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Theorem cflem 10223
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 4000 . . 3 𝐴𝐴
2 ssid 4000 . . . . 5 𝑧𝑧
3 sseq2 4004 . . . . . 6 (𝑤 = 𝑧 → (𝑧𝑤𝑧𝑧))
43rspcev 3609 . . . . 5 ((𝑧𝐴𝑧𝑧) → ∃𝑤𝐴 𝑧𝑤)
52, 4mpan2 689 . . . 4 (𝑧𝐴 → ∃𝑤𝐴 𝑧𝑤)
65rgen 3062 . . 3 𝑧𝐴𝑤𝐴 𝑧𝑤
7 sseq1 4003 . . . . 5 (𝑦 = 𝐴 → (𝑦𝐴𝐴𝐴))
8 rexeq 3320 . . . . . 6 (𝑦 = 𝐴 → (∃𝑤𝑦 𝑧𝑤 ↔ ∃𝑤𝐴 𝑧𝑤))
98ralbidv 3176 . . . . 5 (𝑦 = 𝐴 → (∀𝑧𝐴𝑤𝑦 𝑧𝑤 ↔ ∀𝑧𝐴𝑤𝐴 𝑧𝑤))
107, 9anbi12d 631 . . . 4 (𝑦 = 𝐴 → ((𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤) ↔ (𝐴𝐴 ∧ ∀𝑧𝐴𝑤𝐴 𝑧𝑤)))
1110spcegv 3584 . . 3 (𝐴𝑉 → ((𝐴𝐴 ∧ ∀𝑧𝐴𝑤𝐴 𝑧𝑤) → ∃𝑦(𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
121, 6, 11mp2ani 696 . 2 (𝐴𝑉 → ∃𝑦(𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))
13 fvex 6891 . . . . . 6 (card‘𝑦) ∈ V
1413isseti 3488 . . . . 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 707 . . . 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 1541  wex 1781  wcel 2106  wral 3060  wrex 3069  wss 3944  cfv 6532  cardccrd 9912
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2702  ax-nul 5299
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-sn 4623  df-pr 4625  df-uni 4902  df-iota 6484  df-fv 6540
This theorem is referenced by:  cfval  10224  cff  10225  cff1  10235
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