MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fpwwe Structured version   Visualization version   GIF version

Theorem fpwwe 10159
Description: Given any function 𝐹 from the powerset of 𝐴 to 𝐴, canth2 8733 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 9543. 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 7186 . . . . . 6 (𝑌(𝐹 ∘ 1st )𝑅) = ((𝐹 ∘ 1st )‘⟨𝑌, 𝑅⟩)
2 fo1st 7747 . . . . . . . 8 1st :V–onto→V
3 fofn 6605 . . . . . . . 8 (1st :V–onto→V → 1st Fn V)
42, 3ax-mp 5 . . . . . . 7 1st Fn V
5 opex 5332 . . . . . . 7 𝑌, 𝑅⟩ ∈ V
6 fvco2 6778 . . . . . . 7 ((1st Fn V ∧ ⟨𝑌, 𝑅⟩ ∈ V) → ((𝐹 ∘ 1st )‘⟨𝑌, 𝑅⟩) = (𝐹‘(1st ‘⟨𝑌, 𝑅⟩)))
74, 5, 6mp2an 692 . . . . . 6 ((𝐹 ∘ 1st )‘⟨𝑌, 𝑅⟩) = (𝐹‘(1st ‘⟨𝑌, 𝑅⟩))
81, 7eqtri 2762 . . . . 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 5623 . . . . . . 7 (𝑌𝑊𝑅 → (𝑌 ∈ V ∧ 𝑅 ∈ V))
11 op1stg 7739 . . . . . . 7 ((𝑌 ∈ V ∧ 𝑅 ∈ V) → (1st ‘⟨𝑌, 𝑅⟩) = 𝑌)
1210, 11syl 17 . . . . . 6 (𝑌𝑊𝑅 → (1st ‘⟨𝑌, 𝑅⟩) = 𝑌)
1312fveq2d 6691 . . . . 5 (𝑌𝑊𝑅 → (𝐹‘(1st ‘⟨𝑌, 𝑅⟩)) = (𝐹𝑌))
148, 13syl5eq 2786 . . . 4 (𝑌𝑊𝑅 → (𝑌(𝐹 ∘ 1st )𝑅) = (𝐹𝑌))
1514eleq1d 2818 . . 3 (𝑌𝑊𝑅 → ((𝑌(𝐹 ∘ 1st )𝑅) ∈ 𝑌 ↔ (𝐹𝑌) ∈ 𝑌))
1615pm5.32i 578 . 2 ((𝑌𝑊𝑅 ∧ (𝑌(𝐹 ∘ 1st )𝑅) ∈ 𝑌) ↔ (𝑌𝑊𝑅 ∧ (𝐹𝑌) ∈ 𝑌))
17 vex 3404 . . . . . . . . . . 11 𝑟 ∈ V
1817cnvex 7669 . . . . . . . . . 10 𝑟 ∈ V
1918imaex 7660 . . . . . . . . 9 (𝑟 “ {𝑦}) ∈ V
20 vex 3404 . . . . . . . . . . . 12 𝑢 ∈ V
2117inex1 5195 . . . . . . . . . . . 12 (𝑟 ∩ (𝑢 × 𝑢)) ∈ V
2220, 21algrflem 7858 . . . . . . . . . . 11 (𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = (𝐹𝑢)
23 fveq2 6687 . . . . . . . . . . 11 (𝑢 = (𝑟 “ {𝑦}) → (𝐹𝑢) = (𝐹‘(𝑟 “ {𝑦})))
2422, 23syl5eq 2786 . . . . . . . . . 10 (𝑢 = (𝑟 “ {𝑦}) → (𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = (𝐹‘(𝑟 “ {𝑦})))
2524eqeq1d 2741 . . . . . . . . 9 (𝑢 = (𝑟 “ {𝑦}) → ((𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ (𝐹‘(𝑟 “ {𝑦})) = 𝑦))
2619, 25sbcie 3728 . . . . . . . 8 ([(𝑟 “ {𝑦}) / 𝑢](𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ (𝐹‘(𝑟 “ {𝑦})) = 𝑦)
2726ralbii 3081 . . . . . . 7 (∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦)
2827anbi2i 626 . . . . . 6 ((𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦) ↔ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦))
2928anbi2i 626 . . . . 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 5107 . . . 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 2765 . . 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 3404 . . . . 5 𝑥 ∈ V
3433, 17algrflem 7858 . . . 4 (𝑥(𝐹 ∘ 1st )𝑟) = (𝐹𝑥)
35 simp1 1137 . . . . . . 7 ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑥𝐴)
36 velpw 4503 . . . . . . 7 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
3735, 36sylibr 237 . . . . . 6 ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑥 ∈ 𝒫 𝐴)
38 19.8a 2182 . . . . . . . 8 (𝑟 We 𝑥 → ∃𝑟 𝑟 We 𝑥)
39383ad2ant3 1136 . . . . . . 7 ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → ∃𝑟 𝑟 We 𝑥)
40 ween 9548 . . . . . . 7 (𝑥 ∈ dom card ↔ ∃𝑟 𝑟 We 𝑥)
4139, 40sylibr 237 . . . . . 6 ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑥 ∈ dom card)
4237, 41elind 4094 . . . . 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 596 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝐹𝑥) ∈ 𝐴)
4534, 44eqeltrid 2838 . . 3 ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥(𝐹 ∘ 1st )𝑟) ∈ 𝐴)
46 fpwwe.4 . . 3 𝑋 = dom 𝑊
4731, 32, 45, 46fpwwe2 10156 . 2 (𝜑 → ((𝑌𝑊𝑅 ∧ (𝑌(𝐹 ∘ 1st )𝑅) ∈ 𝑌) ↔ (𝑌 = 𝑋𝑅 = (𝑊𝑋))))
4816, 47bitr3id 288 1 (𝜑 → ((𝑌𝑊𝑅 ∧ (𝐹𝑌) ∈ 𝑌) ↔ (𝑌 = 𝑋𝑅 = (𝑊𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1088   = wceq 1542  wex 1786  wcel 2114  wral 3054  Vcvv 3400  [wsbc 3685  cin 3852  wss 3853  𝒫 cpw 4498  {csn 4526  cop 4532   cuni 4806   class class class wbr 5040  {copab 5102   We wwe 5492   × cxp 5533  ccnv 5534  dom cdm 5535  cima 5538  ccom 5539   Fn wfn 6345  ontowfo 6348  cfv 6350  (class class class)co 7183  1st c1st 7725  cardccrd 9450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7492
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4807  df-int 4847  df-iun 4893  df-br 5041  df-opab 5103  df-mpt 5121  df-tr 5147  df-id 5439  df-eprel 5444  df-po 5452  df-so 5453  df-fr 5493  df-se 5494  df-we 5495  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-pred 6139  df-ord 6186  df-on 6187  df-lim 6188  df-suc 6189  df-iota 6308  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-isom 6359  df-riota 7140  df-ov 7186  df-1st 7727  df-wrecs 7989  df-recs 8050  df-en 8569  df-oi 9060  df-card 9454
This theorem is referenced by:  canth4  10160  canthnumlem  10161  canthp1lem2  10166
  Copyright terms: Public domain W3C validator