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Theorem fpwwe 10402
Description: Given any function 𝐹 from the powerset of 𝐴 to 𝐴, canth2 8917 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 9786. Theorem 1.1 of [KanamoriPincus] p. 415. (Contributed by Mario Carneiro, 18-May-2015.) (Revised by AV, 20-Jul-2024.)
Hypotheses
Ref Expression
fpwwe.1 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦))}
fpwwe.2 (𝜑𝐴𝑉)
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 hints:   𝐴(𝑦)   𝑉(𝑥,𝑦,𝑟)

Proof of Theorem fpwwe
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 df-ov 7278 . . . . . 6 (𝑌(𝐹 ∘ 1st )𝑅) = ((𝐹 ∘ 1st )‘⟨𝑌, 𝑅⟩)
2 fo1st 7851 . . . . . . . 8 1st :V–onto→V
3 fofn 6690 . . . . . . . 8 (1st :V–onto→V → 1st Fn V)
42, 3ax-mp 5 . . . . . . 7 1st Fn V
5 opex 5379 . . . . . . 7 𝑌, 𝑅⟩ ∈ V
6 fvco2 6865 . . . . . . 7 ((1st Fn V ∧ ⟨𝑌, 𝑅⟩ ∈ V) → ((𝐹 ∘ 1st )‘⟨𝑌, 𝑅⟩) = (𝐹‘(1st ‘⟨𝑌, 𝑅⟩)))
74, 5, 6mp2an 689 . . . . . 6 ((𝐹 ∘ 1st )‘⟨𝑌, 𝑅⟩) = (𝐹‘(1st ‘⟨𝑌, 𝑅⟩))
81, 7eqtri 2766 . . . . 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 5678 . . . . . . 7 (𝑌𝑊𝑅 → (𝑌 ∈ V ∧ 𝑅 ∈ V))
11 op1stg 7843 . . . . . . 7 ((𝑌 ∈ V ∧ 𝑅 ∈ V) → (1st ‘⟨𝑌, 𝑅⟩) = 𝑌)
1210, 11syl 17 . . . . . 6 (𝑌𝑊𝑅 → (1st ‘⟨𝑌, 𝑅⟩) = 𝑌)
1312fveq2d 6778 . . . . 5 (𝑌𝑊𝑅 → (𝐹‘(1st ‘⟨𝑌, 𝑅⟩)) = (𝐹𝑌))
148, 13eqtrid 2790 . . . 4 (𝑌𝑊𝑅 → (𝑌(𝐹 ∘ 1st )𝑅) = (𝐹𝑌))
1514eleq1d 2823 . . 3 (𝑌𝑊𝑅 → ((𝑌(𝐹 ∘ 1st )𝑅) ∈ 𝑌 ↔ (𝐹𝑌) ∈ 𝑌))
1615pm5.32i 575 . 2 ((𝑌𝑊𝑅 ∧ (𝑌(𝐹 ∘ 1st )𝑅) ∈ 𝑌) ↔ (𝑌𝑊𝑅 ∧ (𝐹𝑌) ∈ 𝑌))
17 vex 3436 . . . . . . . . . . 11 𝑟 ∈ V
1817cnvex 7772 . . . . . . . . . 10 𝑟 ∈ V
1918imaex 7763 . . . . . . . . 9 (𝑟 “ {𝑦}) ∈ V
20 vex 3436 . . . . . . . . . . . 12 𝑢 ∈ V
2117inex1 5241 . . . . . . . . . . . 12 (𝑟 ∩ (𝑢 × 𝑢)) ∈ V
2220, 21opco1i 7966 . . . . . . . . . . 11 (𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = (𝐹𝑢)
23 fveq2 6774 . . . . . . . . . . 11 (𝑢 = (𝑟 “ {𝑦}) → (𝐹𝑢) = (𝐹‘(𝑟 “ {𝑦})))
2422, 23eqtrid 2790 . . . . . . . . . 10 (𝑢 = (𝑟 “ {𝑦}) → (𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = (𝐹‘(𝑟 “ {𝑦})))
2524eqeq1d 2740 . . . . . . . . 9 (𝑢 = (𝑟 “ {𝑦}) → ((𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ (𝐹‘(𝑟 “ {𝑦})) = 𝑦))
2619, 25sbcie 3759 . . . . . . . 8 ([(𝑟 “ {𝑦}) / 𝑢](𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ (𝐹‘(𝑟 “ {𝑦})) = 𝑦)
2726ralbii 3092 . . . . . . 7 (∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦)
2827anbi2i 623 . . . . . 6 ((𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦) ↔ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦))
2928anbi2i 623 . . . . 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 5141 . . . 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 2769 . . 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 (𝜑𝐴𝑉)
33 vex 3436 . . . . 5 𝑥 ∈ V
3433, 17opco1i 7966 . . . 4 (𝑥(𝐹 ∘ 1st )𝑟) = (𝐹𝑥)
35 simp1 1135 . . . . . . 7 ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑥𝐴)
36 velpw 4538 . . . . . . 7 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
3735, 36sylibr 233 . . . . . 6 ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑥 ∈ 𝒫 𝐴)
38 19.8a 2174 . . . . . . . 8 (𝑟 We 𝑥 → ∃𝑟 𝑟 We 𝑥)
39383ad2ant3 1134 . . . . . . 7 ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → ∃𝑟 𝑟 We 𝑥)
40 ween 9791 . . . . . . 7 (𝑥 ∈ dom card ↔ ∃𝑟 𝑟 We 𝑥)
4139, 40sylibr 233 . . . . . 6 ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑥 ∈ dom card)
4237, 41elind 4128 . . . . 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 593 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝐹𝑥) ∈ 𝐴)
4534, 44eqeltrid 2843 . . 3 ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥(𝐹 ∘ 1st )𝑟) ∈ 𝐴)
46 fpwwe.4 . . 3 𝑋 = dom 𝑊
4731, 32, 45, 46fpwwe2 10399 . 2 (𝜑 → ((𝑌𝑊𝑅 ∧ (𝑌(𝐹 ∘ 1st )𝑅) ∈ 𝑌) ↔ (𝑌 = 𝑋𝑅 = (𝑊𝑋))))
4816, 47bitr3id 285 1 (𝜑 → ((𝑌𝑊𝑅 ∧ (𝐹𝑌) ∈ 𝑌) ↔ (𝑌 = 𝑋𝑅 = (𝑊𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1539  wex 1782  wcel 2106  wral 3064  Vcvv 3432  [wsbc 3716  cin 3886  wss 3887  𝒫 cpw 4533  {csn 4561  cop 4567   cuni 4839   class class class wbr 5074  {copab 5136   We wwe 5543   × cxp 5587  ccnv 5588  dom cdm 5589  cima 5592  ccom 5593   Fn wfn 6428  ontowfo 6431  cfv 6433  (class class class)co 7275  1st c1st 7829  cardccrd 9693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-en 8734  df-oi 9269  df-card 9697
This theorem is referenced by:  canth4  10403  canthnumlem  10404  canthp1lem2  10409
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