Theorem isfne3 36965

Description: The predicate "𝐵 is finer than 𝐴". (Contributed by Jeff Hankins, 11-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)

Hypotheses

Ref	Expression
isfne.1	⊢ 𝑋 = ∪ 𝐴
isfne.2	⊢ 𝑌 = ∪ 𝐵

Assertion

Ref	Expression
isfne3	⊢ (𝐵 ∈ 𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦))))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦

Allowed substitution hints:   𝑋(𝑥, 𝑦)   𝑌(𝑥, 𝑦)

Proof of Theorem isfne3

Step	Hyp	Ref	Expression
1		isfne.1	. . 3 ⊢ 𝑋 = ∪ 𝐴
2		isfne.2	. . 3 ⊢ 𝑌 = ∪ 𝐵
3	1, 2	isfne4 36962	. 2 ⊢ (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)))
4		dfss3 3920	. . . 4 ⊢ (𝐴 ⊆ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ (topGen‘𝐵))
5		eltg3 23190	. . . . 5 ⊢ (𝐵 ∈ 𝐶 → (𝑥 ∈ (topGen‘𝐵) ↔ ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))
6	5	ralbidv 3185	. . . 4 ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ 𝐴 𝑥 ∈ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐴 ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))
7	4, 6	bitrid 286	. . 3 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ⊆ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐴 ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))
8	7	anbi2d 642	. 2 ⊢ (𝐵 ∈ 𝐶 → ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦))))
9	3, 8	bitrid 286	1 ⊢ (𝐵 ∈ 𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∀wral 3076   ⊆ wss 3899  ∪ cuni 4867   class class class wbr 5103  ‘cfv 6533  topGenctg 17525  Fnecfne 36958

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6489  df-fun 6535  df-fv 6541  df-topgen 17531  df-fne 36959

This theorem is used by: (None)