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Theorem ex-sategoelel 35426
Description: Example of a valuation of a simplified satisfaction predicate for a Godel-set of membership. (Contributed by AV, 5-Nov-2023.)
Hypotheses
Ref Expression
sategoelfvb.s 𝐸 = (𝑀 Sat (𝐴𝑔𝐵))
ex-sategoelel.s 𝑆 = (𝑥 ∈ ω ↦ if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅)))
Assertion
Ref Expression
ex-sategoelel (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝑆𝐸)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑀   𝑥,𝑍
Allowed substitution hints:   𝑆(𝑥)   𝐸(𝑥)

Proof of Theorem ex-sategoelel
StepHypRef Expression
1 simpr 484 . . . . . . 7 ((𝑀 ∈ WUni ∧ 𝑍𝑀) → 𝑍𝑀)
2 simpl 482 . . . . . . . . 9 ((𝑀 ∈ WUni ∧ 𝑍𝑀) → 𝑀 ∈ WUni)
32, 1wunpw 10747 . . . . . . . 8 ((𝑀 ∈ WUni ∧ 𝑍𝑀) → 𝒫 𝑍𝑀)
42wun0 10758 . . . . . . . 8 ((𝑀 ∈ WUni ∧ 𝑍𝑀) → ∅ ∈ 𝑀)
53, 4ifcld 4572 . . . . . . 7 ((𝑀 ∈ WUni ∧ 𝑍𝑀) → if(𝑥 = 𝐵, 𝒫 𝑍, ∅) ∈ 𝑀)
61, 5ifcld 4572 . . . . . 6 ((𝑀 ∈ WUni ∧ 𝑍𝑀) → if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅)) ∈ 𝑀)
76adantr 480 . . . . 5 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅)) ∈ 𝑀)
87adantr 480 . . . 4 ((((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) ∧ 𝑥 ∈ ω) → if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅)) ∈ 𝑀)
9 ex-sategoelel.s . . . 4 𝑆 = (𝑥 ∈ ω ↦ if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅)))
108, 9fmptd 7134 . . 3 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝑆:ω⟶𝑀)
112adantr 480 . . . 4 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝑀 ∈ WUni)
12 omex 9683 . . . . 5 ω ∈ V
1312a1i 11 . . . 4 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → ω ∈ V)
1411, 13elmapd 8880 . . 3 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝑆 ∈ (𝑀m ω) ↔ 𝑆:ω⟶𝑀))
1510, 14mpbird 257 . 2 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝑆 ∈ (𝑀m ω))
16 pwidg 4620 . . . . 5 (𝑍𝑀𝑍 ∈ 𝒫 𝑍)
1716adantl 481 . . . 4 ((𝑀 ∈ WUni ∧ 𝑍𝑀) → 𝑍 ∈ 𝒫 𝑍)
1817adantr 480 . . 3 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝑍 ∈ 𝒫 𝑍)
199a1i 11 . . . 4 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝑆 = (𝑥 ∈ ω ↦ if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅))))
20 iftrue 4531 . . . . 5 (𝑥 = 𝐴 → if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅)) = 𝑍)
2120adantl 481 . . . 4 ((((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) ∧ 𝑥 = 𝐴) → if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅)) = 𝑍)
22 simpr1 1195 . . . 4 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝐴 ∈ ω)
231adantr 480 . . . 4 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝑍𝑀)
2419, 21, 22, 23fvmptd 7023 . . 3 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝑆𝐴) = 𝑍)
25 eqeq1 2741 . . . . . . 7 (𝑥 = 𝐵 → (𝑥 = 𝐴𝐵 = 𝐴))
26 eqeq1 2741 . . . . . . . 8 (𝑥 = 𝐵 → (𝑥 = 𝐵𝐵 = 𝐵))
2726ifbid 4549 . . . . . . 7 (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝒫 𝑍, ∅) = if(𝐵 = 𝐵, 𝒫 𝑍, ∅))
2825, 27ifbieq2d 4552 . . . . . 6 (𝑥 = 𝐵 → if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅)) = if(𝐵 = 𝐴, 𝑍, if(𝐵 = 𝐵, 𝒫 𝑍, ∅)))
29 necom 2994 . . . . . . . . 9 (𝐴𝐵𝐵𝐴)
30 ifnefalse 4537 . . . . . . . . 9 (𝐵𝐴 → if(𝐵 = 𝐴, 𝑍, if(𝐵 = 𝐵, 𝒫 𝑍, ∅)) = if(𝐵 = 𝐵, 𝒫 𝑍, ∅))
3129, 30sylbi 217 . . . . . . . 8 (𝐴𝐵 → if(𝐵 = 𝐴, 𝑍, if(𝐵 = 𝐵, 𝒫 𝑍, ∅)) = if(𝐵 = 𝐵, 𝒫 𝑍, ∅))
32313ad2ant3 1136 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → if(𝐵 = 𝐴, 𝑍, if(𝐵 = 𝐵, 𝒫 𝑍, ∅)) = if(𝐵 = 𝐵, 𝒫 𝑍, ∅))
3332adantl 481 . . . . . 6 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → if(𝐵 = 𝐴, 𝑍, if(𝐵 = 𝐵, 𝒫 𝑍, ∅)) = if(𝐵 = 𝐵, 𝒫 𝑍, ∅))
3428, 33sylan9eqr 2799 . . . . 5 ((((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅)) = if(𝐵 = 𝐵, 𝒫 𝑍, ∅))
35 simpr2 1196 . . . . 5 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝐵 ∈ ω)
36 pwexg 5378 . . . . . . . 8 (𝑍𝑀 → 𝒫 𝑍 ∈ V)
3736adantl 481 . . . . . . 7 ((𝑀 ∈ WUni ∧ 𝑍𝑀) → 𝒫 𝑍 ∈ V)
38 0ex 5307 . . . . . . . 8 ∅ ∈ V
3938a1i 11 . . . . . . 7 ((𝑀 ∈ WUni ∧ 𝑍𝑀) → ∅ ∈ V)
4037, 39ifcld 4572 . . . . . 6 ((𝑀 ∈ WUni ∧ 𝑍𝑀) → if(𝐵 = 𝐵, 𝒫 𝑍, ∅) ∈ V)
4140adantr 480 . . . . 5 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → if(𝐵 = 𝐵, 𝒫 𝑍, ∅) ∈ V)
4219, 34, 35, 41fvmptd 7023 . . . 4 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝑆𝐵) = if(𝐵 = 𝐵, 𝒫 𝑍, ∅))
43 eqid 2737 . . . . 5 𝐵 = 𝐵
4443iftruei 4532 . . . 4 if(𝐵 = 𝐵, 𝒫 𝑍, ∅) = 𝒫 𝑍
4542, 44eqtrdi 2793 . . 3 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝑆𝐵) = 𝒫 𝑍)
4618, 24, 453eltr4d 2856 . 2 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝑆𝐴) ∈ (𝑆𝐵))
47 3simpa 1149 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴 ∈ ω ∧ 𝐵 ∈ ω))
48 sategoelfvb.s . . . 4 𝐸 = (𝑀 Sat (𝐴𝑔𝐵))
4948sategoelfvb 35424 . . 3 ((𝑀 ∈ WUni ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝑆𝐸 ↔ (𝑆 ∈ (𝑀m ω) ∧ (𝑆𝐴) ∈ (𝑆𝐵))))
502, 47, 49syl2an 596 . 2 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝑆𝐸 ↔ (𝑆 ∈ (𝑀m ω) ∧ (𝑆𝐴) ∈ (𝑆𝐵))))
5115, 46, 50mpbir2and 713 1 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝑆𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  Vcvv 3480  c0 4333  ifcif 4525  𝒫 cpw 4600  cmpt 5225  wf 6557  cfv 6561  (class class class)co 7431  ωcom 7887  m cmap 8866  WUnicwun 10740  𝑔cgoe 35338   Sat csate 35343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-ac2 10503
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979  df-ac 10156  df-wun 10742  df-goel 35345  df-gona 35346  df-goal 35347  df-sat 35348  df-sate 35349  df-fmla 35350
This theorem is referenced by:  ex-sategoel  35427
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