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Theorem ex-sategoelel 34412
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 486 . . . . . . 7 ((𝑀 ∈ WUni ∧ 𝑍𝑀) → 𝑍𝑀)
2 simpl 484 . . . . . . . . 9 ((𝑀 ∈ WUni ∧ 𝑍𝑀) → 𝑀 ∈ WUni)
32, 1wunpw 10702 . . . . . . . 8 ((𝑀 ∈ WUni ∧ 𝑍𝑀) → 𝒫 𝑍𝑀)
42wun0 10713 . . . . . . . 8 ((𝑀 ∈ WUni ∧ 𝑍𝑀) → ∅ ∈ 𝑀)
53, 4ifcld 4575 . . . . . . 7 ((𝑀 ∈ WUni ∧ 𝑍𝑀) → if(𝑥 = 𝐵, 𝒫 𝑍, ∅) ∈ 𝑀)
61, 5ifcld 4575 . . . . . 6 ((𝑀 ∈ WUni ∧ 𝑍𝑀) → if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅)) ∈ 𝑀)
76adantr 482 . . . . 5 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅)) ∈ 𝑀)
87adantr 482 . . . 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 7114 . . 3 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝑆:ω⟶𝑀)
112adantr 482 . . . 4 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝑀 ∈ WUni)
12 omex 9638 . . . . 5 ω ∈ V
1312a1i 11 . . . 4 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → ω ∈ V)
1411, 13elmapd 8834 . . 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 4623 . . . . 5 (𝑍𝑀𝑍 ∈ 𝒫 𝑍)
1716adantl 483 . . . 4 ((𝑀 ∈ WUni ∧ 𝑍𝑀) → 𝑍 ∈ 𝒫 𝑍)
1817adantr 482 . . 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 4535 . . . . 5 (𝑥 = 𝐴 → if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅)) = 𝑍)
2120adantl 483 . . . 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 482 . . . 4 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝑍𝑀)
2419, 21, 22, 23fvmptd 7006 . . 3 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝑆𝐴) = 𝑍)
25 eqeq1 2737 . . . . . . 7 (𝑥 = 𝐵 → (𝑥 = 𝐴𝐵 = 𝐴))
26 eqeq1 2737 . . . . . . . 8 (𝑥 = 𝐵 → (𝑥 = 𝐵𝐵 = 𝐵))
2726ifbid 4552 . . . . . . 7 (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝒫 𝑍, ∅) = if(𝐵 = 𝐵, 𝒫 𝑍, ∅))
2825, 27ifbieq2d 4555 . . . . . 6 (𝑥 = 𝐵 → if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅)) = if(𝐵 = 𝐴, 𝑍, if(𝐵 = 𝐵, 𝒫 𝑍, ∅)))
29 necom 2995 . . . . . . . . 9 (𝐴𝐵𝐵𝐴)
30 ifnefalse 4541 . . . . . . . . 9 (𝐵𝐴 → if(𝐵 = 𝐴, 𝑍, if(𝐵 = 𝐵, 𝒫 𝑍, ∅)) = if(𝐵 = 𝐵, 𝒫 𝑍, ∅))
3129, 30sylbi 216 . . . . . . . 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 483 . . . . . 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 2795 . . . . 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 5377 . . . . . . . 8 (𝑍𝑀 → 𝒫 𝑍 ∈ V)
3736adantl 483 . . . . . . 7 ((𝑀 ∈ WUni ∧ 𝑍𝑀) → 𝒫 𝑍 ∈ V)
38 0ex 5308 . . . . . . . 8 ∅ ∈ V
3938a1i 11 . . . . . . 7 ((𝑀 ∈ WUni ∧ 𝑍𝑀) → ∅ ∈ V)
4037, 39ifcld 4575 . . . . . 6 ((𝑀 ∈ WUni ∧ 𝑍𝑀) → if(𝐵 = 𝐵, 𝒫 𝑍, ∅) ∈ V)
4140adantr 482 . . . . 5 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → if(𝐵 = 𝐵, 𝒫 𝑍, ∅) ∈ V)
4219, 34, 35, 41fvmptd 7006 . . . 4 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝑆𝐵) = if(𝐵 = 𝐵, 𝒫 𝑍, ∅))
43 eqid 2733 . . . . 5 𝐵 = 𝐵
4443iftruei 4536 . . . 4 if(𝐵 = 𝐵, 𝒫 𝑍, ∅) = 𝒫 𝑍
4542, 44eqtrdi 2789 . . 3 (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝑆𝐵) = 𝒫 𝑍)
4618, 24, 453eltr4d 2849 . 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 34410 . . 3 ((𝑀 ∈ WUni ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝑆𝐸 ↔ (𝑆 ∈ (𝑀m ω) ∧ (𝑆𝐴) ∈ (𝑆𝐵))))
502, 47, 49syl2an 597 . 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 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wne 2941  Vcvv 3475  c0 4323  ifcif 4529  𝒫 cpw 4603  cmpt 5232  wf 6540  cfv 6544  (class class class)co 7409  ωcom 7855  m cmap 8820  WUnicwun 10695  𝑔cgoe 34324   Sat csate 34329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-ac2 10458
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-2o 8467  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9934  df-ac 10111  df-wun 10697  df-goel 34331  df-gona 34332  df-goal 34333  df-sat 34334  df-sate 34335  df-fmla 34336
This theorem is referenced by:  ex-sategoel  34413
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