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Theorem cfflb 9283
Description: If there is a cofinal map from 𝐵 to 𝐴, then 𝐵 is at least (cf‘𝐴). This theorem and cff1 9282 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 6193 . . . . . . 7 (𝑓:𝐵𝐴 → ran 𝑓𝐴)
21adantr 466 . . . . . 6 ((𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → ran 𝑓𝐴)
3 ffn 6185 . . . . . . . . . . . 12 (𝑓:𝐵𝐴𝑓 Fn 𝐵)
4 fnfvelrn 6499 . . . . . . . . . . . 12 ((𝑓 Fn 𝐵𝑤𝐵) → (𝑓𝑤) ∈ ran 𝑓)
53, 4sylan 569 . . . . . . . . . . 11 ((𝑓:𝐵𝐴𝑤𝐵) → (𝑓𝑤) ∈ ran 𝑓)
6 sseq2 3776 . . . . . . . . . . . 12 (𝑠 = (𝑓𝑤) → (𝑧𝑠𝑧 ⊆ (𝑓𝑤)))
76rspcev 3460 . . . . . . . . . . 11 (((𝑓𝑤) ∈ ran 𝑓𝑧 ⊆ (𝑓𝑤)) → ∃𝑠 ∈ ran 𝑓 𝑧𝑠)
85, 7sylan 569 . . . . . . . . . 10 (((𝑓:𝐵𝐴𝑤𝐵) ∧ 𝑧 ⊆ (𝑓𝑤)) → ∃𝑠 ∈ ran 𝑓 𝑧𝑠)
98exp31 406 . . . . . . . . 9 (𝑓:𝐵𝐴 → (𝑤𝐵 → (𝑧 ⊆ (𝑓𝑤) → ∃𝑠 ∈ ran 𝑓 𝑧𝑠)))
109rexlimdv 3178 . . . . . . . 8 (𝑓:𝐵𝐴 → (∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤) → ∃𝑠 ∈ ran 𝑓 𝑧𝑠))
1110ralimdv 3112 . . . . . . 7 (𝑓:𝐵𝐴 → (∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤) → ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠))
1211imp 393 . . . . . 6 ((𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠)
132, 12jca 501 . . . . 5 ((𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → (ran 𝑓𝐴 ∧ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠))
14 fvex 6342 . . . . . 6 (card‘ran 𝑓) ∈ V
15 cfval 9271 . . . . . . . . . . 11 (𝐴 ∈ On → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))})
1615adantr 466 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))})
17163ad2ant2 1128 . . . . . . . . 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 3354 . . . . . . . . . . . . . 14 𝑓 ∈ V
1918rnex 7247 . . . . . . . . . . . . 13 ran 𝑓 ∈ V
20 fveq2 6332 . . . . . . . . . . . . . . 15 (𝑦 = ran 𝑓 → (card‘𝑦) = (card‘ran 𝑓))
2120eqeq2d 2781 . . . . . . . . . . . . . 14 (𝑦 = ran 𝑓 → (𝑥 = (card‘𝑦) ↔ 𝑥 = (card‘ran 𝑓)))
22 sseq1 3775 . . . . . . . . . . . . . . 15 (𝑦 = ran 𝑓 → (𝑦𝐴 ↔ ran 𝑓𝐴))
23 rexeq 3288 . . . . . . . . . . . . . . . 16 (𝑦 = ran 𝑓 → (∃𝑠𝑦 𝑧𝑠 ↔ ∃𝑠 ∈ ran 𝑓 𝑧𝑠))
2423ralbidv 3135 . . . . . . . . . . . . . . 15 (𝑦 = ran 𝑓 → (∀𝑧𝐴𝑠𝑦 𝑧𝑠 ↔ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠))
2522, 24anbi12d 616 . . . . . . . . . . . . . 14 (𝑦 = ran 𝑓 → ((𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠) ↔ (ran 𝑓𝐴 ∧ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠)))
2621, 25anbi12d 616 . . . . . . . . . . . . 13 (𝑦 = ran 𝑓 → ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) ↔ (𝑥 = (card‘ran 𝑓) ∧ (ran 𝑓𝐴 ∧ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠))))
2719, 26spcev 3451 . . . . . . . . . . . 12 ((𝑥 = (card‘ran 𝑓) ∧ (ran 𝑓𝐴 ∧ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠)) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)))
28 abid 2759 . . . . . . . . . . . 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 4626 . . . . . . . . . . 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 1125 . . . . . . . . 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 3788 . . . . . . . 8 ((𝑥 = (card‘ran 𝑓) ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (ran 𝑓𝐴 ∧ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠)) → (cf‘𝐴) ⊆ 𝑥)
34333expib 1116 . . . . . . 7 (𝑥 = (card‘ran 𝑓) → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (ran 𝑓𝐴 ∧ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠)) → (cf‘𝐴) ⊆ 𝑥))
35 sseq2 3776 . . . . . . 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 3433 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (ran 𝑓𝐴 ∧ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠)) → (cf‘𝐴) ⊆ (card‘ran 𝑓))
3813, 37sylan2 580 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → (cf‘𝐴) ⊆ (card‘ran 𝑓))
39 cardidm 8985 . . . . . . 7 (card‘(card‘ran 𝑓)) = (card‘ran 𝑓)
40 onss 7137 . . . . . . . . . . . . . 14 (𝐴 ∈ On → 𝐴 ⊆ On)
411, 40sylan9ssr 3766 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝑓:𝐵𝐴) → ran 𝑓 ⊆ On)
42413adant2 1125 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → ran 𝑓 ⊆ On)
43 onssnum 9063 . . . . . . . . . . . 12 ((ran 𝑓 ∈ V ∧ ran 𝑓 ⊆ On) → ran 𝑓 ∈ dom card)
4419, 42, 43sylancr 575 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → ran 𝑓 ∈ dom card)
45 cardid2 8979 . . . . . . . . . . 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 8975 . . . . . . . . . . . . 13 (𝐵 ∈ On → 𝐵 ∈ dom card)
48 dffn4 6262 . . . . . . . . . . . . . 14 (𝑓 Fn 𝐵𝑓:𝐵onto→ran 𝑓)
493, 48sylib 208 . . . . . . . . . . . . 13 (𝑓:𝐵𝐴𝑓:𝐵onto→ran 𝑓)
50 fodomnum 9080 . . . . . . . . . . . . 13 (𝐵 ∈ dom card → (𝑓:𝐵onto→ran 𝑓 → ran 𝑓𝐵))
5147, 49, 50syl2im 40 . . . . . . . . . . . 12 (𝐵 ∈ On → (𝑓:𝐵𝐴 → ran 𝑓𝐵))
5251imp 393 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → ran 𝑓𝐵)
53523adant1 1124 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → ran 𝑓𝐵)
54 endomtr 8167 . . . . . . . . . 10 (((card‘ran 𝑓) ≈ ran 𝑓 ∧ ran 𝑓𝐵) → (card‘ran 𝑓) ≼ 𝐵)
5546, 53, 54syl2anc 573 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → (card‘ran 𝑓) ≼ 𝐵)
56 cardon 8970 . . . . . . . . . . . 12 (card‘ran 𝑓) ∈ On
57 onenon 8975 . . . . . . . . . . . 12 ((card‘ran 𝑓) ∈ On → (card‘ran 𝑓) ∈ dom card)
5856, 57ax-mp 5 . . . . . . . . . . 11 (card‘ran 𝑓) ∈ dom card
59 carddom2 9003 . . . . . . . . . . 11 (((card‘ran 𝑓) ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘(card‘ran 𝑓)) ⊆ (card‘𝐵) ↔ (card‘ran 𝑓) ≼ 𝐵))
6058, 47, 59sylancr 575 . . . . . . . . . 10 (𝐵 ∈ On → ((card‘(card‘ran 𝑓)) ⊆ (card‘𝐵) ↔ (card‘ran 𝑓) ≼ 𝐵))
61603ad2ant2 1128 . . . . . . . . 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 8983 . . . . . . . . 9 (𝐵 ∈ On → (card‘𝐵) ⊆ 𝐵)
64633ad2ant2 1128 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → (card‘𝐵) ⊆ 𝐵)
6562, 64sstrd 3762 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → (card‘(card‘ran 𝑓)) ⊆ 𝐵)
6639, 65syl5eqssr 3799 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → (card‘ran 𝑓) ⊆ 𝐵)
67663expa 1111 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → (card‘ran 𝑓) ⊆ 𝐵)
6867adantrr 696 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → (card‘ran 𝑓) ⊆ 𝐵)
6938, 68sstrd 3762 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → (cf‘𝐴) ⊆ 𝐵)
7069ex 397 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → (cf‘𝐴) ⊆ 𝐵))
7170exlimdv 2013 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 382  w3a 1071   = wceq 1631  wex 1852  wcel 2145  {cab 2757  wral 3061  wrex 3062  Vcvv 3351  wss 3723   cint 4611   class class class wbr 4786  dom cdm 5249  ran crn 5250  Oncon0 5866   Fn wfn 6026  wf 6027  ontowfo 6029  cfv 6031  cen 8106  cdom 8107  cardccrd 8961  cfccf 8963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-er 7896  df-map 8011  df-en 8110  df-dom 8111  df-sdom 8112  df-card 8965  df-cf 8967  df-acn 8968
This theorem is referenced by:  cfsmolem  9294  cfcoflem  9296  cfcof  9298  inar1  9799
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