Theorem isfne3 36525

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 36522	. 2 ⊢ (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)))
4		dfss3 3910	. . . 4 ⊢ (𝐴 ⊆ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ (topGen‘𝐵))
5		eltg3 22927	. . . . 5 ⊢ (𝐵 ∈ 𝐶 → (𝑥 ∈ (topGen‘𝐵) ↔ ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))
6	5	ralbidv 3160	. . . 4 ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ 𝐴 𝑥 ∈ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐴 ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))
7	4, 6	bitrid 283	. . 3 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ⊆ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐴 ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))
8	7	anbi2d 631	. 2 ⊢ (𝐵 ∈ 𝐶 → ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦))))
9	3, 8	bitrid 283	1 ⊢ (𝐵 ∈ 𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3051   ⊆ wss 3889  ∪ cuni 4850   class class class wbr 5085  ‘cfv 6498  topGenctg 17400  Fnecfne 36518

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6454  df-fun 6500  df-fv 6506  df-topgen 17406  df-fne 36519

This theorem is referenced by: (None)