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Theorem cff1 9118
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 9107 . . . 4 (𝐴 ∈ On → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))})
2 cardon 8808 . . . . . . . . 9 (card‘𝑦) ∈ On
3 eleq1 2718 . . . . . . . . 9 (𝑥 = (card‘𝑦) → (𝑥 ∈ On ↔ (card‘𝑦) ∈ On))
42, 3mpbiri 248 . . . . . . . 8 (𝑥 = (card‘𝑦) → 𝑥 ∈ On)
54adantr 480 . . . . . . 7 ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) → 𝑥 ∈ On)
65exlimiv 1898 . . . . . 6 (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) → 𝑥 ∈ On)
76abssi 3710 . . . . 5 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))} ⊆ On
8 cflem 9106 . . . . . 6 (𝐴 ∈ On → ∃𝑥𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)))
9 abn0 3987 . . . . . 6 ({𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))} ≠ ∅ ↔ ∃𝑥𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)))
108, 9sylibr 224 . . . . 5 (𝐴 ∈ On → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))} ≠ ∅)
11 onint 7037 . . . . 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 696 . . . 4 (𝐴 ∈ On → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))} ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))})
131, 12eqeltrd 2730 . . 3 (𝐴 ∈ On → (cf‘𝐴) ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))})
14 fvex 6239 . . . 4 (cf‘𝐴) ∈ V
15 eqeq1 2655 . . . . . 6 (𝑥 = (cf‘𝐴) → (𝑥 = (card‘𝑦) ↔ (cf‘𝐴) = (card‘𝑦)))
1615anbi1d 741 . . . . 5 (𝑥 = (cf‘𝐴) → ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) ↔ ((cf‘𝐴) = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))))
1716exbidv 1890 . . . 4 (𝑥 = (cf‘𝐴) → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) ↔ ∃𝑦((cf‘𝐴) = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))))
1814, 17elab 3382 . . 3 ((cf‘𝐴) ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))} ↔ ∃𝑦((cf‘𝐴) = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)))
1913, 18sylib 208 . 2 (𝐴 ∈ On → ∃𝑦((cf‘𝐴) = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)))
20 simplr 807 . . . . . 6 (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) → (cf‘𝐴) = (card‘𝑦))
21 onss 7032 . . . . . . . . 9 (𝐴 ∈ On → 𝐴 ⊆ On)
22 sstr 3644 . . . . . . . . 9 ((𝑦𝐴𝐴 ⊆ On) → 𝑦 ⊆ On)
2321, 22sylan2 490 . . . . . . . 8 ((𝑦𝐴𝐴 ∈ On) → 𝑦 ⊆ On)
2423ancoms 468 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑦𝐴) → 𝑦 ⊆ On)
2524ad2ant2r 798 . . . . . 6 (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) → 𝑦 ⊆ On)
26 vex 3234 . . . . . . . . . . 11 𝑦 ∈ V
27 onssnum 8901 . . . . . . . . . . 11 ((𝑦 ∈ V ∧ 𝑦 ⊆ On) → 𝑦 ∈ dom card)
2826, 27mpan 706 . . . . . . . . . 10 (𝑦 ⊆ On → 𝑦 ∈ dom card)
29 cardid2 8817 . . . . . . . . . 10 (𝑦 ∈ dom card → (card‘𝑦) ≈ 𝑦)
3028, 29syl 17 . . . . . . . . 9 (𝑦 ⊆ On → (card‘𝑦) ≈ 𝑦)
3130adantl 481 . . . . . . . 8 (((cf‘𝐴) = (card‘𝑦) ∧ 𝑦 ⊆ On) → (card‘𝑦) ≈ 𝑦)
32 breq1 4688 . . . . . . . . 9 ((cf‘𝐴) = (card‘𝑦) → ((cf‘𝐴) ≈ 𝑦 ↔ (card‘𝑦) ≈ 𝑦))
3332adantr 480 . . . . . . . 8 (((cf‘𝐴) = (card‘𝑦) ∧ 𝑦 ⊆ On) → ((cf‘𝐴) ≈ 𝑦 ↔ (card‘𝑦) ≈ 𝑦))
3431, 33mpbird 247 . . . . . . 7 (((cf‘𝐴) = (card‘𝑦) ∧ 𝑦 ⊆ On) → (cf‘𝐴) ≈ 𝑦)
35 bren 8006 . . . . . . 7 ((cf‘𝐴) ≈ 𝑦 ↔ ∃𝑓 𝑓:(cf‘𝐴)–1-1-onto𝑦)
3634, 35sylib 208 . . . . . 6 (((cf‘𝐴) = (card‘𝑦) ∧ 𝑦 ⊆ On) → ∃𝑓 𝑓:(cf‘𝐴)–1-1-onto𝑦)
3720, 25, 36syl2anc 694 . . . . 5 (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) → ∃𝑓 𝑓:(cf‘𝐴)–1-1-onto𝑦)
38 f1of1 6174 . . . . . . . . . . 11 (𝑓:(cf‘𝐴)–1-1-onto𝑦𝑓:(cf‘𝐴)–1-1𝑦)
39 f1ss 6144 . . . . . . . . . . . 12 ((𝑓:(cf‘𝐴)–1-1𝑦𝑦𝐴) → 𝑓:(cf‘𝐴)–1-1𝐴)
4039ancoms 468 . . . . . . . . . . 11 ((𝑦𝐴𝑓:(cf‘𝐴)–1-1𝑦) → 𝑓:(cf‘𝐴)–1-1𝐴)
4138, 40sylan2 490 . . . . . . . . . 10 ((𝑦𝐴𝑓:(cf‘𝐴)–1-1-onto𝑦) → 𝑓:(cf‘𝐴)–1-1𝐴)
4241adantlr 751 . . . . . . . . 9 (((𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠) ∧ 𝑓:(cf‘𝐴)–1-1-onto𝑦) → 𝑓:(cf‘𝐴)–1-1𝐴)
43423adant1 1099 . . . . . . . 8 (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠) ∧ 𝑓:(cf‘𝐴)–1-1-onto𝑦) → 𝑓:(cf‘𝐴)–1-1𝐴)
44 f1ofo 6182 . . . . . . . . . . . 12 (𝑓:(cf‘𝐴)–1-1-onto𝑦𝑓:(cf‘𝐴)–onto𝑦)
45 foelrn 6418 . . . . . . . . . . . . . . 15 ((𝑓:(cf‘𝐴)–onto𝑦𝑠𝑦) → ∃𝑤 ∈ (cf‘𝐴)𝑠 = (𝑓𝑤))
46 sseq2 3660 . . . . . . . . . . . . . . . . 17 (𝑠 = (𝑓𝑤) → (𝑧𝑠𝑧 ⊆ (𝑓𝑤)))
4746biimpcd 239 . . . . . . . . . . . . . . . 16 (𝑧𝑠 → (𝑠 = (𝑓𝑤) → 𝑧 ⊆ (𝑓𝑤)))
4847reximdv 3045 . . . . . . . . . . . . . . 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 3060 . . . . . . . . . . . . 13 (𝑓:(cf‘𝐴)–onto𝑦 → (∃𝑠𝑦 𝑧𝑠 → ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤)))
5150ralimdv 2992 . . . . . . . . . . . 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 445 . . . . . . . . . 10 ((∀𝑧𝐴𝑠𝑦 𝑧𝑠𝑓:(cf‘𝐴)–1-1-onto𝑦) → ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤))
5453adantll 750 . . . . . . . . 9 (((𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠) ∧ 𝑓:(cf‘𝐴)–1-1-onto𝑦) → ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤))
55543adant1 1099 . . . . . . . 8 (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠) ∧ 𝑓:(cf‘𝐴)–1-1-onto𝑦) → ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤))
5643, 55jca 553 . . . . . . 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 1286 . . . . . 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 1886 . . . . 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 647 . . 3 (𝐴 ∈ On → (((cf‘𝐴) = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) → ∃𝑓(𝑓:(cf‘𝐴)–1-1𝐴 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤))))
6160exlimdv 1901 . 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 196  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wne 2823  wral 2941  wrex 2942  Vcvv 3231  wss 3607  c0 3948   cint 4507   class class class wbr 4685  dom cdm 5143  Oncon0 5761  1-1wf1 5923  ontowfo 5924  1-1-ontowf1o 5925  cfv 5926  cen 7994  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-wrecs 7452  df-recs 7513  df-er 7787  df-en 7998  df-dom 7999  df-card 8803  df-cf 8805
This theorem is referenced by:  cfsmolem  9130  cfcoflem  9132  cfcof  9134  alephreg  9442
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