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Theorem cff1 10295
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 10284 . . . 4 (𝐴 ∈ On → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))})
2 cardon 9981 . . . . . . . . 9 (card‘𝑦) ∈ On
3 eleq1 2826 . . . . . . . . 9 (𝑥 = (card‘𝑦) → (𝑥 ∈ On ↔ (card‘𝑦) ∈ On))
42, 3mpbiri 258 . . . . . . . 8 (𝑥 = (card‘𝑦) → 𝑥 ∈ On)
54adantr 480 . . . . . . 7 ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) → 𝑥 ∈ On)
65exlimiv 1927 . . . . . 6 (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) → 𝑥 ∈ On)
76abssi 4079 . . . . 5 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))} ⊆ On
8 cflem 10282 . . . . . 6 (𝐴 ∈ On → ∃𝑥𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)))
9 abn0 4390 . . . . . 6 ({𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))} ≠ ∅ ↔ ∃𝑥𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)))
108, 9sylibr 234 . . . . 5 (𝐴 ∈ On → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))} ≠ ∅)
11 onint 7809 . . . . 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 2838 . . 3 (𝐴 ∈ On → (cf‘𝐴) ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))})
14 fvex 6919 . . . 4 (cf‘𝐴) ∈ V
15 eqeq1 2738 . . . . . 6 (𝑥 = (cf‘𝐴) → (𝑥 = (card‘𝑦) ↔ (cf‘𝐴) = (card‘𝑦)))
1615anbi1d 631 . . . . 5 (𝑥 = (cf‘𝐴) → ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) ↔ ((cf‘𝐴) = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))))
1716exbidv 1918 . . . 4 (𝑥 = (cf‘𝐴) → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) ↔ ∃𝑦((cf‘𝐴) = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))))
1814, 17elab 3680 . . 3 ((cf‘𝐴) ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))} ↔ ∃𝑦((cf‘𝐴) = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)))
1913, 18sylib 218 . 2 (𝐴 ∈ On → ∃𝑦((cf‘𝐴) = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)))
20 simplr 769 . . . . . 6 (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) → (cf‘𝐴) = (card‘𝑦))
21 onss 7803 . . . . . . . . 9 (𝐴 ∈ On → 𝐴 ⊆ On)
22 sstr 4003 . . . . . . . . 9 ((𝑦𝐴𝐴 ⊆ On) → 𝑦 ⊆ On)
2321, 22sylan2 593 . . . . . . . 8 ((𝑦𝐴𝐴 ∈ On) → 𝑦 ⊆ On)
2423ancoms 458 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑦𝐴) → 𝑦 ⊆ On)
2524ad2ant2r 747 . . . . . 6 (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) → 𝑦 ⊆ On)
26 vex 3481 . . . . . . . . . . 11 𝑦 ∈ V
27 onssnum 10077 . . . . . . . . . . 11 ((𝑦 ∈ V ∧ 𝑦 ⊆ On) → 𝑦 ∈ dom card)
2826, 27mpan 690 . . . . . . . . . 10 (𝑦 ⊆ On → 𝑦 ∈ dom card)
29 cardid2 9990 . . . . . . . . . 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 5150 . . . . . . . . 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 257 . . . . . . 7 (((cf‘𝐴) = (card‘𝑦) ∧ 𝑦 ⊆ On) → (cf‘𝐴) ≈ 𝑦)
35 bren 8993 . . . . . . 7 ((cf‘𝐴) ≈ 𝑦 ↔ ∃𝑓 𝑓:(cf‘𝐴)–1-1-onto𝑦)
3634, 35sylib 218 . . . . . 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 6847 . . . . . . . . . . 11 (𝑓:(cf‘𝐴)–1-1-onto𝑦𝑓:(cf‘𝐴)–1-1𝑦)
39 f1ss 6809 . . . . . . . . . . . 12 ((𝑓:(cf‘𝐴)–1-1𝑦𝑦𝐴) → 𝑓:(cf‘𝐴)–1-1𝐴)
4039ancoms 458 . . . . . . . . . . 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 715 . . . . . . . . 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 6855 . . . . . . . . . . . 12 (𝑓:(cf‘𝐴)–1-1-onto𝑦𝑓:(cf‘𝐴)–onto𝑦)
45 foelrn 7126 . . . . . . . . . . . . . . 15 ((𝑓:(cf‘𝐴)–onto𝑦𝑠𝑦) → ∃𝑤 ∈ (cf‘𝐴)𝑠 = (𝑓𝑤))
46 sseq2 4021 . . . . . . . . . . . . . . . . 17 (𝑠 = (𝑓𝑤) → (𝑧𝑠𝑧 ⊆ (𝑓𝑤)))
4746biimpcd 249 . . . . . . . . . . . . . . . 16 (𝑧𝑠 → (𝑠 = (𝑓𝑤) → 𝑧 ⊆ (𝑓𝑤)))
4847reximdv 3167 . . . . . . . . . . . . . . 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 3152 . . . . . . . . . . . . 13 (𝑓:(cf‘𝐴)–onto𝑦 → (∃𝑠𝑦 𝑧𝑠 → ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤)))
5150ralimdv 3166 . . . . . . . . . . . 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 407 . . . . . . . . . 10 ((∀𝑧𝐴𝑠𝑦 𝑧𝑠𝑓:(cf‘𝐴)–1-1-onto𝑦) → ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤))
5453adantll 714 . . . . . . . . 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 511 . . . . . . 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 1914 . . . . 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 457 . . 3 (𝐴 ∈ On → (((cf‘𝐴) = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) → ∃𝑓(𝑓:(cf‘𝐴)–1-1𝐴 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤))))
6160exlimdv 1930 . 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 206  wa 395  w3a 1086   = wceq 1536  wex 1775  wcel 2105  {cab 2711  wne 2937  wral 3058  wrex 3067  Vcvv 3477  wss 3962  c0 4338   cint 4950   class class class wbr 5147  dom cdm 5688  Oncon0 6385  1-1wf1 6559  ontowfo 6560  1-1-ontowf1o 6561  cfv 6562  cen 8980  cardccrd 9972  cfccf 9974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-er 8743  df-en 8984  df-dom 8985  df-card 9976  df-cf 9978
This theorem is referenced by:  cfsmolem  10307  cfcoflem  10309  cfcof  10311  alephreg  10619
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