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Theorem fpwwe 10056
Description: Given any function 𝐹 from the powerset of 𝐴 to 𝐴, canth2 8658 gives that the function is not injective, but we can say rather more than that. There is a unique well-ordered subset 𝑋, (𝑊𝑋)⟩ which "agrees" with 𝐹 in the sense that each initial segment maps to its upper bound, and such that the entire set maps to an element of the set (so that it cannot be extended without losing the well-ordering). This theorem can be used to prove dfac8a 9444. Theorem 1.1 of [KanamoriPincus] p. 415. (Contributed by Mario Carneiro, 18-May-2015.)
Hypotheses
Ref Expression
fpwwe.1 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦))}
fpwwe.2 (𝜑𝐴 ∈ V)
fpwwe.3 ((𝜑𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → (𝐹𝑥) ∈ 𝐴)
fpwwe.4 𝑋 = dom 𝑊
Assertion
Ref Expression
fpwwe (𝜑 → ((𝑌𝑊𝑅 ∧ (𝐹𝑌) ∈ 𝑌) ↔ (𝑌 = 𝑋𝑅 = (𝑊𝑋))))
Distinct variable groups:   𝑥,𝑟,𝐴   𝑦,𝑟,𝐹,𝑥   𝜑,𝑟,𝑥,𝑦   𝑅,𝑟,𝑥,𝑦   𝑋,𝑟,𝑥,𝑦   𝑌,𝑟,𝑥,𝑦   𝑊,𝑟,𝑥,𝑦
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem fpwwe
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 df-ov 7148 . . . . . 6 (𝑌(𝐹 ∘ 1st )𝑅) = ((𝐹 ∘ 1st )‘⟨𝑌, 𝑅⟩)
2 fo1st 7698 . . . . . . . 8 1st :V–onto→V
3 fofn 6585 . . . . . . . 8 (1st :V–onto→V → 1st Fn V)
42, 3ax-mp 5 . . . . . . 7 1st Fn V
5 opex 5347 . . . . . . 7 𝑌, 𝑅⟩ ∈ V
6 fvco2 6751 . . . . . . 7 ((1st Fn V ∧ ⟨𝑌, 𝑅⟩ ∈ V) → ((𝐹 ∘ 1st )‘⟨𝑌, 𝑅⟩) = (𝐹‘(1st ‘⟨𝑌, 𝑅⟩)))
74, 5, 6mp2an 688 . . . . . 6 ((𝐹 ∘ 1st )‘⟨𝑌, 𝑅⟩) = (𝐹‘(1st ‘⟨𝑌, 𝑅⟩))
81, 7eqtri 2841 . . . . 5 (𝑌(𝐹 ∘ 1st )𝑅) = (𝐹‘(1st ‘⟨𝑌, 𝑅⟩))
9 fpwwe.1 . . . . . . . 8 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦))}
109bropaex12 5635 . . . . . . 7 (𝑌𝑊𝑅 → (𝑌 ∈ V ∧ 𝑅 ∈ V))
11 op1stg 7690 . . . . . . 7 ((𝑌 ∈ V ∧ 𝑅 ∈ V) → (1st ‘⟨𝑌, 𝑅⟩) = 𝑌)
1210, 11syl 17 . . . . . 6 (𝑌𝑊𝑅 → (1st ‘⟨𝑌, 𝑅⟩) = 𝑌)
1312fveq2d 6667 . . . . 5 (𝑌𝑊𝑅 → (𝐹‘(1st ‘⟨𝑌, 𝑅⟩)) = (𝐹𝑌))
148, 13syl5eq 2865 . . . 4 (𝑌𝑊𝑅 → (𝑌(𝐹 ∘ 1st )𝑅) = (𝐹𝑌))
1514eleq1d 2894 . . 3 (𝑌𝑊𝑅 → ((𝑌(𝐹 ∘ 1st )𝑅) ∈ 𝑌 ↔ (𝐹𝑌) ∈ 𝑌))
1615pm5.32i 575 . 2 ((𝑌𝑊𝑅 ∧ (𝑌(𝐹 ∘ 1st )𝑅) ∈ 𝑌) ↔ (𝑌𝑊𝑅 ∧ (𝐹𝑌) ∈ 𝑌))
17 vex 3495 . . . . . . . . . . 11 𝑟 ∈ V
1817cnvex 7619 . . . . . . . . . 10 𝑟 ∈ V
1918imaex 7610 . . . . . . . . 9 (𝑟 “ {𝑦}) ∈ V
20 vex 3495 . . . . . . . . . . . 12 𝑢 ∈ V
2117inex1 5212 . . . . . . . . . . . 12 (𝑟 ∩ (𝑢 × 𝑢)) ∈ V
2220, 21algrflem 7808 . . . . . . . . . . 11 (𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = (𝐹𝑢)
23 fveq2 6663 . . . . . . . . . . 11 (𝑢 = (𝑟 “ {𝑦}) → (𝐹𝑢) = (𝐹‘(𝑟 “ {𝑦})))
2422, 23syl5eq 2865 . . . . . . . . . 10 (𝑢 = (𝑟 “ {𝑦}) → (𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = (𝐹‘(𝑟 “ {𝑦})))
2524eqeq1d 2820 . . . . . . . . 9 (𝑢 = (𝑟 “ {𝑦}) → ((𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ (𝐹‘(𝑟 “ {𝑦})) = 𝑦))
2619, 25sbcie 3809 . . . . . . . 8 ([(𝑟 “ {𝑦}) / 𝑢](𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ (𝐹‘(𝑟 “ {𝑦})) = 𝑦)
2726ralbii 3162 . . . . . . 7 (∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦)
2827anbi2i 622 . . . . . 6 ((𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦) ↔ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦))
2928anbi2i 622 . . . . 5 (((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦)) ↔ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦)))
3029opabbii 5124 . . . 4 {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))} = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦))}
319, 30eqtr4i 2844 . . 3 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
32 fpwwe.2 . . 3 (𝜑𝐴 ∈ V)
33 vex 3495 . . . . 5 𝑥 ∈ V
3433, 17algrflem 7808 . . . 4 (𝑥(𝐹 ∘ 1st )𝑟) = (𝐹𝑥)
35 simp1 1128 . . . . . . 7 ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑥𝐴)
36 velpw 4543 . . . . . . 7 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
3735, 36sylibr 235 . . . . . 6 ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑥 ∈ 𝒫 𝐴)
38 19.8a 2170 . . . . . . . 8 (𝑟 We 𝑥 → ∃𝑟 𝑟 We 𝑥)
39383ad2ant3 1127 . . . . . . 7 ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → ∃𝑟 𝑟 We 𝑥)
40 ween 9449 . . . . . . 7 (𝑥 ∈ dom card ↔ ∃𝑟 𝑟 We 𝑥)
4139, 40sylibr 235 . . . . . 6 ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑥 ∈ dom card)
4237, 41elind 4168 . . . . 5 ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑥 ∈ (𝒫 𝐴 ∩ dom card))
43 fpwwe.3 . . . . 5 ((𝜑𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → (𝐹𝑥) ∈ 𝐴)
4442, 43sylan2 592 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝐹𝑥) ∈ 𝐴)
4534, 44eqeltrid 2914 . . 3 ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥(𝐹 ∘ 1st )𝑟) ∈ 𝐴)
46 fpwwe.4 . . 3 𝑋 = dom 𝑊
4731, 32, 45, 46fpwwe2 10053 . 2 (𝜑 → ((𝑌𝑊𝑅 ∧ (𝑌(𝐹 ∘ 1st )𝑅) ∈ 𝑌) ↔ (𝑌 = 𝑋𝑅 = (𝑊𝑋))))
4816, 47syl5bbr 286 1 (𝜑 → ((𝑌𝑊𝑅 ∧ (𝐹𝑌) ∈ 𝑌) ↔ (𝑌 = 𝑋𝑅 = (𝑊𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wex 1771  wcel 2105  wral 3135  Vcvv 3492  [wsbc 3769  cin 3932  wss 3933  𝒫 cpw 4535  {csn 4557  cop 4563   cuni 4830   class class class wbr 5057  {copab 5119   We wwe 5506   × cxp 5546  ccnv 5547  dom cdm 5548  cima 5551  ccom 5552   Fn wfn 6343  ontowfo 6346  cfv 6348  (class class class)co 7145  1st c1st 7676  cardccrd 9352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-1st 7678  df-wrecs 7936  df-recs 7997  df-en 8498  df-oi 8962  df-card 9356
This theorem is referenced by:  canth4  10057  canthnumlem  10058  canthp1lem2  10063
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