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Theorem fpwwe 9453
Description: Given any function 𝐹 from the powerset of 𝐴 to 𝐴, canth2 8098 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 8838. 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 6638 . . . . . 6 (𝑌(𝐹 ∘ 1st )𝑅) = ((𝐹 ∘ 1st )‘⟨𝑌, 𝑅⟩)
2 fo1st 7173 . . . . . . . 8 1st :V–onto→V
3 fofn 6104 . . . . . . . 8 (1st :V–onto→V → 1st Fn V)
42, 3ax-mp 5 . . . . . . 7 1st Fn V
5 opex 4923 . . . . . . 7 𝑌, 𝑅⟩ ∈ V
6 fvco2 6260 . . . . . . 7 ((1st Fn V ∧ ⟨𝑌, 𝑅⟩ ∈ V) → ((𝐹 ∘ 1st )‘⟨𝑌, 𝑅⟩) = (𝐹‘(1st ‘⟨𝑌, 𝑅⟩)))
74, 5, 6mp2an 707 . . . . . 6 ((𝐹 ∘ 1st )‘⟨𝑌, 𝑅⟩) = (𝐹‘(1st ‘⟨𝑌, 𝑅⟩))
81, 7eqtri 2642 . . . . 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 5182 . . . . . . 7 (𝑌𝑊𝑅 → (𝑌 ∈ V ∧ 𝑅 ∈ V))
11 op1stg 7165 . . . . . . 7 ((𝑌 ∈ V ∧ 𝑅 ∈ V) → (1st ‘⟨𝑌, 𝑅⟩) = 𝑌)
1210, 11syl 17 . . . . . 6 (𝑌𝑊𝑅 → (1st ‘⟨𝑌, 𝑅⟩) = 𝑌)
1312fveq2d 6182 . . . . 5 (𝑌𝑊𝑅 → (𝐹‘(1st ‘⟨𝑌, 𝑅⟩)) = (𝐹𝑌))
148, 13syl5eq 2666 . . . 4 (𝑌𝑊𝑅 → (𝑌(𝐹 ∘ 1st )𝑅) = (𝐹𝑌))
1514eleq1d 2684 . . 3 (𝑌𝑊𝑅 → ((𝑌(𝐹 ∘ 1st )𝑅) ∈ 𝑌 ↔ (𝐹𝑌) ∈ 𝑌))
1615pm5.32i 668 . 2 ((𝑌𝑊𝑅 ∧ (𝑌(𝐹 ∘ 1st )𝑅) ∈ 𝑌) ↔ (𝑌𝑊𝑅 ∧ (𝐹𝑌) ∈ 𝑌))
17 vex 3198 . . . . . . . . . . 11 𝑟 ∈ V
1817cnvex 7098 . . . . . . . . . 10 𝑟 ∈ V
1918imaex 7089 . . . . . . . . 9 (𝑟 “ {𝑦}) ∈ V
20 vex 3198 . . . . . . . . . . . 12 𝑢 ∈ V
2117inex1 4790 . . . . . . . . . . . 12 (𝑟 ∩ (𝑢 × 𝑢)) ∈ V
2220, 21algrflem 7271 . . . . . . . . . . 11 (𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = (𝐹𝑢)
23 fveq2 6178 . . . . . . . . . . 11 (𝑢 = (𝑟 “ {𝑦}) → (𝐹𝑢) = (𝐹‘(𝑟 “ {𝑦})))
2422, 23syl5eq 2666 . . . . . . . . . 10 (𝑢 = (𝑟 “ {𝑦}) → (𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = (𝐹‘(𝑟 “ {𝑦})))
2524eqeq1d 2622 . . . . . . . . 9 (𝑢 = (𝑟 “ {𝑦}) → ((𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ (𝐹‘(𝑟 “ {𝑦})) = 𝑦))
2619, 25sbcie 3464 . . . . . . . 8 ([(𝑟 “ {𝑦}) / 𝑢](𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ (𝐹‘(𝑟 “ {𝑦})) = 𝑦)
2726ralbii 2977 . . . . . . 7 (∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦)
2827anbi2i 729 . . . . . 6 ((𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦) ↔ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦))
2928anbi2i 729 . . . . 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 4708 . . . 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 2645 . . 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 3198 . . . . 5 𝑥 ∈ V
3433, 17algrflem 7271 . . . 4 (𝑥(𝐹 ∘ 1st )𝑟) = (𝐹𝑥)
35 simp1 1059 . . . . . . 7 ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑥𝐴)
36 selpw 4156 . . . . . . 7 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
3735, 36sylibr 224 . . . . . 6 ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑥 ∈ 𝒫 𝐴)
38 19.8a 2050 . . . . . . . 8 (𝑟 We 𝑥 → ∃𝑟 𝑟 We 𝑥)
39383ad2ant3 1082 . . . . . . 7 ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → ∃𝑟 𝑟 We 𝑥)
40 ween 8843 . . . . . . 7 (𝑥 ∈ dom card ↔ ∃𝑟 𝑟 We 𝑥)
4139, 40sylibr 224 . . . . . 6 ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑥 ∈ dom card)
4237, 41elind 3790 . . . . 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 491 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝐹𝑥) ∈ 𝐴)
4534, 44syl5eqel 2703 . . 3 ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥(𝐹 ∘ 1st )𝑟) ∈ 𝐴)
46 fpwwe.4 . . 3 𝑋 = dom 𝑊
4731, 32, 45, 46fpwwe2 9450 . 2 (𝜑 → ((𝑌𝑊𝑅 ∧ (𝑌(𝐹 ∘ 1st )𝑅) ∈ 𝑌) ↔ (𝑌 = 𝑋𝑅 = (𝑊𝑋))))
4816, 47syl5bbr 274 1 (𝜑 → ((𝑌𝑊𝑅 ∧ (𝐹𝑌) ∈ 𝑌) ↔ (𝑌 = 𝑋𝑅 = (𝑊𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1481  wex 1702  wcel 1988  wral 2909  Vcvv 3195  [wsbc 3429  cin 3566  wss 3567  𝒫 cpw 4149  {csn 4168  cop 4174   cuni 4427   class class class wbr 4644  {copab 4703   We wwe 5062   × cxp 5102  ccnv 5103  dom cdm 5104  cima 5107  ccom 5108   Fn wfn 5871  ontowfo 5874  cfv 5876  (class class class)co 6635  1st c1st 7151  cardccrd 8746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-1st 7153  df-wrecs 7392  df-recs 7453  df-en 7941  df-oi 8400  df-card 8750
This theorem is referenced by:  canth4  9454  canthnumlem  9455  canthp1lem2  9460
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