Theorem satfdmlem 34348

Description: Lemma for satfdm 34349. (Contributed by AV, 12-Oct-2023.)

Assertion

Ref	Expression
satfdmlem	⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)) → ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑌)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑌)𝑥 = ((1st ‘𝑎)⊼𝑔(1st ‘𝑏)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑎))))

Distinct variable groups:   𝐸,𝑎,𝑏,𝑖,𝑢,𝑣   𝐹,𝑎,𝑏,𝑖,𝑢,𝑣   𝑀,𝑎,𝑏,𝑖,𝑢,𝑣   𝑁,𝑎,𝑏,𝑖,𝑢,𝑣   𝑉,𝑎,𝑏,𝑖,𝑢,𝑣   𝑊,𝑎,𝑏,𝑖,𝑢,𝑣   𝑌,𝑎,𝑏,𝑖,𝑢,𝑣   𝑥,𝑎,𝑏,𝑖,𝑢,𝑣

Allowed substitution hints:   𝐸(𝑥)   𝐹(𝑥)   𝑀(𝑥)   𝑁(𝑥)   𝑉(𝑥)   𝑊(𝑥)   𝑌(𝑥)

Proof of Theorem satfdmlem

Dummy variables 𝑟 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		satfrel 34347	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑌 ∈ ω) → Rel ((𝑀 Sat 𝐸)‘𝑌))
2	1	adantr 482	. . . 4 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) → Rel ((𝑀 Sat 𝐸)‘𝑌))
3		1stdm 8023	. . . 4 ⊢ ((Rel ((𝑀 Sat 𝐸)‘𝑌) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) → (1st ‘𝑢) ∈ dom ((𝑀 Sat 𝐸)‘𝑌))
4	2, 3	sylan 581	. . 3 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) → (1st ‘𝑢) ∈ dom ((𝑀 Sat 𝐸)‘𝑌))
5		eleq2 2823	. . . . . 6 ⊢ (dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌) → ((1st ‘𝑢) ∈ dom ((𝑀 Sat 𝐸)‘𝑌) ↔ (1st ‘𝑢) ∈ dom ((𝑁 Sat 𝐹)‘𝑌)))
6	5	adantl 483	. . . . 5 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) → ((1st ‘𝑢) ∈ dom ((𝑀 Sat 𝐸)‘𝑌) ↔ (1st ‘𝑢) ∈ dom ((𝑁 Sat 𝐹)‘𝑌)))
7	6	adantr 482	. . . 4 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) → ((1st ‘𝑢) ∈ dom ((𝑀 Sat 𝐸)‘𝑌) ↔ (1st ‘𝑢) ∈ dom ((𝑁 Sat 𝐹)‘𝑌)))
8		fvex 6902	. . . . . 6 ⊢ (1st ‘𝑢) ∈ V
9		eldm2g 5898	. . . . . 6 ⊢ ((1st ‘𝑢) ∈ V → ((1st ‘𝑢) ∈ dom ((𝑁 Sat 𝐹)‘𝑌) ↔ ∃𝑠⟨(1st ‘𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)))
10	8, 9	ax-mp 5	. . . . 5 ⊢ ((1st ‘𝑢) ∈ dom ((𝑁 Sat 𝐹)‘𝑌) ↔ ∃𝑠⟨(1st ‘𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌))
11		simpr 486	. . . . . . . 8 ⊢ (((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st ‘𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) → ⟨(1st ‘𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌))
12	2	ad4antr 731	. . . . . . . . . . . 12 ⊢ (((((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st ‘𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑎 = ⟨(1st ‘𝑢), 𝑠⟩) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)) → Rel ((𝑀 Sat 𝐸)‘𝑌))
13		1stdm 8023	. . . . . . . . . . . 12 ⊢ ((Rel ((𝑀 Sat 𝐸)‘𝑌) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)) → (1st ‘𝑣) ∈ dom ((𝑀 Sat 𝐸)‘𝑌))
14	12, 13	sylancom 589	. . . . . . . . . . 11 ⊢ (((((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st ‘𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑎 = ⟨(1st ‘𝑢), 𝑠⟩) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)) → (1st ‘𝑣) ∈ dom ((𝑀 Sat 𝐸)‘𝑌))
15		eleq2 2823	. . . . . . . . . . . . 13 ⊢ (dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌) → ((1st ‘𝑣) ∈ dom ((𝑀 Sat 𝐸)‘𝑌) ↔ (1st ‘𝑣) ∈ dom ((𝑁 Sat 𝐹)‘𝑌)))
16	15	ad5antlr 734	. . . . . . . . . . . 12 ⊢ (((((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st ‘𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑎 = ⟨(1st ‘𝑢), 𝑠⟩) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)) → ((1st ‘𝑣) ∈ dom ((𝑀 Sat 𝐸)‘𝑌) ↔ (1st ‘𝑣) ∈ dom ((𝑁 Sat 𝐹)‘𝑌)))
17		fvex 6902	. . . . . . . . . . . . . 14 ⊢ (1st ‘𝑣) ∈ V
18		eldm2g 5898	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑣) ∈ V → ((1st ‘𝑣) ∈ dom ((𝑁 Sat 𝐹)‘𝑌) ↔ ∃𝑟⟨(1st ‘𝑣), 𝑟⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)))
19	17, 18	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑣) ∈ dom ((𝑁 Sat 𝐹)‘𝑌) ↔ ∃𝑟⟨(1st ‘𝑣), 𝑟⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌))
20		simpr 486	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st ‘𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑎 = ⟨(1st ‘𝑢), 𝑠⟩) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st ‘𝑣), 𝑟⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) → ⟨(1st ‘𝑣), 𝑟⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌))
21		vex 3479	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑠 ∈ V
22	8, 21	op1std 7982	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = ⟨(1st ‘𝑢), 𝑠⟩ → (1st ‘𝑎) = (1st ‘𝑢))
23	22	eqcomd 2739	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = ⟨(1st ‘𝑢), 𝑠⟩ → (1st ‘𝑢) = (1st ‘𝑎))
24	23	ad3antlr 730	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st ‘𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑎 = ⟨(1st ‘𝑢), 𝑠⟩) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st ‘𝑣), 𝑟⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) → (1st ‘𝑢) = (1st ‘𝑎))
25		vex 3479	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑟 ∈ V
26	17, 25	op1std 7982	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 = ⟨(1st ‘𝑣), 𝑟⟩ → (1st ‘𝑏) = (1st ‘𝑣))
27	26	eqcomd 2739	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = ⟨(1st ‘𝑣), 𝑟⟩ → (1st ‘𝑣) = (1st ‘𝑏))
28	24, 27	oveqan12d 7425	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st ‘𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑎 = ⟨(1st ‘𝑢), 𝑠⟩) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st ‘𝑣), 𝑟⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑏 = ⟨(1st ‘𝑣), 𝑟⟩) → ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) = ((1st ‘𝑎)⊼𝑔(1st ‘𝑏)))
29	28	eqeq2d 2744	. . . . . . . . . . . . . . . . 17 ⊢ (((((((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st ‘𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑎 = ⟨(1st ‘𝑢), 𝑠⟩) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st ‘𝑣), 𝑟⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑏 = ⟨(1st ‘𝑣), 𝑟⟩) → (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ↔ 𝑥 = ((1st ‘𝑎)⊼𝑔(1st ‘𝑏))))
30	29	biimpd 228	. . . . . . . . . . . . . . . 16 ⊢ (((((((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st ‘𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑎 = ⟨(1st ‘𝑢), 𝑠⟩) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st ‘𝑣), 𝑟⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑏 = ⟨(1st ‘𝑣), 𝑟⟩) → (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) → 𝑥 = ((1st ‘𝑎)⊼𝑔(1st ‘𝑏))))
31	20, 30	rspcimedv 3604	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st ‘𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑎 = ⟨(1st ‘𝑢), 𝑠⟩) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st ‘𝑣), 𝑟⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) → (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) → ∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑌)𝑥 = ((1st ‘𝑎)⊼𝑔(1st ‘𝑏))))
32	31	ex 414	. . . . . . . . . . . . . 14 ⊢ (((((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st ‘𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑎 = ⟨(1st ‘𝑢), 𝑠⟩) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)) → (⟨(1st ‘𝑣), 𝑟⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌) → (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) → ∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑌)𝑥 = ((1st ‘𝑎)⊼𝑔(1st ‘𝑏)))))
33	32	exlimdv 1937	. . . . . . . . . . . . 13 ⊢ (((((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st ‘𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑎 = ⟨(1st ‘𝑢), 𝑠⟩) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)) → (∃𝑟⟨(1st ‘𝑣), 𝑟⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌) → (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) → ∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑌)𝑥 = ((1st ‘𝑎)⊼𝑔(1st ‘𝑏)))))
34	19, 33	biimtrid 241	. . . . . . . . . . . 12 ⊢ (((((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st ‘𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑎 = ⟨(1st ‘𝑢), 𝑠⟩) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)) → ((1st ‘𝑣) ∈ dom ((𝑁 Sat 𝐹)‘𝑌) → (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) → ∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑌)𝑥 = ((1st ‘𝑎)⊼𝑔(1st ‘𝑏)))))
35	16, 34	sylbid 239	. . . . . . . . . . 11 ⊢ (((((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st ‘𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑎 = ⟨(1st ‘𝑢), 𝑠⟩) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)) → ((1st ‘𝑣) ∈ dom ((𝑀 Sat 𝐸)‘𝑌) → (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) → ∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑌)𝑥 = ((1st ‘𝑎)⊼𝑔(1st ‘𝑏)))))
36	14, 35	mpd 15	. . . . . . . . . 10 ⊢ (((((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st ‘𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑎 = ⟨(1st ‘𝑢), 𝑠⟩) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)) → (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) → ∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑌)𝑥 = ((1st ‘𝑎)⊼𝑔(1st ‘𝑏))))
37	36	rexlimdva 3156	. . . . . . . . 9 ⊢ ((((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st ‘𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑎 = ⟨(1st ‘𝑢), 𝑠⟩) → (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) → ∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑌)𝑥 = ((1st ‘𝑎)⊼𝑔(1st ‘𝑏))))
38		eqidd 2734	. . . . . . . . . . . . . 14 ⊢ (𝑎 = ⟨(1st ‘𝑢), 𝑠⟩ → 𝑖 = 𝑖)
39	38, 23	goaleq12d 34331	. . . . . . . . . . . . 13 ⊢ (𝑎 = ⟨(1st ‘𝑢), 𝑠⟩ → ∀𝑔𝑖(1st ‘𝑢) = ∀𝑔𝑖(1st ‘𝑎))
40	39	eqeq2d 2744	. . . . . . . . . . . 12 ⊢ (𝑎 = ⟨(1st ‘𝑢), 𝑠⟩ → (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ↔ 𝑥 = ∀𝑔𝑖(1st ‘𝑎)))
41	40	biimpd 228	. . . . . . . . . . 11 ⊢ (𝑎 = ⟨(1st ‘𝑢), 𝑠⟩ → (𝑥 = ∀𝑔𝑖(1st ‘𝑢) → 𝑥 = ∀𝑔𝑖(1st ‘𝑎)))
42	41	adantl 483	. . . . . . . . . 10 ⊢ ((((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st ‘𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑎 = ⟨(1st ‘𝑢), 𝑠⟩) → (𝑥 = ∀𝑔𝑖(1st ‘𝑢) → 𝑥 = ∀𝑔𝑖(1st ‘𝑎)))
43	42	reximdv 3171	. . . . . . . . 9 ⊢ ((((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st ‘𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑎 = ⟨(1st ‘𝑢), 𝑠⟩) → (∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢) → ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑎)))
44	37, 43	orim12d 964	. . . . . . . 8 ⊢ ((((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st ‘𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑎 = ⟨(1st ‘𝑢), 𝑠⟩) → ((∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)) → (∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑌)𝑥 = ((1st ‘𝑎)⊼𝑔(1st ‘𝑏)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑎))))
45	11, 44	rspcimedv 3604	. . . . . . 7 ⊢ (((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st ‘𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) → ((∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)) → ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑌)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑌)𝑥 = ((1st ‘𝑎)⊼𝑔(1st ‘𝑏)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑎))))
46	45	ex 414	. . . . . 6 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) → (⟨(1st ‘𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌) → ((∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)) → ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑌)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑌)𝑥 = ((1st ‘𝑎)⊼𝑔(1st ‘𝑏)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑎)))))
47	46	exlimdv 1937	. . . . 5 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) → (∃𝑠⟨(1st ‘𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌) → ((∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)) → ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑌)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑌)𝑥 = ((1st ‘𝑎)⊼𝑔(1st ‘𝑏)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑎)))))
48	10, 47	biimtrid 241	. . . 4 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) → ((1st ‘𝑢) ∈ dom ((𝑁 Sat 𝐹)‘𝑌) → ((∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)) → ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑌)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑌)𝑥 = ((1st ‘𝑎)⊼𝑔(1st ‘𝑏)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑎)))))
49	7, 48	sylbid 239	. . 3 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) → ((1st ‘𝑢) ∈ dom ((𝑀 Sat 𝐸)‘𝑌) → ((∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)) → ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑌)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑌)𝑥 = ((1st ‘𝑎)⊼𝑔(1st ‘𝑏)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑎)))))
50	4, 49	mpd 15	. 2 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) → ((∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)) → ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑌)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑌)𝑥 = ((1st ‘𝑎)⊼𝑔(1st ‘𝑏)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑎))))
51	50	rexlimdva 3156	1 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)) → ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑌)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑌)𝑥 = ((1st ‘𝑎)⊼𝑔(1st ‘𝑏)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑎))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542  ∃wex 1782   ∈ wcel 2107  ∃wrex 3071  Vcvv 3475  ⟨cop 4634  dom cdm 5676  Rel wrel 5681  ‘cfv 6541  (class class class)co 7406  ωcom 7852  1^st c1st 7970  ⊼_𝑔cgna 34314  ∀_𝑔cgol 34315   Sat csat 34316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-inf2 9633

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-1st 7972  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-goel 34320  df-goal 34322  df-sat 34323

This theorem is referenced by:  satfdm  34349