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Theorem cflem 8827
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 3491 . . 3 𝐴𝐴
2 ssid 3491 . . . . 5 𝑧𝑧
3 sseq2 3494 . . . . . 6 (𝑤 = 𝑧 → (𝑧𝑤𝑧𝑧))
43rspcev 3186 . . . . 5 ((𝑧𝐴𝑧𝑧) → ∃𝑤𝐴 𝑧𝑤)
52, 4mpan2 702 . . . 4 (𝑧𝐴 → ∃𝑤𝐴 𝑧𝑤)
65rgen 2810 . . 3 𝑧𝐴𝑤𝐴 𝑧𝑤
7 sseq1 3493 . . . . 5 (𝑦 = 𝐴 → (𝑦𝐴𝐴𝐴))
8 rexeq 3020 . . . . . 6 (𝑦 = 𝐴 → (∃𝑤𝑦 𝑧𝑤 ↔ ∃𝑤𝐴 𝑧𝑤))
98ralbidv 2873 . . . . 5 (𝑦 = 𝐴 → (∀𝑧𝐴𝑤𝑦 𝑧𝑤 ↔ ∀𝑧𝐴𝑤𝐴 𝑧𝑤))
107, 9anbi12d 742 . . . 4 (𝑦 = 𝐴 → ((𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤) ↔ (𝐴𝐴 ∧ ∀𝑧𝐴𝑤𝐴 𝑧𝑤)))
1110spcegv 3171 . . 3 (𝐴𝑉 → ((𝐴𝐴 ∧ ∀𝑧𝐴𝑤𝐴 𝑧𝑤) → ∃𝑦(𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
121, 6, 11mp2ani 709 . 2 (𝐴𝑉 → ∃𝑦(𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))
13 fvex 5997 . . . . . 6 (card‘𝑦) ∈ V
1413isseti 3086 . . . . 5 𝑥 𝑥 = (card‘𝑦)
15 19.41v 1864 . . . . 5 (∃𝑥(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ (∃𝑥 𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
1614, 15mpbiran 954 . . . 4 (∃𝑥(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))
1716exbii 1752 . . 3 (∃𝑦𝑥(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ ∃𝑦(𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))
18 excom 1978 . . 3 (∃𝑦𝑥(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ ∃𝑥𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
1917, 18bitr3i 264 . 2 (∃𝑦(𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤) ↔ ∃𝑥𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
2012, 19sylib 206 1 (𝐴𝑉 → ∃𝑥𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wex 1694  wcel 1938  wral 2800  wrex 2801  wss 3444  cfv 5689  cardccrd 8520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-nul 4616
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rex 2806  df-v 3079  df-sbc 3307  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-sn 4029  df-pr 4031  df-uni 4271  df-iota 5653  df-fv 5697
This theorem is referenced by:  cfval  8828  cff  8829  cff1  8839
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