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Theorem ex-sategoelel 35406
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 10745 . . . . . . . 8 ((𝑀 ∈ WUni ∧ 𝑍𝑀) → 𝒫 𝑍𝑀)
42wun0 10756 . . . . . . . 8 ((𝑀 ∈ WUni ∧ 𝑍𝑀) → ∅ ∈ 𝑀)
53, 4ifcld 4577 . . . . . . 7 ((𝑀 ∈ WUni ∧ 𝑍𝑀) → if(𝑥 = 𝐵, 𝒫 𝑍, ∅) ∈ 𝑀)
61, 5ifcld 4577 . . . . . 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 9681 . . . . 5 ω ∈ V
1312a1i 11 . . . 4 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → ω ∈ V)
1411, 13elmapd 8879 . . 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 4625 . . . . 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 4537 . . . . 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 1193 . . . 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 2739 . . . . . . 7 (𝑥 = 𝐵 → (𝑥 = 𝐴𝐵 = 𝐴))
26 eqeq1 2739 . . . . . . . 8 (𝑥 = 𝐵 → (𝑥 = 𝐵𝐵 = 𝐵))
2726ifbid 4554 . . . . . . 7 (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝒫 𝑍, ∅) = if(𝐵 = 𝐵, 𝒫 𝑍, ∅))
2825, 27ifbieq2d 4557 . . . . . 6 (𝑥 = 𝐵 → if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅)) = if(𝐵 = 𝐴, 𝑍, if(𝐵 = 𝐵, 𝒫 𝑍, ∅)))
29 necom 2992 . . . . . . . . 9 (𝐴𝐵𝐵𝐴)
30 ifnefalse 4543 . . . . . . . . 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 1134 . . . . . . 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 2797 . . . . 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 1194 . . . . 5 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝐵 ∈ ω)
36 pwexg 5384 . . . . . . . 8 (𝑍𝑀 → 𝒫 𝑍 ∈ V)
3736adantl 481 . . . . . . 7 ((𝑀 ∈ WUni ∧ 𝑍𝑀) → 𝒫 𝑍 ∈ V)
38 0ex 5313 . . . . . . . 8 ∅ ∈ V
3938a1i 11 . . . . . . 7 ((𝑀 ∈ WUni ∧ 𝑍𝑀) → ∅ ∈ V)
4037, 39ifcld 4577 . . . . . 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 2735 . . . . 5 𝐵 = 𝐵
4443iftruei 4538 . . . 4 if(𝐵 = 𝐵, 𝒫 𝑍, ∅) = 𝒫 𝑍
4542, 44eqtrdi 2791 . . 3 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝑆𝐵) = 𝒫 𝑍)
4618, 24, 453eltr4d 2854 . 2 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝑆𝐴) ∈ (𝑆𝐵))
47 3simpa 1147 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴 ∈ ω ∧ 𝐵 ∈ ω))
48 sategoelfvb.s . . . 4 𝐸 = (𝑀 Sat (𝐴𝑔𝐵))
4948sategoelfvb 35404 . . 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 1086   = wceq 1537  wcel 2106  wne 2938  Vcvv 3478  c0 4339  ifcif 4531  𝒫 cpw 4605  cmpt 5231  wf 6559  cfv 6563  (class class class)co 7431  ωcom 7887  m cmap 8865  WUnicwun 10738  𝑔cgoe 35318   Sat csate 35323
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  ax-inf2 9679  ax-ac2 10501
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-nel 3045  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-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-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977  df-ac 10154  df-wun 10740  df-goel 35325  df-gona 35326  df-goal 35327  df-sat 35328  df-sate 35329  df-fmla 35330
This theorem is referenced by:  ex-sategoel  35407
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