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Theorem genpassl 6984
Description: Associativity of lower cuts. Lemma for genpassg 6986. (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 6935 . . . . . . . . 9 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2 elprnql 6941 . . . . . . . . 9 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑓 ∈ (1st𝐴)) → 𝑓Q)
31, 2sylan 277 . . . . . . . 8 ((𝐴P𝑓 ∈ (1st𝐴)) → 𝑓Q)
4 prop 6935 . . . . . . . . . . . . . . . 16 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
5 elprnql 6941 . . . . . . . . . . . . . . . 16 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑔 ∈ (1st𝐵)) → 𝑔Q)
64, 5sylan 277 . . . . . . . . . . . . . . 15 ((𝐵P𝑔 ∈ (1st𝐵)) → 𝑔Q)
7 r19.41v 2516 . . . . . . . . . . . . . . . . 17 (∃ ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (∃ ∈ (1st𝐶)𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)))
8 prop 6935 . . . . . . . . . . . . . . . . . . . . 21 (𝐶P → ⟨(1st𝐶), (2nd𝐶)⟩ ∈ P)
9 elprnql 6941 . . . . . . . . . . . . . . . . . . . . 21 ((⟨(1st𝐶), (2nd𝐶)⟩ ∈ P ∈ (1st𝐶)) → Q)
108, 9sylan 277 . . . . . . . . . . . . . . . . . . . 20 ((𝐶P ∈ (1st𝐶)) → Q)
11 oveq2 5597 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = (𝑔𝐺) → (𝑓𝐺𝑡) = (𝑓𝐺(𝑔𝐺)))
1211adantr 270 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑡 = (𝑔𝐺) ∧ (𝑓Q𝑔QQ)) → (𝑓𝐺𝑡) = (𝑓𝐺(𝑔𝐺)))
13 genpassg.6 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓Q𝑔QQ) → ((𝑓𝐺𝑔)𝐺) = (𝑓𝐺(𝑔𝐺)))
1413adantl 271 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑡 = (𝑔𝐺) ∧ (𝑓Q𝑔QQ)) → ((𝑓𝐺𝑔)𝐺) = (𝑓𝐺(𝑔𝐺)))
1512, 14eqtr4d 2118 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑡 = (𝑔𝐺) ∧ (𝑓Q𝑔QQ)) → (𝑓𝐺𝑡) = ((𝑓𝐺𝑔)𝐺))
1615eqeq2d 2094 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑡 = (𝑔𝐺) ∧ (𝑓Q𝑔QQ)) → (𝑥 = (𝑓𝐺𝑡) ↔ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
1716expcom 114 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑓Q𝑔QQ) → (𝑡 = (𝑔𝐺) → (𝑥 = (𝑓𝐺𝑡) ↔ 𝑥 = ((𝑓𝐺𝑔)𝐺))))
1817pm5.32d 438 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓Q𝑔QQ) → ((𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺))))
19183expa 1139 . . . . . . . . . . . . . . . . . . . 20 (((𝑓Q𝑔Q) ∧ Q) → ((𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺))))
2010, 19sylan2 280 . . . . . . . . . . . . . . . . . . 19 (((𝑓Q𝑔Q) ∧ (𝐶P ∈ (1st𝐶))) → ((𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺))))
2120anassrs 392 . . . . . . . . . . . . . . . . . 18 ((((𝑓Q𝑔Q) ∧ 𝐶P) ∧ ∈ (1st𝐶)) → ((𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺))))
2221rexbidva 2371 . . . . . . . . . . . . . . . . 17 (((𝑓Q𝑔Q) ∧ 𝐶P) → (∃ ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ ∃ ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺))))
237, 22syl5rbbr 193 . . . . . . . . . . . . . . . 16 (((𝑓Q𝑔Q) ∧ 𝐶P) → (∃ ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ (∃ ∈ (1st𝐶)𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡))))
2423an32s 533 . . . . . . . . . . . . . . 15 (((𝑓Q𝐶P) ∧ 𝑔Q) → (∃ ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ (∃ ∈ (1st𝐶)𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡))))
256, 24sylan2 280 . . . . . . . . . . . . . 14 (((𝑓Q𝐶P) ∧ (𝐵P𝑔 ∈ (1st𝐵))) → (∃ ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ (∃ ∈ (1st𝐶)𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡))))
2625anassrs 392 . . . . . . . . . . . . 13 ((((𝑓Q𝐶P) ∧ 𝐵P) ∧ 𝑔 ∈ (1st𝐵)) → (∃ ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ (∃ ∈ (1st𝐶)𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡))))
2726rexbidva 2371 . . . . . . . . . . . 12 (((𝑓Q𝐶P) ∧ 𝐵P) → (∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ ∃𝑔 ∈ (1st𝐵)(∃ ∈ (1st𝐶)𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡))))
28 r19.41v 2516 . . . . . . . . . . . 12 (∃𝑔 ∈ (1st𝐵)(∃ ∈ (1st𝐶)𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡)))
2927, 28syl6bb 194 . . . . . . . . . . 11 (((𝑓Q𝐶P) ∧ 𝐵P) → (∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ (∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡))))
3029an31s 535 . . . . . . . . . 10 (((𝐵P𝐶P) ∧ 𝑓Q) → (∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ (∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡))))
3130exbidv 1748 . . . . . . . . 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 5732 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑔QQ) → (𝑔𝐺) ∈ Q)
34 elisset 2624 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑔𝐺) ∈ Q → ∃𝑡 𝑡 = (𝑔𝐺))
3533, 34syl 14 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑔QQ) → ∃𝑡 𝑡 = (𝑔𝐺))
3635biantrurd 299 . . . . . . . . . . . . . . . . . . . . 21 ((𝑔QQ) → (𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ (∃𝑡 𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺))))
37 19.41v 1825 . . . . . . . . . . . . . . . . . . . . 21 (∃𝑡(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ (∃𝑡 𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
3836, 37syl6bbr 196 . . . . . . . . . . . . . . . . . . . 20 ((𝑔QQ) → (𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺))))
3910, 38sylan2 280 . . . . . . . . . . . . . . . . . . 19 ((𝑔Q ∧ (𝐶P ∈ (1st𝐶))) → (𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺))))
4039anassrs 392 . . . . . . . . . . . . . . . . . 18 (((𝑔Q𝐶P) ∧ ∈ (1st𝐶)) → (𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺))))
4140rexbidva 2371 . . . . . . . . . . . . . . . . 17 ((𝑔Q𝐶P) → (∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃ ∈ (1st𝐶)∃𝑡(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺))))
42 rexcom4 2633 . . . . . . . . . . . . . . . . 17 (∃ ∈ (1st𝐶)∃𝑡(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ ∃𝑡 ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
4341, 42syl6bb 194 . . . . . . . . . . . . . . . 16 ((𝑔Q𝐶P) → (∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡 ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺))))
4443ancoms 264 . . . . . . . . . . . . . . 15 ((𝐶P𝑔Q) → (∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡 ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺))))
456, 44sylan2 280 . . . . . . . . . . . . . 14 ((𝐶P ∧ (𝐵P𝑔 ∈ (1st𝐵))) → (∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡 ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺))))
4645anassrs 392 . . . . . . . . . . . . 13 (((𝐶P𝐵P) ∧ 𝑔 ∈ (1st𝐵)) → (∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡 ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺))))
4746rexbidva 2371 . . . . . . . . . . . 12 ((𝐶P𝐵P) → (∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑔 ∈ (1st𝐵)∃𝑡 ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺))))
4847ancoms 264 . . . . . . . . . . 11 ((𝐵P𝐶P) → (∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑔 ∈ (1st𝐵)∃𝑡 ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺))))
49 rexcom4 2633 . . . . . . . . . . 11 (∃𝑔 ∈ (1st𝐵)∃𝑡 ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ ∃𝑡𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
5048, 49syl6bb 194 . . . . . . . . . 10 ((𝐵P𝐶P) → (∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺))))
5150adantr 270 . . . . . . . . 9 (((𝐵P𝐶P) ∧ 𝑓Q) → (∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)(𝑡 = (𝑔𝐺) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺))))
52 df-rex 2359 . . . . . . . . . . 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 6972 . . . . . . . . . . . . 13 ((𝐵P𝐶P) → (𝑡 ∈ (1st ‘(𝐵𝐹𝐶)) ↔ ∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑡 = (𝑔𝐺)))
5554anbi1d 453 . . . . . . . . . . . 12 ((𝐵P𝐶P) → ((𝑡 ∈ (1st ‘(𝐵𝐹𝐶)) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡))))
5655exbidv 1748 . . . . . . . . . . 11 ((𝐵P𝐶P) → (∃𝑡(𝑡 ∈ (1st ‘(𝐵𝐹𝐶)) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ ∃𝑡(∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡))))
5752, 56syl5bb 190 . . . . . . . . . 10 ((𝐵P𝐶P) → (∃𝑡 ∈ (1st ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑡(∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡))))
5857adantr 270 . . . . . . . . 9 (((𝐵P𝐶P) ∧ 𝑓Q) → (∃𝑡 ∈ (1st ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑡(∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑡 = (𝑔𝐺) ∧ 𝑥 = (𝑓𝐺𝑡))))
5931, 51, 583bitr4rd 219 . . . . . . . 8 (((𝐵P𝐶P) ∧ 𝑓Q) → (∃𝑡 ∈ (1st ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺)))
603, 59sylan2 280 . . . . . . 7 (((𝐵P𝐶P) ∧ (𝐴P𝑓 ∈ (1st𝐴))) → (∃𝑡 ∈ (1st ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺)))
6160anassrs 392 . . . . . 6 ((((𝐵P𝐶P) ∧ 𝐴P) ∧ 𝑓 ∈ (1st𝐴)) → (∃𝑡 ∈ (1st ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺)))
6261rexbidva 2371 . . . . 5 (((𝐵P𝐶P) ∧ 𝐴P) → (∃𝑓 ∈ (1st𝐴)∃𝑡 ∈ (1st ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺)))
6362ancoms 264 . . . 4 ((𝐴P ∧ (𝐵P𝐶P)) → (∃𝑓 ∈ (1st𝐴)∃𝑡 ∈ (1st ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺)))
64633impb 1135 . . 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 5732 . . . . 5 ((𝐵P𝐶P) → (𝐵𝐹𝐶) ∈ P)
6753, 32genpelvl 6972 . . . . 5 ((𝐴P ∧ (𝐵𝐹𝐶) ∈ P) → (𝑥 ∈ (1st ‘(𝐴𝐹(𝐵𝐹𝐶))) ↔ ∃𝑓 ∈ (1st𝐴)∃𝑡 ∈ (1st ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡)))
6866, 67sylan2 280 . . . 4 ((𝐴P ∧ (𝐵P𝐶P)) → (𝑥 ∈ (1st ‘(𝐴𝐹(𝐵𝐹𝐶))) ↔ ∃𝑓 ∈ (1st𝐴)∃𝑡 ∈ (1st ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡)))
69683impb 1135 . . 3 ((𝐴P𝐵P𝐶P) → (𝑥 ∈ (1st ‘(𝐴𝐹(𝐵𝐹𝐶))) ↔ ∃𝑓 ∈ (1st𝐴)∃𝑡 ∈ (1st ‘(𝐵𝐹𝐶))𝑥 = (𝑓𝐺𝑡)))
70 df-rex 2359 . . . . 5 (∃𝑡 ∈ (1st ‘(𝐴𝐹𝐵))∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺) ↔ ∃𝑡(𝑡 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)))
7153, 32genpelvl 6972 . . . . . . . 8 ((𝐴P𝐵P) → (𝑡 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)𝑡 = (𝑓𝐺𝑔)))
72713adant3 959 . . . . . . 7 ((𝐴P𝐵P𝐶P) → (𝑡 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)𝑡 = (𝑓𝐺𝑔)))
7372anbi1d 453 . . . . . 6 ((𝐴P𝐵P𝐶P) → ((𝑡 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)) ↔ (∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺))))
7473exbidv 1748 . . . . 5 ((𝐴P𝐵P𝐶P) → (∃𝑡(𝑡 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)) ↔ ∃𝑡(∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺))))
7570, 74syl5bb 190 . . . 4 ((𝐴P𝐵P𝐶P) → (∃𝑡 ∈ (1st ‘(𝐴𝐹𝐵))∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺) ↔ ∃𝑡(∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺))))
7665caovcl 5732 . . . . . 6 ((𝐴P𝐵P) → (𝐴𝐹𝐵) ∈ P)
7753, 32genpelvl 6972 . . . . . 6 (((𝐴𝐹𝐵) ∈ P𝐶P) → (𝑥 ∈ (1st ‘((𝐴𝐹𝐵)𝐹𝐶)) ↔ ∃𝑡 ∈ (1st ‘(𝐴𝐹𝐵))∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)))
7876, 77sylan 277 . . . . 5 (((𝐴P𝐵P) ∧ 𝐶P) → (𝑥 ∈ (1st ‘((𝐴𝐹𝐵)𝐹𝐶)) ↔ ∃𝑡 ∈ (1st ‘(𝐴𝐹𝐵))∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)))
79783impa 1134 . . . 4 ((𝐴P𝐵P𝐶P) → (𝑥 ∈ (1st ‘((𝐴𝐹𝐵)𝐹𝐶)) ↔ ∃𝑡 ∈ (1st ‘(𝐴𝐹𝐵))∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)))
8032caovcl 5732 . . . . . . . . . . . . . . . . . . 19 ((𝑓Q𝑔Q) → (𝑓𝐺𝑔) ∈ Q)
81 elisset 2624 . . . . . . . . . . . . . . . . . . 19 ((𝑓𝐺𝑔) ∈ Q → ∃𝑡 𝑡 = (𝑓𝐺𝑔))
8280, 81syl 14 . . . . . . . . . . . . . . . . . 18 ((𝑓Q𝑔Q) → ∃𝑡 𝑡 = (𝑓𝐺𝑔))
8382biantrurd 299 . . . . . . . . . . . . . . . . 17 ((𝑓Q𝑔Q) → (∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ (∃𝑡 𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺))))
84 oveq1 5596 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = (𝑓𝐺𝑔) → (𝑡𝐺) = ((𝑓𝐺𝑔)𝐺))
8584eqeq2d 2094 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 = (𝑓𝐺𝑔) → (𝑥 = (𝑡𝐺) ↔ 𝑥 = ((𝑓𝐺𝑔)𝐺)))
8685rexbidv 2375 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = (𝑓𝐺𝑔) → (∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺) ↔ ∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺)))
8786pm5.32i 442 . . . . . . . . . . . . . . . . . . 19 ((𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)) ↔ (𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺)))
8887exbii 1537 . . . . . . . . . . . . . . . . . 18 (∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)) ↔ ∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺)))
89 19.41v 1825 . . . . . . . . . . . . . . . . . 18 (∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺)) ↔ (∃𝑡 𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺)))
9088, 89bitri 182 . . . . . . . . . . . . . . . . 17 (∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)) ↔ (∃𝑡 𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺)))
9183, 90syl6bbr 196 . . . . . . . . . . . . . . . 16 ((𝑓Q𝑔Q) → (∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺))))
926, 91sylan2 280 . . . . . . . . . . . . . . 15 ((𝑓Q ∧ (𝐵P𝑔 ∈ (1st𝐵))) → (∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺))))
9392anassrs 392 . . . . . . . . . . . . . 14 (((𝑓Q𝐵P) ∧ 𝑔 ∈ (1st𝐵)) → (∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺))))
9493rexbidva 2371 . . . . . . . . . . . . 13 ((𝑓Q𝐵P) → (∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑔 ∈ (1st𝐵)∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺))))
95 rexcom4 2633 . . . . . . . . . . . . 13 (∃𝑔 ∈ (1st𝐵)∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)) ↔ ∃𝑡𝑔 ∈ (1st𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)))
9694, 95syl6bb 194 . . . . . . . . . . . 12 ((𝑓Q𝐵P) → (∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡𝑔 ∈ (1st𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺))))
9796ancoms 264 . . . . . . . . . . 11 ((𝐵P𝑓Q) → (∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡𝑔 ∈ (1st𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺))))
983, 97sylan2 280 . . . . . . . . . 10 ((𝐵P ∧ (𝐴P𝑓 ∈ (1st𝐴))) → (∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡𝑔 ∈ (1st𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺))))
9998anassrs 392 . . . . . . . . 9 (((𝐵P𝐴P) ∧ 𝑓 ∈ (1st𝐴)) → (∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡𝑔 ∈ (1st𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺))))
10099rexbidva 2371 . . . . . . . 8 ((𝐵P𝐴P) → (∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑓 ∈ (1st𝐴)∃𝑡𝑔 ∈ (1st𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺))))
101 rexcom4 2633 . . . . . . . 8 (∃𝑓 ∈ (1st𝐴)∃𝑡𝑔 ∈ (1st𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)) ↔ ∃𝑡𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)))
102100, 101syl6bb 194 . . . . . . 7 ((𝐵P𝐴P) → (∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺))))
103 r19.41v 2516 . . . . . . . . . 10 (∃𝑔 ∈ (1st𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)) ↔ (∃𝑔 ∈ (1st𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)))
104103rexbii 2379 . . . . . . . . 9 (∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)) ↔ ∃𝑓 ∈ (1st𝐴)(∃𝑔 ∈ (1st𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)))
105 r19.41v 2516 . . . . . . . . 9 (∃𝑓 ∈ (1st𝐴)(∃𝑔 ∈ (1st𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)) ↔ (∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)))
106104, 105bitri 182 . . . . . . . 8 (∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)) ↔ (∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)))
107106exbii 1537 . . . . . . 7 (∃𝑡𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)(𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)) ↔ ∃𝑡(∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺)))
108102, 107syl6bb 194 . . . . . 6 ((𝐵P𝐴P) → (∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡(∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺))))
109108ancoms 264 . . . . 5 ((𝐴P𝐵P) → (∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡(∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺))))
1101093adant3 959 . . . 4 ((𝐴P𝐵P𝐶P) → (∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺) ↔ ∃𝑡(∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)𝑡 = (𝑓𝐺𝑔) ∧ ∃ ∈ (1st𝐶)𝑥 = (𝑡𝐺))))
11175, 79, 1103bitr4d 218 . . 3 ((𝐴P𝐵P𝐶P) → (𝑥 ∈ (1st ‘((𝐴𝐹𝐵)𝐹𝐶)) ↔ ∃𝑓 ∈ (1st𝐴)∃𝑔 ∈ (1st𝐵)∃ ∈ (1st𝐶)𝑥 = ((𝑓𝐺𝑔)𝐺)))
11264, 69, 1113bitr4rd 219 . 2 ((𝐴P𝐵P𝐶P) → (𝑥 ∈ (1st ‘((𝐴𝐹𝐵)𝐹𝐶)) ↔ 𝑥 ∈ (1st ‘(𝐴𝐹(𝐵𝐹𝐶)))))
113112eqrdv 2081 1 ((𝐴P𝐵P𝐶P) → (1st ‘((𝐴𝐹𝐵)𝐹𝐶)) = (1st ‘(𝐴𝐹(𝐵𝐹𝐶))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 920   = wceq 1285  wex 1422  wcel 1434  wrex 2354  {crab 2357  cop 3425   × cxp 4397  dom cdm 4399  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:  genpassg  6986
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