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Theorem cfflb 9873
Description: If there is a cofinal map from 𝐵 to 𝐴, then 𝐵 is at least (cf‘𝐴). This theorem and cff1 9872 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 6552 . . . . . . 7 (𝑓:𝐵𝐴 → ran 𝑓𝐴)
21adantr 484 . . . . . 6 ((𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → ran 𝑓𝐴)
3 ffn 6545 . . . . . . . . . . 11 (𝑓:𝐵𝐴𝑓 Fn 𝐵)
4 fnfvelrn 6901 . . . . . . . . . . 11 ((𝑓 Fn 𝐵𝑤𝐵) → (𝑓𝑤) ∈ ran 𝑓)
53, 4sylan 583 . . . . . . . . . 10 ((𝑓:𝐵𝐴𝑤𝐵) → (𝑓𝑤) ∈ ran 𝑓)
6 sseq2 3927 . . . . . . . . . . 11 (𝑠 = (𝑓𝑤) → (𝑧𝑠𝑧 ⊆ (𝑓𝑤)))
76rspcev 3537 . . . . . . . . . 10 (((𝑓𝑤) ∈ ran 𝑓𝑧 ⊆ (𝑓𝑤)) → ∃𝑠 ∈ ran 𝑓 𝑧𝑠)
85, 7sylan 583 . . . . . . . . 9 (((𝑓:𝐵𝐴𝑤𝐵) ∧ 𝑧 ⊆ (𝑓𝑤)) → ∃𝑠 ∈ ran 𝑓 𝑧𝑠)
98rexlimdva2 3206 . . . . . . . 8 (𝑓:𝐵𝐴 → (∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤) → ∃𝑠 ∈ ran 𝑓 𝑧𝑠))
109ralimdv 3101 . . . . . . 7 (𝑓:𝐵𝐴 → (∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤) → ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠))
1110imp 410 . . . . . 6 ((𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠)
122, 11jca 515 . . . . 5 ((𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → (ran 𝑓𝐴 ∧ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠))
13 fvex 6730 . . . . . 6 (card‘ran 𝑓) ∈ V
14 cfval 9861 . . . . . . . . . . 11 (𝐴 ∈ On → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))})
1514adantr 484 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))})
16153ad2ant2 1136 . . . . . . . . 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 3412 . . . . . . . . . . . . . 14 𝑓 ∈ V
1817rnex 7690 . . . . . . . . . . . . 13 ran 𝑓 ∈ V
19 fveq2 6717 . . . . . . . . . . . . . . 15 (𝑦 = ran 𝑓 → (card‘𝑦) = (card‘ran 𝑓))
2019eqeq2d 2748 . . . . . . . . . . . . . 14 (𝑦 = ran 𝑓 → (𝑥 = (card‘𝑦) ↔ 𝑥 = (card‘ran 𝑓)))
21 sseq1 3926 . . . . . . . . . . . . . . 15 (𝑦 = ran 𝑓 → (𝑦𝐴 ↔ ran 𝑓𝐴))
22 rexeq 3320 . . . . . . . . . . . . . . . 16 (𝑦 = ran 𝑓 → (∃𝑠𝑦 𝑧𝑠 ↔ ∃𝑠 ∈ ran 𝑓 𝑧𝑠))
2322ralbidv 3118 . . . . . . . . . . . . . . 15 (𝑦 = ran 𝑓 → (∀𝑧𝐴𝑠𝑦 𝑧𝑠 ↔ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠))
2421, 23anbi12d 634 . . . . . . . . . . . . . 14 (𝑦 = ran 𝑓 → ((𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠) ↔ (ran 𝑓𝐴 ∧ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠)))
2520, 24anbi12d 634 . . . . . . . . . . . . 13 (𝑦 = ran 𝑓 → ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) ↔ (𝑥 = (card‘ran 𝑓) ∧ (ran 𝑓𝐴 ∧ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠))))
2618, 25spcev 3521 . . . . . . . . . . . 12 ((𝑥 = (card‘ran 𝑓) ∧ (ran 𝑓𝐴 ∧ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠)) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)))
27 abid 2718 . . . . . . . . . . . 12 (𝑥 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))} ↔ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)))
2826, 27sylibr 237 . . . . . . . . . . 11 ((𝑥 = (card‘ran 𝑓) ∧ (ran 𝑓𝐴 ∧ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠)) → 𝑥 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))})
29 intss1 4874 . . . . . . . . . . 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 1133 . . . . . . . . 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 3939 . . . . . . . 8 ((𝑥 = (card‘ran 𝑓) ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (ran 𝑓𝐴 ∧ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠)) → (cf‘𝐴) ⊆ 𝑥)
33323expib 1124 . . . . . . 7 (𝑥 = (card‘ran 𝑓) → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (ran 𝑓𝐴 ∧ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠)) → (cf‘𝐴) ⊆ 𝑥))
34 sseq2 3927 . . . . . . 7 (𝑥 = (card‘ran 𝑓) → ((cf‘𝐴) ⊆ 𝑥 ↔ (cf‘𝐴) ⊆ (card‘ran 𝑓)))
3533, 34sylibd 242 . . . . . 6 (𝑥 = (card‘ran 𝑓) → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (ran 𝑓𝐴 ∧ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠)) → (cf‘𝐴) ⊆ (card‘ran 𝑓)))
3613, 35vtocle 3500 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (ran 𝑓𝐴 ∧ ∀𝑧𝐴𝑠 ∈ ran 𝑓 𝑧𝑠)) → (cf‘𝐴) ⊆ (card‘ran 𝑓))
3712, 36sylan2 596 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → (cf‘𝐴) ⊆ (card‘ran 𝑓))
38 cardidm 9575 . . . . . . 7 (card‘(card‘ran 𝑓)) = (card‘ran 𝑓)
39 onss 7568 . . . . . . . . . . . . . 14 (𝐴 ∈ On → 𝐴 ⊆ On)
401, 39sylan9ssr 3915 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝑓:𝐵𝐴) → ran 𝑓 ⊆ On)
41403adant2 1133 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → ran 𝑓 ⊆ On)
42 onssnum 9654 . . . . . . . . . . . 12 ((ran 𝑓 ∈ V ∧ ran 𝑓 ⊆ On) → ran 𝑓 ∈ dom card)
4318, 41, 42sylancr 590 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → ran 𝑓 ∈ dom card)
44 cardid2 9569 . . . . . . . . . . 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 9565 . . . . . . . . . . . . 13 (𝐵 ∈ On → 𝐵 ∈ dom card)
47 dffn4 6639 . . . . . . . . . . . . . 14 (𝑓 Fn 𝐵𝑓:𝐵onto→ran 𝑓)
483, 47sylib 221 . . . . . . . . . . . . 13 (𝑓:𝐵𝐴𝑓:𝐵onto→ran 𝑓)
49 fodomnum 9671 . . . . . . . . . . . . 13 (𝐵 ∈ dom card → (𝑓:𝐵onto→ran 𝑓 → ran 𝑓𝐵))
5046, 48, 49syl2im 40 . . . . . . . . . . . 12 (𝐵 ∈ On → (𝑓:𝐵𝐴 → ran 𝑓𝐵))
5150imp 410 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → ran 𝑓𝐵)
52513adant1 1132 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → ran 𝑓𝐵)
53 endomtr 8686 . . . . . . . . . 10 (((card‘ran 𝑓) ≈ ran 𝑓 ∧ ran 𝑓𝐵) → (card‘ran 𝑓) ≼ 𝐵)
5445, 52, 53syl2anc 587 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → (card‘ran 𝑓) ≼ 𝐵)
55 cardon 9560 . . . . . . . . . . . 12 (card‘ran 𝑓) ∈ On
56 onenon 9565 . . . . . . . . . . . 12 ((card‘ran 𝑓) ∈ On → (card‘ran 𝑓) ∈ dom card)
5755, 56ax-mp 5 . . . . . . . . . . 11 (card‘ran 𝑓) ∈ dom card
58 carddom2 9593 . . . . . . . . . . 11 (((card‘ran 𝑓) ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘(card‘ran 𝑓)) ⊆ (card‘𝐵) ↔ (card‘ran 𝑓) ≼ 𝐵))
5957, 46, 58sylancr 590 . . . . . . . . . 10 (𝐵 ∈ On → ((card‘(card‘ran 𝑓)) ⊆ (card‘𝐵) ↔ (card‘ran 𝑓) ≼ 𝐵))
60593ad2ant2 1136 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → ((card‘(card‘ran 𝑓)) ⊆ (card‘𝐵) ↔ (card‘ran 𝑓) ≼ 𝐵))
6154, 60mpbird 260 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → (card‘(card‘ran 𝑓)) ⊆ (card‘𝐵))
62 cardonle 9573 . . . . . . . . 9 (𝐵 ∈ On → (card‘𝐵) ⊆ 𝐵)
63623ad2ant2 1136 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → (card‘𝐵) ⊆ 𝐵)
6461, 63sstrd 3911 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → (card‘(card‘ran 𝑓)) ⊆ 𝐵)
6538, 64eqsstrrid 3950 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑓:𝐵𝐴) → (card‘ran 𝑓) ⊆ 𝐵)
66653expa 1120 . . . . 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 3911 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → (cf‘𝐴) ⊆ 𝐵)
6968ex 416 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → (cf‘𝐴) ⊆ 𝐵))
7069exlimdv 1941 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 209  wa 399  w3a 1089   = wceq 1543  wex 1787  wcel 2110  {cab 2714  wral 3061  wrex 3062  Vcvv 3408  wss 3866   cint 4859   class class class wbr 5053  dom cdm 5551  ran crn 5552  Oncon0 6213   Fn wfn 6375  wf 6376  ontowfo 6378  cfv 6380  cen 8623  cdom 8624  cardccrd 9551  cfccf 9553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-er 8391  df-map 8510  df-en 8627  df-dom 8628  df-sdom 8629  df-card 9555  df-cf 9557  df-acn 9558
This theorem is referenced by:  cfsmolem  9884  cfcoflem  9886  cfcof  9888  inar1  10389
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