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Theorem cfflb 9119
Description: If there is a cofinal map from 𝐵 to 𝐴, then 𝐵 is at least (cf‘𝐴). This theorem and cff1 9118 motivate the picture of (cf‘𝐴) as the greatest lower bound of the domain of cofinal maps into 𝐴. (Contributed by Mario Carneiro, 28-Feb-2013.)
Assertion
Ref Expression
cfflb ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑓(𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → (cf‘𝐴) ⊆ 𝐵))
Distinct variable groups:   𝐴,𝑓,𝑤,𝑧   𝐵,𝑓,𝑤,𝑧

Proof of Theorem cfflb
Dummy variables 𝑠 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frn 6091 . . . . . . 7 (𝑓:𝐵𝐴 → ran 𝑓𝐴)
21adantr 480 . . . . . 6 ((𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → ran 𝑓𝐴)
3 ffn 6083 . . . . . . . . . . . 12 (𝑓:𝐵𝐴𝑓 Fn 𝐵)
4 fnfvelrn 6396 . . . . . . . . . . . 12 ((𝑓 Fn 𝐵𝑤𝐵) → (𝑓𝑤) ∈ ran 𝑓)
53, 4sylan 487 . . . . . . . . . . 11 ((𝑓:𝐵𝐴𝑤𝐵) → (𝑓𝑤) ∈ ran 𝑓)
6 sseq2 3660 . . . . . . . . . . . 12 (𝑠 = (𝑓𝑤) → (𝑧𝑠𝑧 ⊆ (𝑓𝑤)))
76rspcev 3340 . . . . . . . . . . 11 (((𝑓𝑤) ∈ ran 𝑓𝑧 ⊆ (𝑓𝑤)) → ∃𝑠 ∈ ran 𝑓 𝑧𝑠)
85, 7sylan 487 . . . . . . . . . 10 (((𝑓:𝐵𝐴𝑤𝐵) ∧ 𝑧 ⊆ (𝑓𝑤)) → ∃𝑠 ∈ ran 𝑓 𝑧𝑠)
98exp31 629 . . . . . . . . 9 (𝑓:𝐵𝐴 → (𝑤𝐵 → (𝑧 ⊆ (𝑓𝑤) → ∃𝑠 ∈ ran 𝑓 𝑧𝑠)))
109rexlimdv 3059 . . . . . . . 8 (𝑓:𝐵𝐴 → (∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤) → ∃𝑠 ∈ ran 𝑓 𝑧𝑠))
1110ralimdv 2992 . . . . . . 7 (𝑓:𝐵𝐴 → (∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤) → ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠))
1211imp 444 . . . . . 6 ((𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠)
132, 12jca 553 . . . . 5 ((𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → (ran 𝑓𝐴 ∧ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠))
14 fvex 6239 . . . . . 6 (card‘ran 𝑓) ∈ V
15 cfval 9107 . . . . . . . . . . 11 (𝐴 ∈ On → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))})
1615adantr 480 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))})
17163ad2ant2 1103 . . . . . . . . 9 ((𝑥 = (card‘ran 𝑓) ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (ran 𝑓𝐴 ∧ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠)) → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))})
18 vex 3234 . . . . . . . . . . . . . 14 𝑓 ∈ V
1918rnex 7142 . . . . . . . . . . . . 13 ran 𝑓 ∈ V
20 fveq2 6229 . . . . . . . . . . . . . . 15 (𝑦 = ran 𝑓 → (card‘𝑦) = (card‘ran 𝑓))
2120eqeq2d 2661 . . . . . . . . . . . . . 14 (𝑦 = ran 𝑓 → (𝑥 = (card‘𝑦) ↔ 𝑥 = (card‘ran 𝑓)))
22 sseq1 3659 . . . . . . . . . . . . . . 15 (𝑦 = ran 𝑓 → (𝑦𝐴 ↔ ran 𝑓𝐴))
23 rexeq 3169 . . . . . . . . . . . . . . . 16 (𝑦 = ran 𝑓 → (∃𝑠𝑦 𝑧𝑠 ↔ ∃𝑠 ∈ ran 𝑓 𝑧𝑠))
2423ralbidv 3015 . . . . . . . . . . . . . . 15 (𝑦 = ran 𝑓 → (∀𝑧𝐴𝑠𝑦 𝑧𝑠 ↔ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠))
2522, 24anbi12d 747 . . . . . . . . . . . . . 14 (𝑦 = ran 𝑓 → ((𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠) ↔ (ran 𝑓𝐴 ∧ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠)))
2621, 25anbi12d 747 . . . . . . . . . . . . 13 (𝑦 = ran 𝑓 → ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) ↔ (𝑥 = (card‘ran 𝑓) ∧ (ran 𝑓𝐴 ∧ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠))))
2719, 26spcev 3331 . . . . . . . . . . . 12 ((𝑥 = (card‘ran 𝑓) ∧ (ran 𝑓𝐴 ∧ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠)) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)))
28 abid 2639 . . . . . . . . . . . 12 (𝑥 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))} ↔ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)))
2927, 28sylibr 224 . . . . . . . . . . 11 ((𝑥 = (card‘ran 𝑓) ∧ (ran 𝑓𝐴 ∧ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠)) → 𝑥 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))})
30 intss1 4524 . . . . . . . . . . 11 (𝑥 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))} → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))} ⊆ 𝑥)
3129, 30syl 17 . . . . . . . . . 10 ((𝑥 = (card‘ran 𝑓) ∧ (ran 𝑓𝐴 ∧ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠)) → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))} ⊆ 𝑥)
32313adant2 1100 . . . . . . . . 9 ((𝑥 = (card‘ran 𝑓) ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (ran 𝑓𝐴 ∧ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠)) → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))} ⊆ 𝑥)
3317, 32eqsstrd 3672 . . . . . . . 8 ((𝑥 = (card‘ran 𝑓) ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (ran 𝑓𝐴 ∧ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠)) → (cf‘𝐴) ⊆ 𝑥)
34333expib 1287 . . . . . . 7 (𝑥 = (card‘ran 𝑓) → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (ran 𝑓𝐴 ∧ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠)) → (cf‘𝐴) ⊆ 𝑥))
35 sseq2 3660 . . . . . . 7 (𝑥 = (card‘ran 𝑓) → ((cf‘𝐴) ⊆ 𝑥 ↔ (cf‘𝐴) ⊆ (card‘ran 𝑓)))
3634, 35sylibd 229 . . . . . 6 (𝑥 = (card‘ran 𝑓) → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (ran 𝑓𝐴 ∧ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠)) → (cf‘𝐴) ⊆ (card‘ran 𝑓)))
3714, 36vtocle 3313 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (ran 𝑓𝐴 ∧ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠)) → (cf‘𝐴) ⊆ (card‘ran 𝑓))
3813, 37sylan2 490 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → (cf‘𝐴) ⊆ (card‘ran 𝑓))
39 cardidm 8823 . . . . . . 7 (card‘(card‘ran 𝑓)) = (card‘ran 𝑓)
40 onss 7032 . . . . . . . . . . . . . 14 (𝐴 ∈ On → 𝐴 ⊆ On)
411, 40sylan9ssr 3650 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝑓:𝐵𝐴) → ran 𝑓 ⊆ On)
42413adant2 1100 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → ran 𝑓 ⊆ On)
43 onssnum 8901 . . . . . . . . . . . 12 ((ran 𝑓 ∈ V ∧ ran 𝑓 ⊆ On) → ran 𝑓 ∈ dom card)
4419, 42, 43sylancr 696 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → ran 𝑓 ∈ dom card)
45 cardid2 8817 . . . . . . . . . . 11 (ran 𝑓 ∈ dom card → (card‘ran 𝑓) ≈ ran 𝑓)
4644, 45syl 17 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → (card‘ran 𝑓) ≈ ran 𝑓)
47 onenon 8813 . . . . . . . . . . . . 13 (𝐵 ∈ On → 𝐵 ∈ dom card)
48 dffn4 6159 . . . . . . . . . . . . . 14 (𝑓 Fn 𝐵𝑓:𝐵onto→ran 𝑓)
493, 48sylib 208 . . . . . . . . . . . . 13 (𝑓:𝐵𝐴𝑓:𝐵onto→ran 𝑓)
50 fodomnum 8918 . . . . . . . . . . . . 13 (𝐵 ∈ dom card → (𝑓:𝐵onto→ran 𝑓 → ran 𝑓𝐵))
5147, 49, 50syl2im 40 . . . . . . . . . . . 12 (𝐵 ∈ On → (𝑓:𝐵𝐴 → ran 𝑓𝐵))
5251imp 444 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → ran 𝑓𝐵)
53523adant1 1099 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → ran 𝑓𝐵)
54 endomtr 8055 . . . . . . . . . 10 (((card‘ran 𝑓) ≈ ran 𝑓 ∧ ran 𝑓𝐵) → (card‘ran 𝑓) ≼ 𝐵)
5546, 53, 54syl2anc 694 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → (card‘ran 𝑓) ≼ 𝐵)
56 cardon 8808 . . . . . . . . . . . 12 (card‘ran 𝑓) ∈ On
57 onenon 8813 . . . . . . . . . . . 12 ((card‘ran 𝑓) ∈ On → (card‘ran 𝑓) ∈ dom card)
5856, 57ax-mp 5 . . . . . . . . . . 11 (card‘ran 𝑓) ∈ dom card
59 carddom2 8841 . . . . . . . . . . 11 (((card‘ran 𝑓) ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘(card‘ran 𝑓)) ⊆ (card‘𝐵) ↔ (card‘ran 𝑓) ≼ 𝐵))
6058, 47, 59sylancr 696 . . . . . . . . . 10 (𝐵 ∈ On → ((card‘(card‘ran 𝑓)) ⊆ (card‘𝐵) ↔ (card‘ran 𝑓) ≼ 𝐵))
61603ad2ant2 1103 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → ((card‘(card‘ran 𝑓)) ⊆ (card‘𝐵) ↔ (card‘ran 𝑓) ≼ 𝐵))
6255, 61mpbird 247 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → (card‘(card‘ran 𝑓)) ⊆ (card‘𝐵))
63 cardonle 8821 . . . . . . . . 9 (𝐵 ∈ On → (card‘𝐵) ⊆ 𝐵)
64633ad2ant2 1103 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → (card‘𝐵) ⊆ 𝐵)
6562, 64sstrd 3646 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → (card‘(card‘ran 𝑓)) ⊆ 𝐵)
6639, 65syl5eqssr 3683 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → (card‘ran 𝑓) ⊆ 𝐵)
67663expa 1284 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → (card‘ran 𝑓) ⊆ 𝐵)
6867adantrr 753 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → (card‘ran 𝑓) ⊆ 𝐵)
6938, 68sstrd 3646 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → (cf‘𝐴) ⊆ 𝐵)
7069ex 449 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → (cf‘𝐴) ⊆ 𝐵))
7170exlimdv 1901 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑓(𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → (cf‘𝐴) ⊆ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wral 2941  wrex 2942  Vcvv 3231  wss 3607   cint 4507   class class class wbr 4685  dom cdm 5143  ran crn 5144  Oncon0 5761   Fn wfn 5921  wf 5922  ontowfo 5924  cfv 5926  cen 7994  cdom 7995  cardccrd 8799  cfccf 8801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-card 8803  df-cf 8805  df-acn 8806
This theorem is referenced by:  cfsmolem  9130  cfcoflem  9132  cfcof  9134  inar1  9635
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