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Theorem ttukeylem1 9924
Description: Lemma for ttukey 9933. Expand out the property of being an element of a property of finite character. (Contributed by Mario Carneiro, 15-May-2015.)
Hypotheses
Ref Expression
ttukeylem.1 (𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))
ttukeylem.2 (𝜑𝐵𝐴)
ttukeylem.3 (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))
Assertion
Ref Expression
ttukeylem1 (𝜑 → (𝐶𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))
Distinct variable groups:   𝑥,𝐶   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ttukeylem1
StepHypRef Expression
1 elex 3462 . . 3 (𝐶𝐴𝐶 ∈ V)
21a1i 11 . 2 (𝜑 → (𝐶𝐴𝐶 ∈ V))
3 id 22 . . . . 5 ((𝒫 𝐶 ∩ Fin) ⊆ 𝐴 → (𝒫 𝐶 ∩ Fin) ⊆ 𝐴)
4 ssun1 4102 . . . . . . . 8 𝐴 ⊆ ( 𝐴𝐵)
5 undif1 4385 . . . . . . . 8 (( 𝐴𝐵) ∪ 𝐵) = ( 𝐴𝐵)
64, 5sseqtrri 3955 . . . . . . 7 𝐴 ⊆ (( 𝐴𝐵) ∪ 𝐵)
7 fvex 6662 . . . . . . . . 9 (card‘( 𝐴𝐵)) ∈ V
8 ttukeylem.1 . . . . . . . . . 10 (𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))
9 f1ofo 6601 . . . . . . . . . 10 (𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵) → 𝐹:(card‘( 𝐴𝐵))–onto→( 𝐴𝐵))
108, 9syl 17 . . . . . . . . 9 (𝜑𝐹:(card‘( 𝐴𝐵))–onto→( 𝐴𝐵))
11 fornex 7643 . . . . . . . . 9 ((card‘( 𝐴𝐵)) ∈ V → (𝐹:(card‘( 𝐴𝐵))–onto→( 𝐴𝐵) → ( 𝐴𝐵) ∈ V))
127, 10, 11mpsyl 68 . . . . . . . 8 (𝜑 → ( 𝐴𝐵) ∈ V)
13 ttukeylem.2 . . . . . . . 8 (𝜑𝐵𝐴)
14 unexg 7456 . . . . . . . 8 ((( 𝐴𝐵) ∈ V ∧ 𝐵𝐴) → (( 𝐴𝐵) ∪ 𝐵) ∈ V)
1512, 13, 14syl2anc 587 . . . . . . 7 (𝜑 → (( 𝐴𝐵) ∪ 𝐵) ∈ V)
16 ssexg 5194 . . . . . . 7 (( 𝐴 ⊆ (( 𝐴𝐵) ∪ 𝐵) ∧ (( 𝐴𝐵) ∪ 𝐵) ∈ V) → 𝐴 ∈ V)
176, 15, 16sylancr 590 . . . . . 6 (𝜑 𝐴 ∈ V)
18 uniexb 7470 . . . . . 6 (𝐴 ∈ V ↔ 𝐴 ∈ V)
1917, 18sylibr 237 . . . . 5 (𝜑𝐴 ∈ V)
20 ssexg 5194 . . . . 5 (((𝒫 𝐶 ∩ Fin) ⊆ 𝐴𝐴 ∈ V) → (𝒫 𝐶 ∩ Fin) ∈ V)
213, 19, 20syl2anr 599 . . . 4 ((𝜑 ∧ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴) → (𝒫 𝐶 ∩ Fin) ∈ V)
22 infpwfidom 9443 . . . 4 ((𝒫 𝐶 ∩ Fin) ∈ V → 𝐶 ≼ (𝒫 𝐶 ∩ Fin))
23 reldom 8502 . . . . 5 Rel ≼
2423brrelex1i 5576 . . . 4 (𝐶 ≼ (𝒫 𝐶 ∩ Fin) → 𝐶 ∈ V)
2521, 22, 243syl 18 . . 3 ((𝜑 ∧ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴) → 𝐶 ∈ V)
2625ex 416 . 2 (𝜑 → ((𝒫 𝐶 ∩ Fin) ⊆ 𝐴𝐶 ∈ V))
27 ttukeylem.3 . . 3 (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))
28 eleq1 2880 . . . . 5 (𝑥 = 𝐶 → (𝑥𝐴𝐶𝐴))
29 pweq 4516 . . . . . . 7 (𝑥 = 𝐶 → 𝒫 𝑥 = 𝒫 𝐶)
3029ineq1d 4141 . . . . . 6 (𝑥 = 𝐶 → (𝒫 𝑥 ∩ Fin) = (𝒫 𝐶 ∩ Fin))
3130sseq1d 3949 . . . . 5 (𝑥 = 𝐶 → ((𝒫 𝑥 ∩ Fin) ⊆ 𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))
3228, 31bibi12d 349 . . . 4 (𝑥 = 𝐶 → ((𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴) ↔ (𝐶𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴)))
3332spcgv 3546 . . 3 (𝐶 ∈ V → (∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴) → (𝐶𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴)))
3427, 33syl5com 31 . 2 (𝜑 → (𝐶 ∈ V → (𝐶𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴)))
352, 26, 34pm5.21ndd 384 1 (𝜑 → (𝐶𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536   = wceq 1538  wcel 2112  Vcvv 3444  cdif 3881  cun 3882  cin 3883  wss 3884  𝒫 cpw 4500   cuni 4803   class class class wbr 5033  ontowfo 6326  1-1-ontowf1o 6327  cfv 6328  cdom 8494  Fincfn 8496  cardccrd 9352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-om 7565  df-1o 8089  df-en 8497  df-dom 8498  df-fin 8500
This theorem is referenced by:  ttukeylem2  9925  ttukeylem6  9929
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