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Theorem canthnumlem 10686
Description: Lemma for canthnum 10687. (Contributed by Mario Carneiro, 19-May-2015.)
Hypotheses
Ref Expression
canth4.1 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦))}
canth4.2 𝐵 = dom 𝑊
canth4.3 𝐶 = ((𝑊𝐵) “ {(𝐹𝐵)})
Assertion
Ref Expression
canthnumlem (𝐴𝑉 → ¬ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴)
Distinct variable groups:   𝑥,𝑟,𝑦,𝐴   𝐵,𝑟,𝑥,𝑦   𝐹,𝑟,𝑥,𝑦   𝑉,𝑟,𝑥,𝑦   𝑦,𝐶   𝑊,𝑟,𝑥,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑟)

Proof of Theorem canthnumlem
StepHypRef Expression
1 f1f 6805 . . . . 5 (𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴𝐹:(𝒫 𝐴 ∩ dom card)⟶𝐴)
2 ssid 4018 . . . . . 6 (𝒫 𝐴 ∩ dom card) ⊆ (𝒫 𝐴 ∩ dom card)
3 canth4.1 . . . . . . 7 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦))}
4 canth4.2 . . . . . . 7 𝐵 = dom 𝑊
5 canth4.3 . . . . . . 7 𝐶 = ((𝑊𝐵) “ {(𝐹𝐵)})
63, 4, 5canth4 10685 . . . . . 6 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ (𝒫 𝐴 ∩ dom card)) → (𝐵𝐴𝐶𝐵 ∧ (𝐹𝐵) = (𝐹𝐶)))
72, 6mp3an3 1449 . . . . 5 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)⟶𝐴) → (𝐵𝐴𝐶𝐵 ∧ (𝐹𝐵) = (𝐹𝐶)))
81, 7sylan2 593 . . . 4 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → (𝐵𝐴𝐶𝐵 ∧ (𝐹𝐵) = (𝐹𝐶)))
98simp3d 1143 . . 3 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → (𝐹𝐵) = (𝐹𝐶))
10 simpr 484 . . . 4 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴)
118simp1d 1141 . . . . . 6 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐵𝐴)
12 elpw2g 5339 . . . . . . 7 (𝐴𝑉 → (𝐵 ∈ 𝒫 𝐴𝐵𝐴))
1312adantr 480 . . . . . 6 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → (𝐵 ∈ 𝒫 𝐴𝐵𝐴))
1411, 13mpbird 257 . . . . 5 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐵 ∈ 𝒫 𝐴)
15 eqid 2735 . . . . . . . . . . . 12 𝐵 = 𝐵
16 eqid 2735 . . . . . . . . . . . 12 (𝑊𝐵) = (𝑊𝐵)
1715, 16pm3.2i 470 . . . . . . . . . . 11 (𝐵 = 𝐵 ∧ (𝑊𝐵) = (𝑊𝐵))
18 simpl 482 . . . . . . . . . . . 12 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐴𝑉)
1910, 1syl 17 . . . . . . . . . . . . 13 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐹:(𝒫 𝐴 ∩ dom card)⟶𝐴)
2019ffvelcdmda 7104 . . . . . . . . . . . 12 (((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → (𝐹𝑥) ∈ 𝐴)
213, 18, 20, 4fpwwe 10684 . . . . . . . . . . 11 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → ((𝐵𝑊(𝑊𝐵) ∧ (𝐹𝐵) ∈ 𝐵) ↔ (𝐵 = 𝐵 ∧ (𝑊𝐵) = (𝑊𝐵))))
2217, 21mpbiri 258 . . . . . . . . . 10 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → (𝐵𝑊(𝑊𝐵) ∧ (𝐹𝐵) ∈ 𝐵))
2322simpld 494 . . . . . . . . 9 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐵𝑊(𝑊𝐵))
243, 18fpwwelem 10683 . . . . . . . . 9 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → (𝐵𝑊(𝑊𝐵) ↔ ((𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵)) ∧ ((𝑊𝐵) We 𝐵 ∧ ∀𝑦𝐵 (𝐹‘((𝑊𝐵) “ {𝑦})) = 𝑦))))
2523, 24mpbid 232 . . . . . . . 8 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → ((𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵)) ∧ ((𝑊𝐵) We 𝐵 ∧ ∀𝑦𝐵 (𝐹‘((𝑊𝐵) “ {𝑦})) = 𝑦)))
2625simprld 772 . . . . . . 7 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → (𝑊𝐵) We 𝐵)
27 fvex 6920 . . . . . . . 8 (𝑊𝐵) ∈ V
28 weeq1 5676 . . . . . . . 8 (𝑟 = (𝑊𝐵) → (𝑟 We 𝐵 ↔ (𝑊𝐵) We 𝐵))
2927, 28spcev 3606 . . . . . . 7 ((𝑊𝐵) We 𝐵 → ∃𝑟 𝑟 We 𝐵)
3026, 29syl 17 . . . . . 6 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → ∃𝑟 𝑟 We 𝐵)
31 ween 10073 . . . . . 6 (𝐵 ∈ dom card ↔ ∃𝑟 𝑟 We 𝐵)
3230, 31sylibr 234 . . . . 5 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐵 ∈ dom card)
3314, 32elind 4210 . . . 4 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐵 ∈ (𝒫 𝐴 ∩ dom card))
348simp2d 1142 . . . . . . . 8 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐶𝐵)
3534pssssd 4110 . . . . . . 7 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐶𝐵)
3635, 11sstrd 4006 . . . . . 6 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐶𝐴)
37 elpw2g 5339 . . . . . . 7 (𝐴𝑉 → (𝐶 ∈ 𝒫 𝐴𝐶𝐴))
3837adantr 480 . . . . . 6 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → (𝐶 ∈ 𝒫 𝐴𝐶𝐴))
3936, 38mpbird 257 . . . . 5 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐶 ∈ 𝒫 𝐴)
40 ssnum 10077 . . . . . 6 ((𝐵 ∈ dom card ∧ 𝐶𝐵) → 𝐶 ∈ dom card)
4132, 35, 40syl2anc 584 . . . . 5 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐶 ∈ dom card)
4239, 41elind 4210 . . . 4 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐶 ∈ (𝒫 𝐴 ∩ dom card))
43 f1fveq 7282 . . . 4 ((𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴 ∧ (𝐵 ∈ (𝒫 𝐴 ∩ dom card) ∧ 𝐶 ∈ (𝒫 𝐴 ∩ dom card))) → ((𝐹𝐵) = (𝐹𝐶) ↔ 𝐵 = 𝐶))
4410, 33, 42, 43syl12anc 837 . . 3 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → ((𝐹𝐵) = (𝐹𝐶) ↔ 𝐵 = 𝐶))
459, 44mpbid 232 . 2 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐵 = 𝐶)
4634pssned 4111 . . . 4 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐶𝐵)
4746necomd 2994 . . 3 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐵𝐶)
4847neneqd 2943 . 2 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → ¬ 𝐵 = 𝐶)
4945, 48pm2.65da 817 1 (𝐴𝑉 → ¬ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  wral 3059  cin 3962  wss 3963  wpss 3964  𝒫 cpw 4605  {csn 4631   cuni 4912   class class class wbr 5148  {copab 5210   We wwe 5640   × cxp 5687  ccnv 5688  dom cdm 5689  cima 5692  wf 6559  1-1wf1 6560  cfv 6563  cardccrd 9973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-er 8744  df-en 8985  df-dom 8986  df-oi 9548  df-card 9977
This theorem is referenced by:  canthnum  10687
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