MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cff1 Structured version   Visualization version   GIF version

Theorem cff1 10115
Description: There is always a map from (cf‘𝐴) to 𝐴 (this is a stronger condition than the definition, which only presupposes a map from some 𝑦 ≈ (cf‘𝐴). (Contributed by Mario Carneiro, 28-Feb-2013.)
Assertion
Ref Expression
cff1 (𝐴 ∈ On → ∃𝑓(𝑓:(cf‘𝐴)–1-1𝐴 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤)))
Distinct variable group:   𝐴,𝑓,𝑤,𝑧

Proof of Theorem cff1
Dummy variables 𝑠 𝑦 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cfval 10104 . . . 4 (𝐴 ∈ On → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))})
2 cardon 9801 . . . . . . . . 9 (card‘𝑦) ∈ On
3 eleq1 2824 . . . . . . . . 9 (𝑥 = (card‘𝑦) → (𝑥 ∈ On ↔ (card‘𝑦) ∈ On))
42, 3mpbiri 257 . . . . . . . 8 (𝑥 = (card‘𝑦) → 𝑥 ∈ On)
54adantr 481 . . . . . . 7 ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) → 𝑥 ∈ On)
65exlimiv 1932 . . . . . 6 (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) → 𝑥 ∈ On)
76abssi 4015 . . . . 5 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))} ⊆ On
8 cflem 10103 . . . . . 6 (𝐴 ∈ On → ∃𝑥𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)))
9 abn0 4327 . . . . . 6 ({𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))} ≠ ∅ ↔ ∃𝑥𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)))
108, 9sylibr 233 . . . . 5 (𝐴 ∈ On → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))} ≠ ∅)
11 onint 7703 . . . . 5 (({𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))} ⊆ On ∧ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))} ≠ ∅) → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))} ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))})
127, 10, 11sylancr 587 . . . 4 (𝐴 ∈ On → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))} ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))})
131, 12eqeltrd 2837 . . 3 (𝐴 ∈ On → (cf‘𝐴) ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))})
14 fvex 6838 . . . 4 (cf‘𝐴) ∈ V
15 eqeq1 2740 . . . . . 6 (𝑥 = (cf‘𝐴) → (𝑥 = (card‘𝑦) ↔ (cf‘𝐴) = (card‘𝑦)))
1615anbi1d 630 . . . . 5 (𝑥 = (cf‘𝐴) → ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) ↔ ((cf‘𝐴) = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))))
1716exbidv 1923 . . . 4 (𝑥 = (cf‘𝐴) → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) ↔ ∃𝑦((cf‘𝐴) = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))))
1814, 17elab 3619 . . 3 ((cf‘𝐴) ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))} ↔ ∃𝑦((cf‘𝐴) = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)))
1913, 18sylib 217 . 2 (𝐴 ∈ On → ∃𝑦((cf‘𝐴) = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)))
20 simplr 766 . . . . . 6 (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) → (cf‘𝐴) = (card‘𝑦))
21 onss 7697 . . . . . . . . 9 (𝐴 ∈ On → 𝐴 ⊆ On)
22 sstr 3940 . . . . . . . . 9 ((𝑦𝐴𝐴 ⊆ On) → 𝑦 ⊆ On)
2321, 22sylan2 593 . . . . . . . 8 ((𝑦𝐴𝐴 ∈ On) → 𝑦 ⊆ On)
2423ancoms 459 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑦𝐴) → 𝑦 ⊆ On)
2524ad2ant2r 744 . . . . . 6 (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) → 𝑦 ⊆ On)
26 vex 3445 . . . . . . . . . . 11 𝑦 ∈ V
27 onssnum 9897 . . . . . . . . . . 11 ((𝑦 ∈ V ∧ 𝑦 ⊆ On) → 𝑦 ∈ dom card)
2826, 27mpan 687 . . . . . . . . . 10 (𝑦 ⊆ On → 𝑦 ∈ dom card)
29 cardid2 9810 . . . . . . . . . 10 (𝑦 ∈ dom card → (card‘𝑦) ≈ 𝑦)
3028, 29syl 17 . . . . . . . . 9 (𝑦 ⊆ On → (card‘𝑦) ≈ 𝑦)
3130adantl 482 . . . . . . . 8 (((cf‘𝐴) = (card‘𝑦) ∧ 𝑦 ⊆ On) → (card‘𝑦) ≈ 𝑦)
32 breq1 5095 . . . . . . . . 9 ((cf‘𝐴) = (card‘𝑦) → ((cf‘𝐴) ≈ 𝑦 ↔ (card‘𝑦) ≈ 𝑦))
3332adantr 481 . . . . . . . 8 (((cf‘𝐴) = (card‘𝑦) ∧ 𝑦 ⊆ On) → ((cf‘𝐴) ≈ 𝑦 ↔ (card‘𝑦) ≈ 𝑦))
3431, 33mpbird 256 . . . . . . 7 (((cf‘𝐴) = (card‘𝑦) ∧ 𝑦 ⊆ On) → (cf‘𝐴) ≈ 𝑦)
35 bren 8814 . . . . . . 7 ((cf‘𝐴) ≈ 𝑦 ↔ ∃𝑓 𝑓:(cf‘𝐴)–1-1-onto𝑦)
3634, 35sylib 217 . . . . . 6 (((cf‘𝐴) = (card‘𝑦) ∧ 𝑦 ⊆ On) → ∃𝑓 𝑓:(cf‘𝐴)–1-1-onto𝑦)
3720, 25, 36syl2anc 584 . . . . 5 (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) → ∃𝑓 𝑓:(cf‘𝐴)–1-1-onto𝑦)
38 f1of1 6766 . . . . . . . . . . 11 (𝑓:(cf‘𝐴)–1-1-onto𝑦𝑓:(cf‘𝐴)–1-1𝑦)
39 f1ss 6727 . . . . . . . . . . . 12 ((𝑓:(cf‘𝐴)–1-1𝑦𝑦𝐴) → 𝑓:(cf‘𝐴)–1-1𝐴)
4039ancoms 459 . . . . . . . . . . 11 ((𝑦𝐴𝑓:(cf‘𝐴)–1-1𝑦) → 𝑓:(cf‘𝐴)–1-1𝐴)
4138, 40sylan2 593 . . . . . . . . . 10 ((𝑦𝐴𝑓:(cf‘𝐴)–1-1-onto𝑦) → 𝑓:(cf‘𝐴)–1-1𝐴)
4241adantlr 712 . . . . . . . . 9 (((𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠) ∧ 𝑓:(cf‘𝐴)–1-1-onto𝑦) → 𝑓:(cf‘𝐴)–1-1𝐴)
43423adant1 1129 . . . . . . . 8 (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠) ∧ 𝑓:(cf‘𝐴)–1-1-onto𝑦) → 𝑓:(cf‘𝐴)–1-1𝐴)
44 f1ofo 6774 . . . . . . . . . . . 12 (𝑓:(cf‘𝐴)–1-1-onto𝑦𝑓:(cf‘𝐴)–onto𝑦)
45 foelrn 7038 . . . . . . . . . . . . . . 15 ((𝑓:(cf‘𝐴)–onto𝑦𝑠𝑦) → ∃𝑤 ∈ (cf‘𝐴)𝑠 = (𝑓𝑤))
46 sseq2 3958 . . . . . . . . . . . . . . . . 17 (𝑠 = (𝑓𝑤) → (𝑧𝑠𝑧 ⊆ (𝑓𝑤)))
4746biimpcd 248 . . . . . . . . . . . . . . . 16 (𝑧𝑠 → (𝑠 = (𝑓𝑤) → 𝑧 ⊆ (𝑓𝑤)))
4847reximdv 3163 . . . . . . . . . . . . . . 15 (𝑧𝑠 → (∃𝑤 ∈ (cf‘𝐴)𝑠 = (𝑓𝑤) → ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤)))
4945, 48syl5com 31 . . . . . . . . . . . . . 14 ((𝑓:(cf‘𝐴)–onto𝑦𝑠𝑦) → (𝑧𝑠 → ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤)))
5049rexlimdva 3148 . . . . . . . . . . . . 13 (𝑓:(cf‘𝐴)–onto𝑦 → (∃𝑠𝑦 𝑧𝑠 → ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤)))
5150ralimdv 3162 . . . . . . . . . . . 12 (𝑓:(cf‘𝐴)–onto𝑦 → (∀𝑧𝐴𝑠𝑦 𝑧𝑠 → ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤)))
5244, 51syl 17 . . . . . . . . . . 11 (𝑓:(cf‘𝐴)–1-1-onto𝑦 → (∀𝑧𝐴𝑠𝑦 𝑧𝑠 → ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤)))
5352impcom 408 . . . . . . . . . 10 ((∀𝑧𝐴𝑠𝑦 𝑧𝑠𝑓:(cf‘𝐴)–1-1-onto𝑦) → ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤))
5453adantll 711 . . . . . . . . 9 (((𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠) ∧ 𝑓:(cf‘𝐴)–1-1-onto𝑦) → ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤))
55543adant1 1129 . . . . . . . 8 (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠) ∧ 𝑓:(cf‘𝐴)–1-1-onto𝑦) → ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤))
5643, 55jca 512 . . . . . . 7 (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠) ∧ 𝑓:(cf‘𝐴)–1-1-onto𝑦) → (𝑓:(cf‘𝐴)–1-1𝐴 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤)))
57563expia 1120 . . . . . 6 (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) → (𝑓:(cf‘𝐴)–1-1-onto𝑦 → (𝑓:(cf‘𝐴)–1-1𝐴 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤))))
5857eximdv 1919 . . . . 5 (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) → (∃𝑓 𝑓:(cf‘𝐴)–1-1-onto𝑦 → ∃𝑓(𝑓:(cf‘𝐴)–1-1𝐴 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤))))
5937, 58mpd 15 . . . 4 (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) → ∃𝑓(𝑓:(cf‘𝐴)–1-1𝐴 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤)))
6059expl 458 . . 3 (𝐴 ∈ On → (((cf‘𝐴) = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) → ∃𝑓(𝑓:(cf‘𝐴)–1-1𝐴 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤))))
6160exlimdv 1935 . 2 (𝐴 ∈ On → (∃𝑦((cf‘𝐴) = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) → ∃𝑓(𝑓:(cf‘𝐴)–1-1𝐴 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤))))
6219, 61mpd 15 1 (𝐴 ∈ On → ∃𝑓(𝑓:(cf‘𝐴)–1-1𝐴 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1540  wex 1780  wcel 2105  {cab 2713  wne 2940  wral 3061  wrex 3070  Vcvv 3441  wss 3898  c0 4269   cint 4894   class class class wbr 5092  dom cdm 5620  Oncon0 6302  1-1wf1 6476  ontowfo 6477  1-1-ontowf1o 6478  cfv 6479  cen 8801  cardccrd 9792  cfccf 9794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-int 4895  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-id 5518  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-se 5576  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-ord 6305  df-on 6306  df-suc 6308  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-isom 6488  df-riota 7293  df-ov 7340  df-2nd 7900  df-frecs 8167  df-wrecs 8198  df-recs 8272  df-er 8569  df-en 8805  df-dom 8806  df-card 9796  df-cf 9798
This theorem is referenced by:  cfsmolem  10127  cfcoflem  10129  cfcof  10131  alephreg  10439
  Copyright terms: Public domain W3C validator