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Theorem ressval3d 17132
Description: Value of structure restriction, deduction version. (Contributed by AV, 14-Mar-2020.) (Revised by AV, 3-Jul-2022.) (Proof shortened by AV, 17-Oct-2024.)
Hypotheses
Ref Expression
ressval3d.r 𝑅 = (𝑆 β†Ύs 𝐴)
ressval3d.b 𝐡 = (Baseβ€˜π‘†)
ressval3d.e 𝐸 = (Baseβ€˜ndx)
ressval3d.s (πœ‘ β†’ 𝑆 ∈ 𝑉)
ressval3d.f (πœ‘ β†’ Fun 𝑆)
ressval3d.d (πœ‘ β†’ 𝐸 ∈ dom 𝑆)
ressval3d.u (πœ‘ β†’ 𝐴 βŠ† 𝐡)
Assertion
Ref Expression
ressval3d (πœ‘ β†’ 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))

Proof of Theorem ressval3d
StepHypRef Expression
1 ressval3d.u . 2 (πœ‘ β†’ 𝐴 βŠ† 𝐡)
2 sspss 4060 . . . 4 (𝐴 βŠ† 𝐡 ↔ (𝐴 ⊊ 𝐡 ∨ 𝐴 = 𝐡))
3 dfpss3 4047 . . . . 5 (𝐴 ⊊ 𝐡 ↔ (𝐴 βŠ† 𝐡 ∧ Β¬ 𝐡 βŠ† 𝐴))
43orbi1i 913 . . . 4 ((𝐴 ⊊ 𝐡 ∨ 𝐴 = 𝐡) ↔ ((𝐴 βŠ† 𝐡 ∧ Β¬ 𝐡 βŠ† 𝐴) ∨ 𝐴 = 𝐡))
52, 4bitri 275 . . 3 (𝐴 βŠ† 𝐡 ↔ ((𝐴 βŠ† 𝐡 ∧ Β¬ 𝐡 βŠ† 𝐴) ∨ 𝐴 = 𝐡))
6 simplr 768 . . . . . . 7 (((𝐴 βŠ† 𝐡 ∧ Β¬ 𝐡 βŠ† 𝐴) ∧ πœ‘) β†’ Β¬ 𝐡 βŠ† 𝐴)
7 ressval3d.s . . . . . . . 8 (πœ‘ β†’ 𝑆 ∈ 𝑉)
87adantl 483 . . . . . . 7 (((𝐴 βŠ† 𝐡 ∧ Β¬ 𝐡 βŠ† 𝐴) ∧ πœ‘) β†’ 𝑆 ∈ 𝑉)
9 simpl 484 . . . . . . . 8 ((𝐴 βŠ† 𝐡 ∧ Β¬ 𝐡 βŠ† 𝐴) β†’ 𝐴 βŠ† 𝐡)
10 ressval3d.b . . . . . . . . . 10 𝐡 = (Baseβ€˜π‘†)
1110fvexi 6857 . . . . . . . . 9 𝐡 ∈ V
1211a1i 11 . . . . . . . 8 (πœ‘ β†’ 𝐡 ∈ V)
13 ssexg 5281 . . . . . . . 8 ((𝐴 βŠ† 𝐡 ∧ 𝐡 ∈ V) β†’ 𝐴 ∈ V)
149, 12, 13syl2an 597 . . . . . . 7 (((𝐴 βŠ† 𝐡 ∧ Β¬ 𝐡 βŠ† 𝐴) ∧ πœ‘) β†’ 𝐴 ∈ V)
15 ressval3d.r . . . . . . . 8 𝑅 = (𝑆 β†Ύs 𝐴)
1615, 10ressval2 17122 . . . . . . 7 ((Β¬ 𝐡 βŠ† 𝐴 ∧ 𝑆 ∈ 𝑉 ∧ 𝐴 ∈ V) β†’ 𝑅 = (𝑆 sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ 𝐡)⟩))
176, 8, 14, 16syl3anc 1372 . . . . . 6 (((𝐴 βŠ† 𝐡 ∧ Β¬ 𝐡 βŠ† 𝐴) ∧ πœ‘) β†’ 𝑅 = (𝑆 sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ 𝐡)⟩))
18 ressval3d.e . . . . . . . . . 10 𝐸 = (Baseβ€˜ndx)
1918a1i 11 . . . . . . . . 9 (((𝐴 βŠ† 𝐡 ∧ Β¬ 𝐡 βŠ† 𝐴) ∧ πœ‘) β†’ 𝐸 = (Baseβ€˜ndx))
20 df-ss 3928 . . . . . . . . . . . . 13 (𝐴 βŠ† 𝐡 ↔ (𝐴 ∩ 𝐡) = 𝐴)
2120biimpi 215 . . . . . . . . . . . 12 (𝐴 βŠ† 𝐡 β†’ (𝐴 ∩ 𝐡) = 𝐴)
2221eqcomd 2739 . . . . . . . . . . 11 (𝐴 βŠ† 𝐡 β†’ 𝐴 = (𝐴 ∩ 𝐡))
2322adantr 482 . . . . . . . . . 10 ((𝐴 βŠ† 𝐡 ∧ Β¬ 𝐡 βŠ† 𝐴) β†’ 𝐴 = (𝐴 ∩ 𝐡))
2423adantr 482 . . . . . . . . 9 (((𝐴 βŠ† 𝐡 ∧ Β¬ 𝐡 βŠ† 𝐴) ∧ πœ‘) β†’ 𝐴 = (𝐴 ∩ 𝐡))
2519, 24opeq12d 4839 . . . . . . . 8 (((𝐴 βŠ† 𝐡 ∧ Β¬ 𝐡 βŠ† 𝐴) ∧ πœ‘) β†’ ⟨𝐸, 𝐴⟩ = ⟨(Baseβ€˜ndx), (𝐴 ∩ 𝐡)⟩)
2625eqcomd 2739 . . . . . . 7 (((𝐴 βŠ† 𝐡 ∧ Β¬ 𝐡 βŠ† 𝐴) ∧ πœ‘) β†’ ⟨(Baseβ€˜ndx), (𝐴 ∩ 𝐡)⟩ = ⟨𝐸, 𝐴⟩)
2726oveq2d 7374 . . . . . 6 (((𝐴 βŠ† 𝐡 ∧ Β¬ 𝐡 βŠ† 𝐴) ∧ πœ‘) β†’ (𝑆 sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ 𝐡)⟩) = (𝑆 sSet ⟨𝐸, 𝐴⟩))
2817, 27eqtrd 2773 . . . . 5 (((𝐴 βŠ† 𝐡 ∧ Β¬ 𝐡 βŠ† 𝐴) ∧ πœ‘) β†’ 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
2928ex 414 . . . 4 ((𝐴 βŠ† 𝐡 ∧ Β¬ 𝐡 βŠ† 𝐴) β†’ (πœ‘ β†’ 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩)))
3015a1i 11 . . . . . . 7 ((𝐴 = 𝐡 ∧ πœ‘) β†’ 𝑅 = (𝑆 β†Ύs 𝐴))
31 oveq2 7366 . . . . . . . 8 (𝐴 = 𝐡 β†’ (𝑆 β†Ύs 𝐴) = (𝑆 β†Ύs 𝐡))
3231adantr 482 . . . . . . 7 ((𝐴 = 𝐡 ∧ πœ‘) β†’ (𝑆 β†Ύs 𝐴) = (𝑆 β†Ύs 𝐡))
337adantl 483 . . . . . . . 8 ((𝐴 = 𝐡 ∧ πœ‘) β†’ 𝑆 ∈ 𝑉)
3410ressid 17130 . . . . . . . 8 (𝑆 ∈ 𝑉 β†’ (𝑆 β†Ύs 𝐡) = 𝑆)
3533, 34syl 17 . . . . . . 7 ((𝐴 = 𝐡 ∧ πœ‘) β†’ (𝑆 β†Ύs 𝐡) = 𝑆)
3630, 32, 353eqtrd 2777 . . . . . 6 ((𝐴 = 𝐡 ∧ πœ‘) β†’ 𝑅 = 𝑆)
37 baseid 17091 . . . . . . . 8 Base = Slot (Baseβ€˜ndx)
38 ressval3d.f . . . . . . . 8 (πœ‘ β†’ Fun 𝑆)
39 ressval3d.d . . . . . . . . 9 (πœ‘ β†’ 𝐸 ∈ dom 𝑆)
4018, 39eqeltrrid 2839 . . . . . . . 8 (πœ‘ β†’ (Baseβ€˜ndx) ∈ dom 𝑆)
4137, 7, 38, 40setsidvald 17076 . . . . . . 7 (πœ‘ β†’ 𝑆 = (𝑆 sSet ⟨(Baseβ€˜ndx), (Baseβ€˜π‘†)⟩))
4241adantl 483 . . . . . 6 ((𝐴 = 𝐡 ∧ πœ‘) β†’ 𝑆 = (𝑆 sSet ⟨(Baseβ€˜ndx), (Baseβ€˜π‘†)⟩))
4318a1i 11 . . . . . . . . 9 ((𝐴 = 𝐡 ∧ πœ‘) β†’ 𝐸 = (Baseβ€˜ndx))
44 simpl 484 . . . . . . . . . 10 ((𝐴 = 𝐡 ∧ πœ‘) β†’ 𝐴 = 𝐡)
4544, 10eqtrdi 2789 . . . . . . . . 9 ((𝐴 = 𝐡 ∧ πœ‘) β†’ 𝐴 = (Baseβ€˜π‘†))
4643, 45opeq12d 4839 . . . . . . . 8 ((𝐴 = 𝐡 ∧ πœ‘) β†’ ⟨𝐸, 𝐴⟩ = ⟨(Baseβ€˜ndx), (Baseβ€˜π‘†)⟩)
4746eqcomd 2739 . . . . . . 7 ((𝐴 = 𝐡 ∧ πœ‘) β†’ ⟨(Baseβ€˜ndx), (Baseβ€˜π‘†)⟩ = ⟨𝐸, 𝐴⟩)
4847oveq2d 7374 . . . . . 6 ((𝐴 = 𝐡 ∧ πœ‘) β†’ (𝑆 sSet ⟨(Baseβ€˜ndx), (Baseβ€˜π‘†)⟩) = (𝑆 sSet ⟨𝐸, 𝐴⟩))
4936, 42, 483eqtrd 2777 . . . . 5 ((𝐴 = 𝐡 ∧ πœ‘) β†’ 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
5049ex 414 . . . 4 (𝐴 = 𝐡 β†’ (πœ‘ β†’ 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩)))
5129, 50jaoi 856 . . 3 (((𝐴 βŠ† 𝐡 ∧ Β¬ 𝐡 βŠ† 𝐴) ∨ 𝐴 = 𝐡) β†’ (πœ‘ β†’ 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩)))
525, 51sylbi 216 . 2 (𝐴 βŠ† 𝐡 β†’ (πœ‘ β†’ 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩)))
531, 52mpcom 38 1 (πœ‘ β†’ 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107  Vcvv 3444   ∩ cin 3910   βŠ† wss 3911   ⊊ wpss 3912  βŸ¨cop 4593  dom cdm 5634  Fun wfun 6491  β€˜cfv 6497  (class class class)co 7358   sSet csts 17040  ndxcnx 17070  Basecbs 17088   β†Ύs cress 17117
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-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-1cn 11114  ax-addcl 11116
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 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-nn 12159  df-sets 17041  df-slot 17059  df-ndx 17071  df-base 17089  df-ress 17118
This theorem is referenced by:  estrres  18032
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