2ndcdisj - Metamath Proof Explorer

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Theorem 2ndcdisj 21259

Description: Any disjoint family of open sets in a second-countable space is countable. (The sets are required to be nonempty because otherwise there could be many empty sets in the family.) (Contributed by Mario Carneiro, 21-Mar-2015.) (Proof shortened by Mario Carneiro, 9-Apr-2015.) (Revised by NM, 17-Jun-2017.)

**Assertion**
Ref	Expression
2ndcdisj	⊢ ((𝐽 ∈ 2^nd𝜔 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (𝐽 ∖ {∅}) ∧ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → 𝐴 ≼ ω)

Distinct variable groups: 𝑥,𝑦,𝐴 𝑦,𝐵 𝑥,𝐽 Allowed substitution hints: 𝐵(𝑥) 𝐽(𝑦)

Proof of Theorem 2ndcdisj Dummy variables 𝑓 𝑏 𝑤 𝑧 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		is2ndc 21249	. . 3 ⊢ (𝐽 ∈ 2^nd𝜔 ↔ ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽))
2		omex 8540	. . . . . . 7 ⊢ ω ∈ V
3	2	brdom 7967	. . . . . 6 ⊢ (𝑏 ≼ ω ↔ ∃𝑓 𝑓:𝑏–1-1→ω)
4		ssrab2 3687	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} ⊆ ran 𝑓
5		f1f 6101	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓:𝑏–1-1→ω → 𝑓:𝑏⟶ω)
6		frn 6053	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓:𝑏⟶ω → ran 𝑓 ⊆ ω)
7	5, 6	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓:𝑏–1-1→ω → ran 𝑓 ⊆ ω)
8	7	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) → ran 𝑓 ⊆ ω)
9	4, 8	syl5ss 3614	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) → {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} ⊆ ω)
10	9	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ((topGen‘𝑏) ∖ {∅}))) → {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} ⊆ ω)
11		eldifsn 4317	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐵 ∈ ((topGen‘𝑏) ∖ {∅}) ↔ (𝐵 ∈ (topGen‘𝑏) ∧ 𝐵 ≠ ∅))
12		n0 3931	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐵)
13		tg2 20769	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐵 ∈ (topGen‘𝑏) ∧ 𝑦 ∈ 𝐵) → ∃𝑧 ∈ 𝑏 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝐵))
14		omsson 7069	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ω ⊆ On
15	9, 14	syl6ss 3615	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) → {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} ⊆ On)
16	15	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ 𝑥 ∈ 𝐴) ∧ (𝑧 ∈ 𝑏 ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝐵))) → {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} ⊆ On)
17		f1fn 6102	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑓:𝑏–1-1→ω → 𝑓 Fn 𝑏)
18	17	ad3antlr 767	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ 𝑥 ∈ 𝐴) ∧ (𝑧 ∈ 𝑏 ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝐵))) → 𝑓 Fn 𝑏)
19		simprl 794	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ 𝑥 ∈ 𝐴) ∧ (𝑧 ∈ 𝑏 ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝐵))) → 𝑧 ∈ 𝑏)
20		fnfvelrn 6356	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑓 Fn 𝑏 ∧ 𝑧 ∈ 𝑏) → (𝑓‘𝑧) ∈ ran 𝑓)
21	18, 19, 20	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ 𝑥 ∈ 𝐴) ∧ (𝑧 ∈ 𝑏 ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝐵))) → (𝑓‘𝑧) ∈ ran 𝑓)
22		f1f1orn 6148	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑓:𝑏–1-1→ω → 𝑓:𝑏–1-1-onto→ran 𝑓)
23	22	ad3antlr 767	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ 𝑥 ∈ 𝐴) ∧ (𝑧 ∈ 𝑏 ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝐵))) → 𝑓:𝑏–1-1-onto→ran 𝑓)
24		f1ocnvfv1 6532	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑓:𝑏–1-1-onto→ran 𝑓 ∧ 𝑧 ∈ 𝑏) → (^◡𝑓‘(𝑓‘𝑧)) = 𝑧)
25	23, 19, 24	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ 𝑥 ∈ 𝐴) ∧ (𝑧 ∈ 𝑏 ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝐵))) → (^◡𝑓‘(𝑓‘𝑧)) = 𝑧)
26		simprrr 805	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ 𝑥 ∈ 𝐴) ∧ (𝑧 ∈ 𝑏 ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝐵))) → 𝑧 ⊆ 𝐵)
27		selpw 4165	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑧 ∈ 𝒫 𝐵 ↔ 𝑧 ⊆ 𝐵)
28	26, 27	sylibr 224	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ 𝑥 ∈ 𝐴) ∧ (𝑧 ∈ 𝑏 ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝐵))) → 𝑧 ∈ 𝒫 𝐵)
29		simprrl 804	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ 𝑥 ∈ 𝐴) ∧ (𝑧 ∈ 𝑏 ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝐵))) → 𝑦 ∈ 𝑧)
30		ne0i 3921	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 ∈ 𝑧 → 𝑧 ≠ ∅)
31	29, 30	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ 𝑥 ∈ 𝐴) ∧ (𝑧 ∈ 𝑏 ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝐵))) → 𝑧 ≠ ∅)
32		eldifsn 4317	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑧 ∈ (𝒫 𝐵 ∖ {∅}) ↔ (𝑧 ∈ 𝒫 𝐵 ∧ 𝑧 ≠ ∅))
33	28, 31, 32	sylanbrc 698	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ 𝑥 ∈ 𝐴) ∧ (𝑧 ∈ 𝑏 ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝐵))) → 𝑧 ∈ (𝒫 𝐵 ∖ {∅}))
34	25, 33	eqeltrd 2701	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ 𝑥 ∈ 𝐴) ∧ (𝑧 ∈ 𝑏 ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝐵))) → (^◡𝑓‘(𝑓‘𝑧)) ∈ (𝒫 𝐵 ∖ {∅}))
35		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑛 = (𝑓‘𝑧) → (^◡𝑓‘𝑛) = (^◡𝑓‘(𝑓‘𝑧)))
36	35	eleq1d 2686	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑛 = (𝑓‘𝑧) → ((^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅}) ↔ (^◡𝑓‘(𝑓‘𝑧)) ∈ (𝒫 𝐵 ∖ {∅})))
37	36	rspcev 3309	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓‘𝑧) ∈ ran 𝑓 ∧ (^◡𝑓‘(𝑓‘𝑧)) ∈ (𝒫 𝐵 ∖ {∅})) → ∃𝑛 ∈ ran 𝑓(^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅}))
38	21, 34, 37	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ 𝑥 ∈ 𝐴) ∧ (𝑧 ∈ 𝑏 ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝐵))) → ∃𝑛 ∈ ran 𝑓(^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅}))
39		rabn0 3958	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ({𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} ≠ ∅ ↔ ∃𝑛 ∈ ran 𝑓(^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅}))
40	38, 39	sylibr 224	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ 𝑥 ∈ 𝐴) ∧ (𝑧 ∈ 𝑏 ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝐵))) → {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} ≠ ∅)
41		onint 6995	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (({𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} ⊆ On ∧ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} ≠ ∅) → ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} ∈ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})})
42	16, 40, 41	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ 𝑥 ∈ 𝐴) ∧ (𝑧 ∈ 𝑏 ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝐵))) → ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} ∈ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})})
43	42	rexlimdvaa 3032	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ 𝑥 ∈ 𝐴) → (∃𝑧 ∈ 𝑏 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝐵) → ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} ∈ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})}))
44	13, 43	syl5 34	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ 𝑥 ∈ 𝐴) → ((𝐵 ∈ (topGen‘𝑏) ∧ 𝑦 ∈ 𝐵) → ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} ∈ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})}))
45	44	expdimp 453	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ 𝑥 ∈ 𝐴) ∧ 𝐵 ∈ (topGen‘𝑏)) → (𝑦 ∈ 𝐵 → ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} ∈ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})}))
46	45	exlimdv 1861	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ 𝑥 ∈ 𝐴) ∧ 𝐵 ∈ (topGen‘𝑏)) → (∃𝑦 𝑦 ∈ 𝐵 → ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} ∈ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})}))
47	12, 46	syl5bi 232	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ 𝑥 ∈ 𝐴) ∧ 𝐵 ∈ (topGen‘𝑏)) → (𝐵 ≠ ∅ → ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} ∈ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})}))
48	47	expimpd 629	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ 𝑥 ∈ 𝐴) → ((𝐵 ∈ (topGen‘𝑏) ∧ 𝐵 ≠ ∅) → ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} ∈ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})}))
49	11, 48	syl5bi 232	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ 𝑥 ∈ 𝐴) → (𝐵 ∈ ((topGen‘𝑏) ∖ {∅}) → ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} ∈ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})}))
50	49	impr 649	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ((topGen‘𝑏) ∖ {∅}))) → ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} ∈ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})})
51	10, 50	sseldd 3604	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ((topGen‘𝑏) ∖ {∅}))) → ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} ∈ ω)
52	51	expr 643	. . . . . . . . . . . . . . . 16 ⊢ (((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ 𝑥 ∈ 𝐴) → (𝐵 ∈ ((topGen‘𝑏) ∖ {∅}) → ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} ∈ ω))
53	52	ralimdva 2962	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) → (∀𝑥 ∈ 𝐴 𝐵 ∈ ((topGen‘𝑏) ∖ {∅}) → ∀𝑥 ∈ 𝐴 ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} ∈ ω))
54	53	imp 445	. . . . . . . . . . . . . 14 ⊢ (((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ ((topGen‘𝑏) ∖ {∅})) → ∀𝑥 ∈ 𝐴 ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} ∈ ω)
55	54	adantrr 753	. . . . . . . . . . . . 13 ⊢ (((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ (∀𝑥 ∈ 𝐴 𝐵 ∈ ((topGen‘𝑏) ∖ {∅}) ∧ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)) → ∀𝑥 ∈ 𝐴 ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} ∈ ω)
56		eqid 2622	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐴 ↦ ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})}) = (𝑥 ∈ 𝐴 ↦ ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})})
57	56	fmpt 6381	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝐴 ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} ∈ ω ↔ (𝑥 ∈ 𝐴 ↦ ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})}):𝐴⟶ω)
58	55, 57	sylib 208	. . . . . . . . . . . 12 ⊢ (((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ (∀𝑥 ∈ 𝐴 𝐵 ∈ ((topGen‘𝑏) ∖ {∅}) ∧ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)) → (𝑥 ∈ 𝐴 ↦ ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})}):𝐴⟶ω)
59		neeq1 2856	. . . . . . . . . . . . . . . . . . 19 ⊢ ((^◡𝑓‘𝑧) = if((^◡𝑓‘𝑧) ≠ ∅, (^◡𝑓‘𝑧), 1_𝑜) → ((^◡𝑓‘𝑧) ≠ ∅ ↔ if((^◡𝑓‘𝑧) ≠ ∅, (^◡𝑓‘𝑧), 1_𝑜) ≠ ∅))
60		neeq1 2856	. . . . . . . . . . . . . . . . . . 19 ⊢ (1_𝑜 = if((^◡𝑓‘𝑧) ≠ ∅, (^◡𝑓‘𝑧), 1_𝑜) → (1_𝑜 ≠ ∅ ↔ if((^◡𝑓‘𝑧) ≠ ∅, (^◡𝑓‘𝑧), 1_𝑜) ≠ ∅))
61		1n0 7575	. . . . . . . . . . . . . . . . . . 19 ⊢ 1_𝑜 ≠ ∅
62	59, 60, 61	elimhyp 4146	. . . . . . . . . . . . . . . . . 18 ⊢ if((^◡𝑓‘𝑧) ≠ ∅, (^◡𝑓‘𝑧), 1_𝑜) ≠ ∅
63		n0 3931	. . . . . . . . . . . . . . . . . 18 ⊢ (if((^◡𝑓‘𝑧) ≠ ∅, (^◡𝑓‘𝑧), 1_𝑜) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ if((^◡𝑓‘𝑧) ≠ ∅, (^◡𝑓‘𝑧), 1_𝑜))
64	62, 63	mpbi 220	. . . . . . . . . . . . . . . . 17 ⊢ ∃𝑦 𝑦 ∈ if((^◡𝑓‘𝑧) ≠ ∅, (^◡𝑓‘𝑧), 1_𝑜)
65		19.29r 1802	. . . . . . . . . . . . . . . . 17 ⊢ ((∃𝑦 𝑦 ∈ if((^◡𝑓‘𝑧) ≠ ∅, (^◡𝑓‘𝑧), 1_𝑜) ∧ ∀𝑦∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → ∃𝑦(𝑦 ∈ if((^◡𝑓‘𝑧) ≠ ∅, (^◡𝑓‘𝑧), 1_𝑜) ∧ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
66	64, 65	mpan 706	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑦∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → ∃𝑦(𝑦 ∈ if((^◡𝑓‘𝑧) ≠ ∅, (^◡𝑓‘𝑧), 1_𝑜) ∧ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
67		eleq1 2689	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑧 = ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} → (𝑧 ∈ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} ↔ ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} ∈ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})}))
68	50, 67	syl5ibrcom 237	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ((topGen‘𝑏) ∖ {∅}))) → (𝑧 = ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} → 𝑧 ∈ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})}))
69	68	imp 445	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ((topGen‘𝑏) ∖ {∅}))) ∧ 𝑧 = ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})}) → 𝑧 ∈ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})})
70		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑛 = 𝑧 → (^◡𝑓‘𝑛) = (^◡𝑓‘𝑧))
71	70	eleq1d 2686	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑛 = 𝑧 → ((^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅}) ↔ (^◡𝑓‘𝑧) ∈ (𝒫 𝐵 ∖ {∅})))
72	71	elrab 3363	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑧 ∈ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} ↔ (𝑧 ∈ ran 𝑓 ∧ (^◡𝑓‘𝑧) ∈ (𝒫 𝐵 ∖ {∅})))
73	72	simprbi 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑧 ∈ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} → (^◡𝑓‘𝑧) ∈ (𝒫 𝐵 ∖ {∅}))
74	69, 73	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ((topGen‘𝑏) ∖ {∅}))) ∧ 𝑧 = ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})}) → (^◡𝑓‘𝑧) ∈ (𝒫 𝐵 ∖ {∅}))
75		eldifsn 4317	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((^◡𝑓‘𝑧) ∈ (𝒫 𝐵 ∖ {∅}) ↔ ((^◡𝑓‘𝑧) ∈ 𝒫 𝐵 ∧ (^◡𝑓‘𝑧) ≠ ∅))
76	74, 75	sylib 208	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ((topGen‘𝑏) ∖ {∅}))) ∧ 𝑧 = ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})}) → ((^◡𝑓‘𝑧) ∈ 𝒫 𝐵 ∧ (^◡𝑓‘𝑧) ≠ ∅))
77	76	simprd 479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ((topGen‘𝑏) ∖ {∅}))) ∧ 𝑧 = ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})}) → (^◡𝑓‘𝑧) ≠ ∅)
78	77	iftrued 4094	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ((topGen‘𝑏) ∖ {∅}))) ∧ 𝑧 = ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})}) → if((^◡𝑓‘𝑧) ≠ ∅, (^◡𝑓‘𝑧), 1_𝑜) = (^◡𝑓‘𝑧))
79	76	simpld 475	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ((topGen‘𝑏) ∖ {∅}))) ∧ 𝑧 = ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})}) → (^◡𝑓‘𝑧) ∈ 𝒫 𝐵)
80	79	elpwid 4170	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ((topGen‘𝑏) ∖ {∅}))) ∧ 𝑧 = ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})}) → (^◡𝑓‘𝑧) ⊆ 𝐵)
81	78, 80	eqsstrd 3639	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ((topGen‘𝑏) ∖ {∅}))) ∧ 𝑧 = ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})}) → if((^◡𝑓‘𝑧) ≠ ∅, (^◡𝑓‘𝑧), 1_𝑜) ⊆ 𝐵)
82	81	sseld 3602	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ((topGen‘𝑏) ∖ {∅}))) ∧ 𝑧 = ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})}) → (𝑦 ∈ if((^◡𝑓‘𝑧) ≠ ∅, (^◡𝑓‘𝑧), 1_𝑜) → 𝑦 ∈ 𝐵))
83	82	exp31 630	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) → ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ((topGen‘𝑏) ∖ {∅})) → (𝑧 = ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} → (𝑦 ∈ if((^◡𝑓‘𝑧) ≠ ∅, (^◡𝑓‘𝑧), 1_𝑜) → 𝑦 ∈ 𝐵))))
84	83	com23 86	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) → (𝑧 = ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} → ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ((topGen‘𝑏) ∖ {∅})) → (𝑦 ∈ if((^◡𝑓‘𝑧) ≠ ∅, (^◡𝑓‘𝑧), 1_𝑜) → 𝑦 ∈ 𝐵))))
85	84	exp4a 633	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) → (𝑧 = ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} → (𝑥 ∈ 𝐴 → (𝐵 ∈ ((topGen‘𝑏) ∖ {∅}) → (𝑦 ∈ if((^◡𝑓‘𝑧) ≠ ∅, (^◡𝑓‘𝑧), 1_𝑜) → 𝑦 ∈ 𝐵)))))
86	85	com25 99	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) → (𝑦 ∈ if((^◡𝑓‘𝑧) ≠ ∅, (^◡𝑓‘𝑧), 1_𝑜) → (𝑥 ∈ 𝐴 → (𝐵 ∈ ((topGen‘𝑏) ∖ {∅}) → (𝑧 = ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} → 𝑦 ∈ 𝐵)))))
87	86	imp31 448	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ 𝑦 ∈ if((^◡𝑓‘𝑧) ≠ ∅, (^◡𝑓‘𝑧), 1_𝑜)) ∧ 𝑥 ∈ 𝐴) → (𝐵 ∈ ((topGen‘𝑏) ∖ {∅}) → (𝑧 = ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} → 𝑦 ∈ 𝐵)))
88	87	ralimdva 2962	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ 𝑦 ∈ if((^◡𝑓‘𝑧) ≠ ∅, (^◡𝑓‘𝑧), 1_𝑜)) → (∀𝑥 ∈ 𝐴 𝐵 ∈ ((topGen‘𝑏) ∖ {∅}) → ∀𝑥 ∈ 𝐴 (𝑧 = ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} → 𝑦 ∈ 𝐵)))
89	88	imp 445	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ 𝑦 ∈ if((^◡𝑓‘𝑧) ≠ ∅, (^◡𝑓‘𝑧), 1_𝑜)) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ ((topGen‘𝑏) ∖ {∅})) → ∀𝑥 ∈ 𝐴 (𝑧 = ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} → 𝑦 ∈ 𝐵))
90	89	an32s 846	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ ((topGen‘𝑏) ∖ {∅})) ∧ 𝑦 ∈ if((^◡𝑓‘𝑧) ≠ ∅, (^◡𝑓‘𝑧), 1_𝑜)) → ∀𝑥 ∈ 𝐴 (𝑧 = ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} → 𝑦 ∈ 𝐵))
91		rmoim 3407	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑥 ∈ 𝐴 (𝑧 = ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} → 𝑦 ∈ 𝐵) → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝑧 = ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})}))
92	90, 91	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ ((topGen‘𝑏) ∖ {∅})) ∧ 𝑦 ∈ if((^◡𝑓‘𝑧) ≠ ∅, (^◡𝑓‘𝑧), 1_𝑜)) → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝑧 = ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})}))
93	92	expimpd 629	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ ((topGen‘𝑏) ∖ {∅})) → ((𝑦 ∈ if((^◡𝑓‘𝑧) ≠ ∅, (^◡𝑓‘𝑧), 1_𝑜) ∧ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑧 = ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})}))
94	93	exlimdv 1861	. . . . . . . . . . . . . . . 16 ⊢ (((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ ((topGen‘𝑏) ∖ {∅})) → (∃𝑦(𝑦 ∈ if((^◡𝑓‘𝑧) ≠ ∅, (^◡𝑓‘𝑧), 1_𝑜) ∧ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑧 = ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})}))
95	66, 94	syl5 34	. . . . . . . . . . . . . . 15 ⊢ (((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ ((topGen‘𝑏) ∖ {∅})) → (∀𝑦∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝑧 = ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})}))
96	95	impr 649	. . . . . . . . . . . . . 14 ⊢ (((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ (∀𝑥 ∈ 𝐴 𝐵 ∈ ((topGen‘𝑏) ∖ {∅}) ∧ ∀𝑦∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)) → ∃𝑥 ∈ 𝐴 𝑧 = ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})})
97		nfcv 2764	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥𝑤
98		nfmpt1 4747	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})})
99		nfcv 2764	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥𝑧
100	97, 98, 99	nfbr 4699	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥 𝑤(𝑥 ∈ 𝐴 ↦ ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})})𝑧
101		nfv 1843	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑤(𝑥 ∈ 𝐴 ∧ 𝑧 = ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})})
102		breq1 4656	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑥 → (𝑤(𝑥 ∈ 𝐴 ↦ ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})})𝑧 ↔ 𝑥(𝑥 ∈ 𝐴 ↦ ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})})𝑧))
103		df-br 4654	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥(𝑥 ∈ 𝐴 ↦ ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})})𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ (𝑥 ∈ 𝐴 ↦ ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})}))
104		df-mpt 4730	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝐴 ↦ ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})}) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})})}
105	104	eleq2i 2693	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨𝑥, 𝑧⟩ ∈ (𝑥 ∈ 𝐴 ↦ ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})}) ↔ ⟨𝑥, 𝑧⟩ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})})})
106		opabid 4982	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨𝑥, 𝑧⟩ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})})} ↔ (𝑥 ∈ 𝐴 ∧ 𝑧 = ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})}))
107	103, 105, 106	3bitri 286	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥(𝑥 ∈ 𝐴 ↦ ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})})𝑧 ↔ (𝑥 ∈ 𝐴 ∧ 𝑧 = ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})}))
108	102, 107	syl6bb 276	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑥 → (𝑤(𝑥 ∈ 𝐴 ↦ ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})})𝑧 ↔ (𝑥 ∈ 𝐴 ∧ 𝑧 = ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})})))
109	100, 101, 108	cbvmo 2506	. . . . . . . . . . . . . . 15 ⊢ (∃𝑤 𝑤(𝑥 ∈ 𝐴 ↦ ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})})𝑧 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 = ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})}))
110		df-rmo 2920	. . . . . . . . . . . . . . 15 ⊢ (∃𝑥 ∈ 𝐴 𝑧 = ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})} ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 = ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})}))
111	109, 110	bitr4i 267	. . . . . . . . . . . . . 14 ⊢ (∃𝑤 𝑤(𝑥 ∈ 𝐴 ↦ ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})})𝑧 ↔ ∃𝑥 ∈ 𝐴 𝑧 = ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})})
112	96, 111	sylibr 224	. . . . . . . . . . . . 13 ⊢ (((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ (∀𝑥 ∈ 𝐴 𝐵 ∈ ((topGen‘𝑏) ∖ {∅}) ∧ ∀𝑦∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)) → ∃𝑤 𝑤(𝑥 ∈ 𝐴 ↦ ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})})𝑧)
113	112	alrimiv 1855	. . . . . . . . . . . 12 ⊢ (((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ (∀𝑥 ∈ 𝐴 𝐵 ∈ ((topGen‘𝑏) ∖ {∅}) ∧ ∀𝑦∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)) → ∀𝑧∃𝑤 𝑤(𝑥 ∈ 𝐴 ↦ ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})})𝑧)
114		dff12 6100	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ↦ ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})}):𝐴–1-1→ω ↔ ((𝑥 ∈ 𝐴 ↦ ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})}):𝐴⟶ω ∧ ∀𝑧∃*𝑤 𝑤(𝑥 ∈ 𝐴 ↦ ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})})𝑧))
115	58, 113, 114	sylanbrc 698	. . . . . . . . . . 11 ⊢ (((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ (∀𝑥 ∈ 𝐴 𝐵 ∈ ((topGen‘𝑏) ∖ {∅}) ∧ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)) → (𝑥 ∈ 𝐴 ↦ ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})}):𝐴–1-1→ω)
116		f1domg 7975	. . . . . . . . . . 11 ⊢ (ω ∈ V → ((𝑥 ∈ 𝐴 ↦ ∩ {𝑛 ∈ ran 𝑓 ∣ (^◡𝑓‘𝑛) ∈ (𝒫 𝐵 ∖ {∅})}):𝐴–1-1→ω → 𝐴 ≼ ω))
117	2, 115, 116	mpsyl 68	. . . . . . . . . 10 ⊢ (((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) ∧ (∀𝑥 ∈ 𝐴 𝐵 ∈ ((topGen‘𝑏) ∖ {∅}) ∧ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)) → 𝐴 ≼ ω)
118	117	ex 450	. . . . . . . . 9 ⊢ ((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) → ((∀𝑥 ∈ 𝐴 𝐵 ∈ ((topGen‘𝑏) ∖ {∅}) ∧ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → 𝐴 ≼ ω))
119		difeq1 3721	. . . . . . . . . . . . 13 ⊢ ((topGen‘𝑏) = 𝐽 → ((topGen‘𝑏) ∖ {∅}) = (𝐽 ∖ {∅}))
120	119	eleq2d 2687	. . . . . . . . . . . 12 ⊢ ((topGen‘𝑏) = 𝐽 → (𝐵 ∈ ((topGen‘𝑏) ∖ {∅}) ↔ 𝐵 ∈ (𝐽 ∖ {∅})))
121	120	ralbidv 2986	. . . . . . . . . . 11 ⊢ ((topGen‘𝑏) = 𝐽 → (∀𝑥 ∈ 𝐴 𝐵 ∈ ((topGen‘𝑏) ∖ {∅}) ↔ ∀𝑥 ∈ 𝐴 𝐵 ∈ (𝐽 ∖ {∅})))
122	121	anbi1d 741	. . . . . . . . . 10 ⊢ ((topGen‘𝑏) = 𝐽 → ((∀𝑥 ∈ 𝐴 𝐵 ∈ ((topGen‘𝑏) ∖ {∅}) ∧ ∀𝑦∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) ↔ (∀𝑥 ∈ 𝐴 𝐵 ∈ (𝐽 ∖ {∅}) ∧ ∀𝑦∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)))
123	122	imbi1d 331	. . . . . . . . 9 ⊢ ((topGen‘𝑏) = 𝐽 → (((∀𝑥 ∈ 𝐴 𝐵 ∈ ((topGen‘𝑏) ∖ {∅}) ∧ ∀𝑦∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → 𝐴 ≼ ω) ↔ ((∀𝑥 ∈ 𝐴 𝐵 ∈ (𝐽 ∖ {∅}) ∧ ∀𝑦∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → 𝐴 ≼ ω)))
124	118, 123	syl5ibcom 235	. . . . . . . 8 ⊢ ((𝑏 ∈ TopBases ∧ 𝑓:𝑏–1-1→ω) → ((topGen‘𝑏) = 𝐽 → ((∀𝑥 ∈ 𝐴 𝐵 ∈ (𝐽 ∖ {∅}) ∧ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → 𝐴 ≼ ω)))
125	124	ex 450	. . . . . . 7 ⊢ (𝑏 ∈ TopBases → (𝑓:𝑏–1-1→ω → ((topGen‘𝑏) = 𝐽 → ((∀𝑥 ∈ 𝐴 𝐵 ∈ (𝐽 ∖ {∅}) ∧ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → 𝐴 ≼ ω))))
126	125	exlimdv 1861	. . . . . 6 ⊢ (𝑏 ∈ TopBases → (∃𝑓 𝑓:𝑏–1-1→ω → ((topGen‘𝑏) = 𝐽 → ((∀𝑥 ∈ 𝐴 𝐵 ∈ (𝐽 ∖ {∅}) ∧ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → 𝐴 ≼ ω))))
127	3, 126	syl5bi 232	. . . . 5 ⊢ (𝑏 ∈ TopBases → (𝑏 ≼ ω → ((topGen‘𝑏) = 𝐽 → ((∀𝑥 ∈ 𝐴 𝐵 ∈ (𝐽 ∖ {∅}) ∧ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → 𝐴 ≼ ω))))
128	127	impd 447	. . . 4 ⊢ (𝑏 ∈ TopBases → ((𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽) → ((∀𝑥 ∈ 𝐴 𝐵 ∈ (𝐽 ∖ {∅}) ∧ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → 𝐴 ≼ ω)))
129	128	rexlimiv 3027	. . 3 ⊢ (∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽) → ((∀𝑥 ∈ 𝐴 𝐵 ∈ (𝐽 ∖ {∅}) ∧ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → 𝐴 ≼ ω))
130	1, 129	sylbi 207	. 2 ⊢ (𝐽 ∈ 2^nd𝜔 → ((∀𝑥 ∈ 𝐴 𝐵 ∈ (𝐽 ∖ {∅}) ∧ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → 𝐴 ≼ ω))
131	130	3impib 1262	1 ⊢ ((𝐽 ∈ 2^nd𝜔 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (𝐽 ∖ {∅}) ∧ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → 𝐴 ≼ ω)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 ∀wal 1481 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ∃*wmo 2471 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 ∃*wrmo 2915 {crab 2916 Vcvv 3200 ∖ cdif 3571 ⊆ wss 3574 ∅c0 3915 ifcif 4086 𝒫 cpw 4158 {csn 4177 ⟨cop 4183 ∩ cint 4475 class class class wbr 4653 {copab 4712 ↦ cmpt 4729 ^◡ccnv 5113 ran crn 5115 Oncon0 5723 Fn wfn 5883 ⟶wf 5884 –1-1→wf1 5885 –1-1-onto→wf1o 5887 ‘cfv 5888 ωcom 7065 1_𝑜c1o 7553 ≼ cdom 7953 topGenctg 16098 TopBasesctb 20749 2^nd𝜔c2ndc 21241

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-om 7066 df-1o 7560 df-dom 7957 df-topgen 16104 df-2ndc 21243

This theorem is referenced by: 2ndcdisj2 21260

W3C validator