Theorem satfdmlem 35435

Description: Lemma for satfdm 35436. (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 35434	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑌 ∈ ω) → Rel ((𝑀 Sat 𝐸)‘𝑌))
2	1	adantr 480	. . . 4 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) → Rel ((𝑀 Sat 𝐸)‘𝑌))
3		1stdm 7980	. . . 4 ⊢ ((Rel ((𝑀 Sat 𝐸)‘𝑌) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) → (1st ‘𝑢) ∈ dom ((𝑀 Sat 𝐸)‘𝑌))
4	2, 3	sylan 580	. . 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 2822	. . . . . 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 481	. . . . 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 480	. . . 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 6843	. . . . . 6 ⊢ (1st ‘𝑢) ∈ V
9		eldm2g 5845	. . . . . 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 484	. . . . . . . 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 732	. . . . . . . . . . . 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 7980	. . . . . . . . . . . 12 ⊢ ((Rel ((𝑀 Sat 𝐸)‘𝑌) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)) → (1st ‘𝑣) ∈ dom ((𝑀 Sat 𝐸)‘𝑌))
14	12, 13	sylancom 588	. . . . . . . . . . 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 2822	. . . . . . . . . . . . 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 735	. . . . . . . . . . . 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 6843	. . . . . . . . . . . . . 14 ⊢ (1st ‘𝑣) ∈ V
18		eldm2g 5845	. . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . 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 3441	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑠 ∈ V
22	8, 21	op1std 7939	. . . . . . . . . . . . . . . . . . . . 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 731	. . . . . . . . . . . . . . . . . . 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 3441	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑟 ∈ V
26	17, 25	op1std 7939	. . . . . . . . . . . . . . . . . . . 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 7373	. . . . . . . . . . . . . . . . . 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 229	. . . . . . . . . . . . . . . 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 3564	. . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . 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 1934	. . . . . . . . . . . . 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 242	. . . . . . . . . . . 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 240	. . . . . . . . . . 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 3134	. . . . . . . . 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 35418	. . . . . . . . . . . . 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 229	. . . . . . . . . . 11 ⊢ (𝑎 = ⟨(1st ‘𝑢), 𝑠⟩ → (𝑥 = ∀𝑔𝑖(1st ‘𝑢) → 𝑥 = ∀𝑔𝑖(1st ‘𝑎)))
42	41	adantl 481	. . . . . . . . . 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 3148	. . . . . . . . 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 966	. . . . . . . 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 3564	. . . . . . 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 412	. . . . . 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 1934	. . . . 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 242	. . . 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 240	. . 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 3134	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 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∃wrex 3057  Vcvv 3437  ⟨cop 4583  dom cdm 5621  Rel wrel 5626  ‘cfv 6488  (class class class)co 7354  ωcom 7804  1^st c1st 7927  ⊼_𝑔cgna 35401  ∀_𝑔cgol 35402   Sat csat 35403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7676  ax-inf2 9540

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-int 4900  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6255  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-ov 7357  df-oprab 7358  df-mpo 7359  df-om 7805  df-1st 7929  df-2nd 7930  df-frecs 8219  df-wrecs 8250  df-recs 8299  df-rdg 8337  df-goel 35407  df-goal 35409  df-sat 35410

This theorem is referenced by:  satfdm  35436