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Theorem ex-sategoelel 32793
 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 488 . . . . . . 7 ((𝑀 ∈ WUni ∧ 𝑍𝑀) → 𝑍𝑀)
2 simpl 486 . . . . . . . . 9 ((𝑀 ∈ WUni ∧ 𝑍𝑀) → 𝑀 ∈ WUni)
32, 1wunpw 10120 . . . . . . . 8 ((𝑀 ∈ WUni ∧ 𝑍𝑀) → 𝒫 𝑍𝑀)
42wun0 10131 . . . . . . . 8 ((𝑀 ∈ WUni ∧ 𝑍𝑀) → ∅ ∈ 𝑀)
53, 4ifcld 4470 . . . . . . 7 ((𝑀 ∈ WUni ∧ 𝑍𝑀) → if(𝑥 = 𝐵, 𝒫 𝑍, ∅) ∈ 𝑀)
61, 5ifcld 4470 . . . . . 6 ((𝑀 ∈ WUni ∧ 𝑍𝑀) → if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅)) ∈ 𝑀)
76adantr 484 . . . . 5 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅)) ∈ 𝑀)
87adantr 484 . . . 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 6855 . . 3 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝑆:ω⟶𝑀)
112adantr 484 . . . 4 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝑀 ∈ WUni)
12 omex 9092 . . . . 5 ω ∈ V
1312a1i 11 . . . 4 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → ω ∈ V)
1411, 13elmapd 8405 . . 3 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝑆 ∈ (𝑀m ω) ↔ 𝑆:ω⟶𝑀))
1510, 14mpbird 260 . 2 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝑆 ∈ (𝑀m ω))
16 pwidg 4519 . . . . 5 (𝑍𝑀𝑍 ∈ 𝒫 𝑍)
1716adantl 485 . . . 4 ((𝑀 ∈ WUni ∧ 𝑍𝑀) → 𝑍 ∈ 𝒫 𝑍)
1817adantr 484 . . 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 4431 . . . . 5 (𝑥 = 𝐴 → if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅)) = 𝑍)
2120adantl 485 . . . 4 ((((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) ∧ 𝑥 = 𝐴) → if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅)) = 𝑍)
22 simpr1 1191 . . . 4 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝐴 ∈ ω)
231adantr 484 . . . 4 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝑍𝑀)
2419, 21, 22, 23fvmptd 6752 . . 3 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝑆𝐴) = 𝑍)
25 eqeq1 2802 . . . . . . 7 (𝑥 = 𝐵 → (𝑥 = 𝐴𝐵 = 𝐴))
26 eqeq1 2802 . . . . . . . 8 (𝑥 = 𝐵 → (𝑥 = 𝐵𝐵 = 𝐵))
2726ifbid 4447 . . . . . . 7 (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝒫 𝑍, ∅) = if(𝐵 = 𝐵, 𝒫 𝑍, ∅))
2825, 27ifbieq2d 4450 . . . . . 6 (𝑥 = 𝐵 → if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅)) = if(𝐵 = 𝐴, 𝑍, if(𝐵 = 𝐵, 𝒫 𝑍, ∅)))
29 necom 3040 . . . . . . . . 9 (𝐴𝐵𝐵𝐴)
30 ifnefalse 4437 . . . . . . . . 9 (𝐵𝐴 → if(𝐵 = 𝐴, 𝑍, if(𝐵 = 𝐵, 𝒫 𝑍, ∅)) = if(𝐵 = 𝐵, 𝒫 𝑍, ∅))
3129, 30sylbi 220 . . . . . . . 8 (𝐴𝐵 → if(𝐵 = 𝐴, 𝑍, if(𝐵 = 𝐵, 𝒫 𝑍, ∅)) = if(𝐵 = 𝐵, 𝒫 𝑍, ∅))
32313ad2ant3 1132 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → if(𝐵 = 𝐴, 𝑍, if(𝐵 = 𝐵, 𝒫 𝑍, ∅)) = if(𝐵 = 𝐵, 𝒫 𝑍, ∅))
3332adantl 485 . . . . . 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 2855 . . . . 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 1192 . . . . 5 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝐵 ∈ ω)
36 pwexg 5244 . . . . . . . 8 (𝑍𝑀 → 𝒫 𝑍 ∈ V)
3736adantl 485 . . . . . . 7 ((𝑀 ∈ WUni ∧ 𝑍𝑀) → 𝒫 𝑍 ∈ V)
38 0ex 5175 . . . . . . . 8 ∅ ∈ V
3938a1i 11 . . . . . . 7 ((𝑀 ∈ WUni ∧ 𝑍𝑀) → ∅ ∈ V)
4037, 39ifcld 4470 . . . . . 6 ((𝑀 ∈ WUni ∧ 𝑍𝑀) → if(𝐵 = 𝐵, 𝒫 𝑍, ∅) ∈ V)
4140adantr 484 . . . . 5 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → if(𝐵 = 𝐵, 𝒫 𝑍, ∅) ∈ V)
4219, 34, 35, 41fvmptd 6752 . . . 4 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝑆𝐵) = if(𝐵 = 𝐵, 𝒫 𝑍, ∅))
43 eqid 2798 . . . . 5 𝐵 = 𝐵
4443iftruei 4432 . . . 4 if(𝐵 = 𝐵, 𝒫 𝑍, ∅) = 𝒫 𝑍
4542, 44eqtrdi 2849 . . 3 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝑆𝐵) = 𝒫 𝑍)
4618, 24, 453eltr4d 2905 . 2 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝑆𝐴) ∈ (𝑆𝐵))
47 3simpa 1145 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴 ∈ ω ∧ 𝐵 ∈ ω))
48 sategoelfvb.s . . . 4 𝐸 = (𝑀 Sat (𝐴𝑔𝐵))
4948sategoelfvb 32791 . . 3 ((𝑀 ∈ WUni ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝑆𝐸 ↔ (𝑆 ∈ (𝑀m ω) ∧ (𝑆𝐴) ∈ (𝑆𝐵))))
502, 47, 49syl2an 598 . 2 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝑆𝐸 ↔ (𝑆 ∈ (𝑀m ω) ∧ (𝑆𝐴) ∈ (𝑆𝐵))))
5115, 46, 50mpbir2and 712 1 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝑆𝐸)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  Vcvv 3441  ∅c0 4243  ifcif 4425  𝒫 cpw 4497   ↦ cmpt 5110  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135  ωcom 7562   ↑m cmap 8391  WUnicwun 10113  ∈𝑔cgoe 32705   Sat∈ csate 32710 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443  ax-inf2 9090  ax-ac2 9876 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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7563  df-1st 7673  df-2nd 7674  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-1o 8087  df-2o 8088  df-er 8274  df-map 8393  df-en 8495  df-dom 8496  df-sdom 8497  df-card 9354  df-ac 9529  df-wun 10115  df-goel 32712  df-gona 32713  df-goal 32714  df-sat 32715  df-sate 32716  df-fmla 32717 This theorem is referenced by:  ex-sategoel  32794
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