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Theorem cff1 10301
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 10290 . . . 4 (𝐴 ∈ On → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))})
2 cardon 9987 . . . . . . . . 9 (card‘𝑦) ∈ On
3 eleq1 2814 . . . . . . . . 9 (𝑥 = (card‘𝑦) → (𝑥 ∈ On ↔ (card‘𝑦) ∈ On))
42, 3mpbiri 257 . . . . . . . 8 (𝑥 = (card‘𝑦) → 𝑥 ∈ On)
54adantr 479 . . . . . . 7 ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) → 𝑥 ∈ On)
65exlimiv 1926 . . . . . 6 (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) → 𝑥 ∈ On)
76abssi 4066 . . . . 5 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))} ⊆ On
8 cflem 10289 . . . . . 6 (𝐴 ∈ On → ∃𝑥𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)))
9 abn0 4385 . . . . . 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 7799 . . . . 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 585 . . . 4 (𝐴 ∈ On → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))} ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))})
131, 12eqeltrd 2826 . . 3 (𝐴 ∈ On → (cf‘𝐴) ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))})
14 fvex 6914 . . . 4 (cf‘𝐴) ∈ V
15 eqeq1 2730 . . . . . 6 (𝑥 = (cf‘𝐴) → (𝑥 = (card‘𝑦) ↔ (cf‘𝐴) = (card‘𝑦)))
1615anbi1d 629 . . . . 5 (𝑥 = (cf‘𝐴) → ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) ↔ ((cf‘𝐴) = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))))
1716exbidv 1917 . . . 4 (𝑥 = (cf‘𝐴) → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) ↔ ∃𝑦((cf‘𝐴) = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))))
1814, 17elab 3666 . . 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 767 . . . . . 6 (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) → (cf‘𝐴) = (card‘𝑦))
21 onss 7793 . . . . . . . . 9 (𝐴 ∈ On → 𝐴 ⊆ On)
22 sstr 3988 . . . . . . . . 9 ((𝑦𝐴𝐴 ⊆ On) → 𝑦 ⊆ On)
2321, 22sylan2 591 . . . . . . . 8 ((𝑦𝐴𝐴 ∈ On) → 𝑦 ⊆ On)
2423ancoms 457 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑦𝐴) → 𝑦 ⊆ On)
2524ad2ant2r 745 . . . . . 6 (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) → 𝑦 ⊆ On)
26 vex 3466 . . . . . . . . . . 11 𝑦 ∈ V
27 onssnum 10083 . . . . . . . . . . 11 ((𝑦 ∈ V ∧ 𝑦 ⊆ On) → 𝑦 ∈ dom card)
2826, 27mpan 688 . . . . . . . . . 10 (𝑦 ⊆ On → 𝑦 ∈ dom card)
29 cardid2 9996 . . . . . . . . . 10 (𝑦 ∈ dom card → (card‘𝑦) ≈ 𝑦)
3028, 29syl 17 . . . . . . . . 9 (𝑦 ⊆ On → (card‘𝑦) ≈ 𝑦)
3130adantl 480 . . . . . . . 8 (((cf‘𝐴) = (card‘𝑦) ∧ 𝑦 ⊆ On) → (card‘𝑦) ≈ 𝑦)
32 breq1 5156 . . . . . . . . 9 ((cf‘𝐴) = (card‘𝑦) → ((cf‘𝐴) ≈ 𝑦 ↔ (card‘𝑦) ≈ 𝑦))
3332adantr 479 . . . . . . . 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 8984 . . . . . . 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 582 . . . . 5 (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) → ∃𝑓 𝑓:(cf‘𝐴)–1-1-onto𝑦)
38 f1of1 6842 . . . . . . . . . . 11 (𝑓:(cf‘𝐴)–1-1-onto𝑦𝑓:(cf‘𝐴)–1-1𝑦)
39 f1ss 6803 . . . . . . . . . . . 12 ((𝑓:(cf‘𝐴)–1-1𝑦𝑦𝐴) → 𝑓:(cf‘𝐴)–1-1𝐴)
4039ancoms 457 . . . . . . . . . . 11 ((𝑦𝐴𝑓:(cf‘𝐴)–1-1𝑦) → 𝑓:(cf‘𝐴)–1-1𝐴)
4138, 40sylan2 591 . . . . . . . . . 10 ((𝑦𝐴𝑓:(cf‘𝐴)–1-1-onto𝑦) → 𝑓:(cf‘𝐴)–1-1𝐴)
4241adantlr 713 . . . . . . . . 9 (((𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠) ∧ 𝑓:(cf‘𝐴)–1-1-onto𝑦) → 𝑓:(cf‘𝐴)–1-1𝐴)
43423adant1 1127 . . . . . . . 8 (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠) ∧ 𝑓:(cf‘𝐴)–1-1-onto𝑦) → 𝑓:(cf‘𝐴)–1-1𝐴)
44 f1ofo 6850 . . . . . . . . . . . 12 (𝑓:(cf‘𝐴)–1-1-onto𝑦𝑓:(cf‘𝐴)–onto𝑦)
45 foelrn 7121 . . . . . . . . . . . . . . 15 ((𝑓:(cf‘𝐴)–onto𝑦𝑠𝑦) → ∃𝑤 ∈ (cf‘𝐴)𝑠 = (𝑓𝑤))
46 sseq2 4006 . . . . . . . . . . . . . . . . 17 (𝑠 = (𝑓𝑤) → (𝑧𝑠𝑧 ⊆ (𝑓𝑤)))
4746biimpcd 248 . . . . . . . . . . . . . . . 16 (𝑧𝑠 → (𝑠 = (𝑓𝑤) → 𝑧 ⊆ (𝑓𝑤)))
4847reximdv 3160 . . . . . . . . . . . . . . 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 3145 . . . . . . . . . . . . 13 (𝑓:(cf‘𝐴)–onto𝑦 → (∃𝑠𝑦 𝑧𝑠 → ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤)))
5150ralimdv 3159 . . . . . . . . . . . 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 406 . . . . . . . . . 10 ((∀𝑧𝐴𝑠𝑦 𝑧𝑠𝑓:(cf‘𝐴)–1-1-onto𝑦) → ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤))
5453adantll 712 . . . . . . . . 9 (((𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠) ∧ 𝑓:(cf‘𝐴)–1-1-onto𝑦) → ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤))
55543adant1 1127 . . . . . . . 8 (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠) ∧ 𝑓:(cf‘𝐴)–1-1-onto𝑦) → ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤))
5643, 55jca 510 . . . . . . 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 1118 . . . . . 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 1913 . . . . 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 456 . . 3 (𝐴 ∈ On → (((cf‘𝐴) = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) → ∃𝑓(𝑓:(cf‘𝐴)–1-1𝐴 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤))))
6160exlimdv 1929 . 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 394  w3a 1084   = wceq 1534  wex 1774  wcel 2099  {cab 2703  wne 2930  wral 3051  wrex 3060  Vcvv 3462  wss 3947  c0 4325   cint 4954   class class class wbr 5153  dom cdm 5682  Oncon0 6376  1-1wf1 6551  ontowfo 6552  1-1-ontowf1o 6553  cfv 6554  cen 8971  cardccrd 9978  cfccf 9980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-isom 6563  df-riota 7380  df-ov 7427  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-er 8734  df-en 8975  df-dom 8976  df-card 9982  df-cf 9984
This theorem is referenced by:  cfsmolem  10313  cfcoflem  10315  cfcof  10317  alephreg  10625
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