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Theorem cflem 10002
Description: A lemma used to simplify cofinality computations, showing the existence of the cardinal of an unbounded subset of a set 𝐴. (Contributed by NM, 24-Apr-2004.)
Assertion
Ref Expression
cflem (𝐴𝑉 → ∃𝑥𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝐴
Allowed substitution hints:   𝑉(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem cflem
StepHypRef Expression
1 ssid 3943 . . 3 𝐴𝐴
2 ssid 3943 . . . . 5 𝑧𝑧
3 sseq2 3947 . . . . . 6 (𝑤 = 𝑧 → (𝑧𝑤𝑧𝑧))
43rspcev 3561 . . . . 5 ((𝑧𝐴𝑧𝑧) → ∃𝑤𝐴 𝑧𝑤)
52, 4mpan2 688 . . . 4 (𝑧𝐴 → ∃𝑤𝐴 𝑧𝑤)
65rgen 3074 . . 3 𝑧𝐴𝑤𝐴 𝑧𝑤
7 sseq1 3946 . . . . 5 (𝑦 = 𝐴 → (𝑦𝐴𝐴𝐴))
8 rexeq 3343 . . . . . 6 (𝑦 = 𝐴 → (∃𝑤𝑦 𝑧𝑤 ↔ ∃𝑤𝐴 𝑧𝑤))
98ralbidv 3112 . . . . 5 (𝑦 = 𝐴 → (∀𝑧𝐴𝑤𝑦 𝑧𝑤 ↔ ∀𝑧𝐴𝑤𝐴 𝑧𝑤))
107, 9anbi12d 631 . . . 4 (𝑦 = 𝐴 → ((𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤) ↔ (𝐴𝐴 ∧ ∀𝑧𝐴𝑤𝐴 𝑧𝑤)))
1110spcegv 3536 . . 3 (𝐴𝑉 → ((𝐴𝐴 ∧ ∀𝑧𝐴𝑤𝐴 𝑧𝑤) → ∃𝑦(𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
121, 6, 11mp2ani 695 . 2 (𝐴𝑉 → ∃𝑦(𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))
13 fvex 6787 . . . . . 6 (card‘𝑦) ∈ V
1413isseti 3447 . . . . 5 𝑥 𝑥 = (card‘𝑦)
15 19.41v 1953 . . . . 5 (∃𝑥(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ (∃𝑥 𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
1614, 15mpbiran 706 . . . 4 (∃𝑥(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))
1716exbii 1850 . . 3 (∃𝑦𝑥(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ ∃𝑦(𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))
18 excom 2162 . . 3 (∃𝑦𝑥(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ ∃𝑥𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
1917, 18bitr3i 276 . 2 (∃𝑦(𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤) ↔ ∃𝑥𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
2012, 19sylib 217 1 (𝐴𝑉 → ∃𝑥𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wex 1782  wcel 2106  wral 3064  wrex 3065  wss 3887  cfv 6433  cardccrd 9693
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-nul 5230
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-sn 4562  df-pr 4564  df-uni 4840  df-iota 6391  df-fv 6441
This theorem is referenced by:  cfval  10003  cff  10004  cff1  10014
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