𝐹𝐶))) ↔ ∃𝑓 ∈ (1st𝐴)∃𝑡 ∈ (1st ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡)))
6866, 67sylan2 270 . . . 4 ((𝐴P ∧ (𝐵P𝐶P)) → (𝑥 ∈ (1st ‘(𝐴𝐹(𝐵𝐹𝐶))) ↔ ∃𝑓 ∈ (1st𝐴)∃𝑡 ∈ (1st ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡)))
69683impb 1100 . . 3 ((𝐴P𝐵P𝐶P) → (𝑥 ∈ (1st ‘(𝐴𝐹(𝐵𝐹𝐶))) ↔ ∃𝑓 ∈ (1st𝐴)∃𝑡 ∈ (1st ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡)))
70 df-rex 2312 . . . . 5 (∃𝑡 ∈ (1st ‘(𝐴𝐹𝐵))∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺) ↔ ∃𝑡(𝑡 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)))
7153, 32genpelvl 6610 . . . . . . . 8 ((𝐴P𝐵P) → (𝑡 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)𝑡 = (𝑓𝐺𝑔)))
72713adant3 924 . . . . . . 7 ((𝐴P𝐵P𝐶P) → (𝑡 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)𝑡 = (𝑓𝐺𝑔)))
7372anbi1d 438 . . . . . 6 ((𝐴P𝐵P𝐶P) → ((𝑡 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)) ↔ (∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺))))
7473exbidv 1706 . . . . 5 ((𝐴P𝐵P𝐶P) → (∃𝑡(𝑡 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)) ↔ ∃𝑡(∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺))))
7570, 74syl5bb 181 . . . 4 ((𝐴P𝐵P𝐶P) → (∃𝑡 ∈ (1st ‘(𝐴𝐹𝐵))∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺) ↔ ∃𝑡(∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺))))
7665caovcl 5655 . . . . . 6 ((𝐴P𝐵P) → (𝐴𝐹𝐵) ∈ P)
7753, 32genpelvl 6610 . . . . . 6 (((𝐴𝐹𝐵) ∈ P𝐶P) → (𝑥 ∈ (1st ‘((𝐴𝐹𝐵)𝐹𝐶)) ↔ ∃𝑡 ∈ (1st ‘(𝐴𝐹𝐵))∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)))
7876, 77sylan 267 . . . . 5 (((𝐴P𝐵P) ∧ 𝐶P) → (𝑥 ∈ (1st ‘((𝐴𝐹𝐵)𝐹𝐶)) ↔ ∃𝑡 ∈ (1st ‘(𝐴𝐹𝐵))∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)))
79783impa 1099 . . . 4 ((𝐴P𝐵P𝐶P) → (𝑥 ∈ (1st ‘((𝐴𝐹𝐵)𝐹𝐶)) ↔ ∃𝑡 ∈ (1st ‘(𝐴𝐹𝐵))∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)))
8032caovcl 5655 . . . . . . . . . . . . . . . . . . 19 ((𝑓Q𝑔Q) → (𝑓𝐺𝑔) ∈ Q)
81 elisset 2568 . . . . . . . . . . . . . . . . . . 19 ((𝑓𝐺𝑔) ∈ Q → ∃𝑡 𝑡 = (𝑓𝐺𝑔))
8280, 81syl 14 . . . . . . . . . . . . . . . . . 18 ((𝑓Q𝑔Q) → ∃𝑡 𝑡 = (𝑓𝐺𝑔))
8382biantrurd 289 . . . . . . . . . . . . . . . . 17 ((𝑓Q𝑔Q) → (∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ (∃𝑡 𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺))))
84 oveq1 5519 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = (𝑓𝐺𝑔) → (𝑡𝐺) = ((𝑓𝐺𝑔)𝐺))
8584eqeq2d 2051 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 = (𝑓𝐺𝑔) → (𝑥 = (𝑡𝐺) ↔ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
8685rexbidv 2327 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = (𝑓𝐺𝑔) → (∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺) ↔ ∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺)))
8786pm5.32i 427 . . . . . . . . . . . . . . . . . . 19 ((𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)) ↔ (𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺)))
8887exbii 1496 . . . . . . . . . . . . . . . . . 18 (∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)) ↔ ∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺)))
89 19.41v 1782 . . . . . . . . . . . . . . . . . 18 (∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ (∃𝑡 𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺)))
9088, 89bitri 173 . . . . . . . . . . . . . . . . 17 (∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)) ↔ (∃𝑡 𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺)))
9183, 90syl6bbr 187 . . . . . . . . . . . . . . . 16 ((𝑓Q𝑔Q) → (∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺))))
926, 91sylan2 270 . . . . . . . . . . . . . . 15 ((𝑓Q ∧ (𝐵P𝑔 ∈ (1st𝐵))) → (∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺))))
9392anassrs 380 . . . . . . . . . . . . . 14 (((𝑓Q𝐵P) ∧ 𝑔 ∈ (1st𝐵)) → (∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺))))
9493rexbidva 2323 . . . . . . . . . . . . 13 ((𝑓Q𝐵P) → (∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑔 ∈ (1st𝐵)∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺))))
95 rexcom4 2577 . . . . . . . . . . . . 13 (∃𝑔 ∈ (1st𝐵)∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)) ↔ ∃𝑡𝑔 ∈ (1st𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)))
9694, 95syl6bb 185 . . . . . . . . . . . 12 ((𝑓Q𝐵P) → (∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡𝑔 ∈ (1st𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺))))
9796ancoms 255 . . . . . . . . . . 11 ((𝐵P𝑓Q) → (∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡𝑔 ∈ (1st𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺))))
983, 97sylan2 270 . . . . . . . . . 10 ((𝐵P ∧ (𝐴P𝑓 ∈ (1st𝐴))) → (∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡𝑔 ∈ (1st𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺))))
9998anassrs 380 . . . . . . . . 9 (((𝐵P𝐴P) ∧ 𝑓 ∈ (1st𝐴)) → (∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡𝑔 ∈ (1st𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺))))
10099rexbidva 2323 . . . . . . . 8 ((𝐵P𝐴P) → (∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑓 ∈ (1st𝐴)∃𝑡𝑔 ∈ (1st𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺))))
101 rexcom4 2577 . . . . . . . 8 (∃𝑓 ∈ (1st𝐴)∃𝑡𝑔 ∈ (1st𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)) ↔ ∃𝑡𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)))
102100, 101syl6bb 185 . . . . . . 7 ((𝐵P𝐴P) → (∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺))))
103 r19.41v 2466 . . . . . . . . . 10 (∃𝑔 ∈ (1st𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)) ↔ (∃𝑔 ∈ (1st𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)))
104103rexbii 2331 . . . . . . . . 9 (∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)) ↔ ∃𝑓 ∈ (1st𝐴)(∃𝑔 ∈ (1st𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)))
105 r19.41v 2466 . . . . . . . . 9 (∃𝑓 ∈ (1st𝐴)(∃𝑔 ∈ (1st𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)) ↔ (∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)))
106104, 105bitri 173 . . . . . . . 8 (∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)) ↔ (∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)))
107106exbii 1496 . . . . . . 7 (∃𝑡𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)) ↔ ∃𝑡(∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)))
108102, 107syl6bb 185 . . . . . 6 ((𝐵P𝐴P) → (∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡(∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺))))
109108ancoms 255 . . . . 5 ((𝐴P𝐵P) → (∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡(∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺))))
1101093adant3 924 . . . 4 ((𝐴P𝐵P𝐶P) → (∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡(∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺))))
11175, 79, 1103bitr4d 209 . . 3 ((𝐴P𝐵P𝐶P) → (𝑥 ∈ (1st ‘((𝐴𝐹𝐵)𝐹𝐶)) ↔ ∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺)))
11264, 69, 1113bitr4rd 210 . 2 ((𝐴P𝐵P𝐶P) → (𝑥 ∈ (1st ‘((𝐴𝐹𝐵)𝐹𝐶)) ↔ 𝑥 ∈ (1st ‘(𝐴𝐹(𝐵𝐹𝐶)))))
113112eqrdv 2038 1 ((𝐴P𝐵P𝐶P) → (1st ‘((𝐴𝐹𝐵)𝐹𝐶)) = (1st ‘(𝐴𝐹(𝐵𝐹𝐶))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 885   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ∃wrex 2307  {crab 2310  ⟨cop 3378   × cxp 4343  dom cdm 4345  ‘cfv 4902  (class class class)co 5512   ↦ cmpt2 5514  1st c1st 5765  2nd c2nd 5766  Qcnq 6378  Pcnp 6389 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 6402  df-nqqs 6446  df-inp 6564 This theorem is referenced by:  genpassg  6624
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