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Theorem cfflb 10297
Description: If there is a cofinal map from 𝐵 to 𝐴, then 𝐵 is at least (cf‘𝐴). This theorem and cff1 10296 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 6744 . . . . . . 7 (𝑓:𝐵𝐴 → ran 𝑓𝐴)
21adantr 480 . . . . . 6 ((𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → ran 𝑓𝐴)
3 ffn 6737 . . . . . . . . . . 11 (𝑓:𝐵𝐴𝑓 Fn 𝐵)
4 fnfvelrn 7100 . . . . . . . . . . 11 ((𝑓 Fn 𝐵𝑤𝐵) → (𝑓𝑤) ∈ ran 𝑓)
53, 4sylan 580 . . . . . . . . . 10 ((𝑓:𝐵𝐴𝑤𝐵) → (𝑓𝑤) ∈ ran 𝑓)
6 sseq2 4022 . . . . . . . . . . 11 (𝑠 = (𝑓𝑤) → (𝑧𝑠𝑧 ⊆ (𝑓𝑤)))
76rspcev 3622 . . . . . . . . . 10 (((𝑓𝑤) ∈ ran 𝑓𝑧 ⊆ (𝑓𝑤)) → ∃𝑠 ∈ ran 𝑓 𝑧𝑠)
85, 7sylan 580 . . . . . . . . 9 (((𝑓:𝐵𝐴𝑤𝐵) ∧ 𝑧 ⊆ (𝑓𝑤)) → ∃𝑠 ∈ ran 𝑓 𝑧𝑠)
98rexlimdva2 3155 . . . . . . . 8 (𝑓:𝐵𝐴 → (∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤) → ∃𝑠 ∈ ran 𝑓 𝑧𝑠))
109ralimdv 3167 . . . . . . 7 (𝑓:𝐵𝐴 → (∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤) → ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠))
1110imp 406 . . . . . 6 ((𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠)
122, 11jca 511 . . . . 5 ((𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → (ran 𝑓𝐴 ∧ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠))
13 fvex 6920 . . . . . 6 (card‘ran 𝑓) ∈ V
14 cfval 10285 . . . . . . . . . . 11 (𝐴 ∈ On → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))})
1514adantr 480 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))})
16153ad2ant2 1133 . . . . . . . . 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))})
17 vex 3482 . . . . . . . . . . . . . 14 𝑓 ∈ V
1817rnex 7933 . . . . . . . . . . . . 13 ran 𝑓 ∈ V
19 fveq2 6907 . . . . . . . . . . . . . . 15 (𝑦 = ran 𝑓 → (card‘𝑦) = (card‘ran 𝑓))
2019eqeq2d 2746 . . . . . . . . . . . . . 14 (𝑦 = ran 𝑓 → (𝑥 = (card‘𝑦) ↔ 𝑥 = (card‘ran 𝑓)))
21 sseq1 4021 . . . . . . . . . . . . . . 15 (𝑦 = ran 𝑓 → (𝑦𝐴 ↔ ran 𝑓𝐴))
22 rexeq 3320 . . . . . . . . . . . . . . . 16 (𝑦 = ran 𝑓 → (∃𝑠𝑦 𝑧𝑠 ↔ ∃𝑠 ∈ ran 𝑓 𝑧𝑠))
2322ralbidv 3176 . . . . . . . . . . . . . . 15 (𝑦 = ran 𝑓 → (∀𝑧𝐴𝑠𝑦 𝑧𝑠 ↔ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠))
2421, 23anbi12d 632 . . . . . . . . . . . . . 14 (𝑦 = ran 𝑓 → ((𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠) ↔ (ran 𝑓𝐴 ∧ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠)))
2520, 24anbi12d 632 . . . . . . . . . . . . 13 (𝑦 = ran 𝑓 → ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) ↔ (𝑥 = (card‘ran 𝑓) ∧ (ran 𝑓𝐴 ∧ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠))))
2618, 25spcev 3606 . . . . . . . . . . . 12 ((𝑥 = (card‘ran 𝑓) ∧ (ran 𝑓𝐴 ∧ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠)) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)))
27 abid 2716 . . . . . . . . . . . 12 (𝑥 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))} ↔ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)))
2826, 27sylibr 234 . . . . . . . . . . 11 ((𝑥 = (card‘ran 𝑓) ∧ (ran 𝑓𝐴 ∧ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠)) → 𝑥 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))})
29 intss1 4968 . . . . . . . . . . 11 (𝑥 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))} → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))} ⊆ 𝑥)
3028, 29syl 17 . . . . . . . . . 10 ((𝑥 = (card‘ran 𝑓) ∧ (ran 𝑓𝐴 ∧ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠)) → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))} ⊆ 𝑥)
31303adant2 1130 . . . . . . . . 9 ((𝑥 = (card‘ran 𝑓) ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (ran 𝑓𝐴 ∧ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠)) → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))} ⊆ 𝑥)
3216, 31eqsstrd 4034 . . . . . . . 8 ((𝑥 = (card‘ran 𝑓) ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (ran 𝑓𝐴 ∧ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠)) → (cf‘𝐴) ⊆ 𝑥)
33323expib 1121 . . . . . . 7 (𝑥 = (card‘ran 𝑓) → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (ran 𝑓𝐴 ∧ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠)) → (cf‘𝐴) ⊆ 𝑥))
34 sseq2 4022 . . . . . . 7 (𝑥 = (card‘ran 𝑓) → ((cf‘𝐴) ⊆ 𝑥 ↔ (cf‘𝐴) ⊆ (card‘ran 𝑓)))
3533, 34sylibd 239 . . . . . 6 (𝑥 = (card‘ran 𝑓) → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (ran 𝑓𝐴 ∧ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠)) → (cf‘𝐴) ⊆ (card‘ran 𝑓)))
3613, 35vtocle 3555 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (ran 𝑓𝐴 ∧ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠)) → (cf‘𝐴) ⊆ (card‘ran 𝑓))
3712, 36sylan2 593 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → (cf‘𝐴) ⊆ (card‘ran 𝑓))
38 cardidm 9997 . . . . . . 7 (card‘(card‘ran 𝑓)) = (card‘ran 𝑓)
39 onss 7804 . . . . . . . . . . . . . 14 (𝐴 ∈ On → 𝐴 ⊆ On)
401, 39sylan9ssr 4010 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝑓:𝐵𝐴) → ran 𝑓 ⊆ On)
41403adant2 1130 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → ran 𝑓 ⊆ On)
42 onssnum 10078 . . . . . . . . . . . 12 ((ran 𝑓 ∈ V ∧ ran 𝑓 ⊆ On) → ran 𝑓 ∈ dom card)
4318, 41, 42sylancr 587 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → ran 𝑓 ∈ dom card)
44 cardid2 9991 . . . . . . . . . . 11 (ran 𝑓 ∈ dom card → (card‘ran 𝑓) ≈ ran 𝑓)
4543, 44syl 17 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → (card‘ran 𝑓) ≈ ran 𝑓)
46 onenon 9987 . . . . . . . . . . . . 13 (𝐵 ∈ On → 𝐵 ∈ dom card)
47 dffn4 6827 . . . . . . . . . . . . . 14 (𝑓 Fn 𝐵𝑓:𝐵onto→ran 𝑓)
483, 47sylib 218 . . . . . . . . . . . . 13 (𝑓:𝐵𝐴𝑓:𝐵onto→ran 𝑓)
49 fodomnum 10095 . . . . . . . . . . . . 13 (𝐵 ∈ dom card → (𝑓:𝐵onto→ran 𝑓 → ran 𝑓𝐵))
5046, 48, 49syl2im 40 . . . . . . . . . . . 12 (𝐵 ∈ On → (𝑓:𝐵𝐴 → ran 𝑓𝐵))
5150imp 406 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → ran 𝑓𝐵)
52513adant1 1129 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → ran 𝑓𝐵)
53 endomtr 9051 . . . . . . . . . 10 (((card‘ran 𝑓) ≈ ran 𝑓 ∧ ran 𝑓𝐵) → (card‘ran 𝑓) ≼ 𝐵)
5445, 52, 53syl2anc 584 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → (card‘ran 𝑓) ≼ 𝐵)
55 cardon 9982 . . . . . . . . . . . 12 (card‘ran 𝑓) ∈ On
56 onenon 9987 . . . . . . . . . . . 12 ((card‘ran 𝑓) ∈ On → (card‘ran 𝑓) ∈ dom card)
5755, 56ax-mp 5 . . . . . . . . . . 11 (card‘ran 𝑓) ∈ dom card
58 carddom2 10015 . . . . . . . . . . 11 (((card‘ran 𝑓) ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘(card‘ran 𝑓)) ⊆ (card‘𝐵) ↔ (card‘ran 𝑓) ≼ 𝐵))
5957, 46, 58sylancr 587 . . . . . . . . . 10 (𝐵 ∈ On → ((card‘(card‘ran 𝑓)) ⊆ (card‘𝐵) ↔ (card‘ran 𝑓) ≼ 𝐵))
60593ad2ant2 1133 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → ((card‘(card‘ran 𝑓)) ⊆ (card‘𝐵) ↔ (card‘ran 𝑓) ≼ 𝐵))
6154, 60mpbird 257 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → (card‘(card‘ran 𝑓)) ⊆ (card‘𝐵))
62 cardonle 9995 . . . . . . . . 9 (𝐵 ∈ On → (card‘𝐵) ⊆ 𝐵)
63623ad2ant2 1133 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → (card‘𝐵) ⊆ 𝐵)
6461, 63sstrd 4006 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → (card‘(card‘ran 𝑓)) ⊆ 𝐵)
6538, 64eqsstrrid 4045 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → (card‘ran 𝑓) ⊆ 𝐵)
66653expa 1117 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → (card‘ran 𝑓) ⊆ 𝐵)
6766adantrr 717 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → (card‘ran 𝑓) ⊆ 𝐵)
6837, 67sstrd 4006 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → (cf‘𝐴) ⊆ 𝐵)
6968ex 412 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → (cf‘𝐴) ⊆ 𝐵))
7069exlimdv 1931 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 206  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wral 3059  wrex 3068  Vcvv 3478  wss 3963   cint 4951   class class class wbr 5148  dom cdm 5689  ran crn 5690  Oncon0 6386   Fn wfn 6558  wf 6559  ontowfo 6561  cfv 6563  cen 8981  cdom 8982  cardccrd 9973  cfccf 9975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-card 9977  df-cf 9979  df-acn 9980
This theorem is referenced by:  cfsmolem  10308  cfcoflem  10310  cfcof  10312  inar1  10813
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