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Theorem cff1 8941
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 8930 . . . 4 (𝐴 ∈ On → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))})
2 cardon 8631 . . . . . . . . 9 (card‘𝑦) ∈ On
3 eleq1 2675 . . . . . . . . 9 (𝑥 = (card‘𝑦) → (𝑥 ∈ On ↔ (card‘𝑦) ∈ On))
42, 3mpbiri 246 . . . . . . . 8 (𝑥 = (card‘𝑦) → 𝑥 ∈ On)
54adantr 479 . . . . . . 7 ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) → 𝑥 ∈ On)
65exlimiv 1844 . . . . . 6 (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) → 𝑥 ∈ On)
76abssi 3639 . . . . 5 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))} ⊆ On
8 cflem 8929 . . . . . 6 (𝐴 ∈ On → ∃𝑥𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)))
9 abn0 3907 . . . . . 6 ({𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))} ≠ ∅ ↔ ∃𝑥𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)))
108, 9sylibr 222 . . . . 5 (𝐴 ∈ On → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))} ≠ ∅)
11 onint 6865 . . . . 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 693 . . . 4 (𝐴 ∈ On → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))} ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))})
131, 12eqeltrd 2687 . . 3 (𝐴 ∈ On → (cf‘𝐴) ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))})
14 fvex 6098 . . . 4 (cf‘𝐴) ∈ V
15 eqeq1 2613 . . . . . 6 (𝑥 = (cf‘𝐴) → (𝑥 = (card‘𝑦) ↔ (cf‘𝐴) = (card‘𝑦)))
1615anbi1d 736 . . . . 5 (𝑥 = (cf‘𝐴) → ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) ↔ ((cf‘𝐴) = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))))
1716exbidv 1836 . . . 4 (𝑥 = (cf‘𝐴) → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) ↔ ∃𝑦((cf‘𝐴) = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))))
1814, 17elab 3318 . . 3 ((cf‘𝐴) ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠))} ↔ ∃𝑦((cf‘𝐴) = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)))
1913, 18sylib 206 . 2 (𝐴 ∈ On → ∃𝑦((cf‘𝐴) = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)))
20 simplr 787 . . . . . 6 (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) → (cf‘𝐴) = (card‘𝑦))
21 onss 6860 . . . . . . . . 9 (𝐴 ∈ On → 𝐴 ⊆ On)
22 sstr 3575 . . . . . . . . 9 ((𝑦𝐴𝐴 ⊆ On) → 𝑦 ⊆ On)
2321, 22sylan2 489 . . . . . . . 8 ((𝑦𝐴𝐴 ∈ On) → 𝑦 ⊆ On)
2423ancoms 467 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑦𝐴) → 𝑦 ⊆ On)
2524ad2ant2r 778 . . . . . 6 (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) → 𝑦 ⊆ On)
26 vex 3175 . . . . . . . . . . 11 𝑦 ∈ V
27 onssnum 8724 . . . . . . . . . . 11 ((𝑦 ∈ V ∧ 𝑦 ⊆ On) → 𝑦 ∈ dom card)
2826, 27mpan 701 . . . . . . . . . 10 (𝑦 ⊆ On → 𝑦 ∈ dom card)
29 cardid2 8640 . . . . . . . . . 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 4580 . . . . . . . . 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 245 . . . . . . 7 (((cf‘𝐴) = (card‘𝑦) ∧ 𝑦 ⊆ On) → (cf‘𝐴) ≈ 𝑦)
35 bren 7828 . . . . . . 7 ((cf‘𝐴) ≈ 𝑦 ↔ ∃𝑓 𝑓:(cf‘𝐴)–1-1-onto𝑦)
3634, 35sylib 206 . . . . . 6 (((cf‘𝐴) = (card‘𝑦) ∧ 𝑦 ⊆ On) → ∃𝑓 𝑓:(cf‘𝐴)–1-1-onto𝑦)
3720, 25, 36syl2anc 690 . . . . 5 (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) → ∃𝑓 𝑓:(cf‘𝐴)–1-1-onto𝑦)
38 f1of1 6034 . . . . . . . . . . 11 (𝑓:(cf‘𝐴)–1-1-onto𝑦𝑓:(cf‘𝐴)–1-1𝑦)
39 f1ss 6004 . . . . . . . . . . . 12 ((𝑓:(cf‘𝐴)–1-1𝑦𝑦𝐴) → 𝑓:(cf‘𝐴)–1-1𝐴)
4039ancoms 467 . . . . . . . . . . 11 ((𝑦𝐴𝑓:(cf‘𝐴)–1-1𝑦) → 𝑓:(cf‘𝐴)–1-1𝐴)
4138, 40sylan2 489 . . . . . . . . . 10 ((𝑦𝐴𝑓:(cf‘𝐴)–1-1-onto𝑦) → 𝑓:(cf‘𝐴)–1-1𝐴)
4241adantlr 746 . . . . . . . . 9 (((𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠) ∧ 𝑓:(cf‘𝐴)–1-1-onto𝑦) → 𝑓:(cf‘𝐴)–1-1𝐴)
43423adant1 1071 . . . . . . . 8 (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠) ∧ 𝑓:(cf‘𝐴)–1-1-onto𝑦) → 𝑓:(cf‘𝐴)–1-1𝐴)
44 f1ofo 6042 . . . . . . . . . . . 12 (𝑓:(cf‘𝐴)–1-1-onto𝑦𝑓:(cf‘𝐴)–onto𝑦)
45 foelrn 6271 . . . . . . . . . . . . . . 15 ((𝑓:(cf‘𝐴)–onto𝑦𝑠𝑦) → ∃𝑤 ∈ (cf‘𝐴)𝑠 = (𝑓𝑤))
46 sseq2 3589 . . . . . . . . . . . . . . . . 17 (𝑠 = (𝑓𝑤) → (𝑧𝑠𝑧 ⊆ (𝑓𝑤)))
4746biimpcd 237 . . . . . . . . . . . . . . . 16 (𝑧𝑠 → (𝑠 = (𝑓𝑤) → 𝑧 ⊆ (𝑓𝑤)))
4847reximdv 2998 . . . . . . . . . . . . . . 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 3012 . . . . . . . . . . . . 13 (𝑓:(cf‘𝐴)–onto𝑦 → (∃𝑠𝑦 𝑧𝑠 → ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤)))
5150ralimdv 2945 . . . . . . . . . . . 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 444 . . . . . . . . . 10 ((∀𝑧𝐴𝑠𝑦 𝑧𝑠𝑓:(cf‘𝐴)–1-1-onto𝑦) → ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤))
5453adantll 745 . . . . . . . . 9 (((𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠) ∧ 𝑓:(cf‘𝐴)–1-1-onto𝑦) → ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤))
55543adant1 1071 . . . . . . . 8 (((𝐴 ∈ On ∧ (cf‘𝐴) = (card‘𝑦)) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠) ∧ 𝑓:(cf‘𝐴)–1-1-onto𝑦) → ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤))
5643, 55jca 552 . . . . . . 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 1258 . . . . . 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 1832 . . . . 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 645 . . 3 (𝐴 ∈ On → (((cf‘𝐴) = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑠𝑦 𝑧𝑠)) → ∃𝑓(𝑓:(cf‘𝐴)–1-1𝐴 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤))))
6160exlimdv 1847 . 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 194  wa 382  w3a 1030   = wceq 1474  wex 1694  wcel 1976  {cab 2595  wne 2779  wral 2895  wrex 2896  Vcvv 3172  wss 3539  c0 3873   cint 4404   class class class wbr 4577  dom cdm 5028  Oncon0 5626  1-1wf1 5787  ontowfo 5788  1-1-ontowf1o 5789  cfv 5790  cen 7816  cardccrd 8622  cfccf 8624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-wrecs 7272  df-recs 7333  df-er 7607  df-en 7820  df-dom 7821  df-card 8626  df-cf 8628
This theorem is referenced by:  cfsmolem  8953  cfcoflem  8955  cfcof  8957  alephreg  9261
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