MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mptmpoopabbrdOLDOLD Structured version   Visualization version   GIF version

Theorem mptmpoopabbrdOLDOLD 8082
Description: Obsolete version of mptmpoopabbrd 8079 as of 13-Dec-2024. (Contributed by Alexander van Vekens, 8-Nov-2017.) (Revised by AV, 15-Jan-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
mptmpoopabbrdOLD.g (𝜑𝐺𝑊)
mptmpoopabbrdOLD.x (𝜑𝑋 ∈ (𝐴𝐺))
mptmpoopabbrdOLD.y (𝜑𝑌 ∈ (𝐵𝐺))
mptmpoopabbrdOLD.v (𝜑 → {⟨𝑓, ⟩ ∣ 𝜓} ∈ 𝑉)
mptmpoopabbrdOLD.r ((𝜑𝑓(𝐷𝐺)) → 𝜓)
mptmpoopabbrdOLD.1 ((𝑎 = 𝑋𝑏 = 𝑌) → (𝜏𝜃))
mptmpoopabbrdOLD.2 (𝑔 = 𝐺 → (𝜒𝜏))
mptmpoopabbrdOLD.m 𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴𝑔), 𝑏 ∈ (𝐵𝑔) ↦ {⟨𝑓, ⟩ ∣ (𝜒𝑓(𝐷𝑔))}))
Assertion
Ref Expression
mptmpoopabbrdOLDOLD (𝜑 → (𝑋(𝑀𝐺)𝑌) = {⟨𝑓, ⟩ ∣ (𝜃𝑓(𝐷𝐺))})
Distinct variable groups:   𝐴,𝑎,𝑏,𝑔   𝐵,𝑎,𝑏,𝑔   𝐷,𝑎,𝑏,𝑔   𝐺,𝑎,𝑏,𝑓,𝑔,   𝑔,𝑊   𝑋,𝑎,𝑏,𝑓,𝑔,   𝑌,𝑎,𝑏,𝑓,𝑔,   𝜑,𝑓,   𝜏,𝑔   𝜃,𝑎,𝑏
Allowed substitution hints:   𝜑(𝑔,𝑎,𝑏)   𝜓(𝑓,𝑔,,𝑎,𝑏)   𝜒(𝑓,𝑔,,𝑎,𝑏)   𝜃(𝑓,𝑔,)   𝜏(𝑓,,𝑎,𝑏)   𝐴(𝑓,)   𝐵(𝑓,)   𝐷(𝑓,)   𝑀(𝑓,𝑔,,𝑎,𝑏)   𝑉(𝑓,𝑔,,𝑎,𝑏)   𝑊(𝑓,,𝑎,𝑏)

Proof of Theorem mptmpoopabbrdOLDOLD
StepHypRef Expression
1 mptmpoopabbrdOLD.g . . . 4 (𝜑𝐺𝑊)
2 mptmpoopabbrdOLD.m . . . . 5 𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴𝑔), 𝑏 ∈ (𝐵𝑔) ↦ {⟨𝑓, ⟩ ∣ (𝜒𝑓(𝐷𝑔))}))
3 fveq2 6891 . . . . . 6 (𝑔 = 𝐺 → (𝐴𝑔) = (𝐴𝐺))
4 fveq2 6891 . . . . . 6 (𝑔 = 𝐺 → (𝐵𝑔) = (𝐵𝐺))
5 mptmpoopabbrdOLD.2 . . . . . . . 8 (𝑔 = 𝐺 → (𝜒𝜏))
6 fveq2 6891 . . . . . . . . 9 (𝑔 = 𝐺 → (𝐷𝑔) = (𝐷𝐺))
76breqd 5153 . . . . . . . 8 (𝑔 = 𝐺 → (𝑓(𝐷𝑔)𝑓(𝐷𝐺)))
85, 7anbi12d 630 . . . . . . 7 (𝑔 = 𝐺 → ((𝜒𝑓(𝐷𝑔)) ↔ (𝜏𝑓(𝐷𝐺))))
98opabbidv 5208 . . . . . 6 (𝑔 = 𝐺 → {⟨𝑓, ⟩ ∣ (𝜒𝑓(𝐷𝑔))} = {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))})
103, 4, 9mpoeq123dv 7489 . . . . 5 (𝑔 = 𝐺 → (𝑎 ∈ (𝐴𝑔), 𝑏 ∈ (𝐵𝑔) ↦ {⟨𝑓, ⟩ ∣ (𝜒𝑓(𝐷𝑔))}) = (𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))}))
11 elex 3488 . . . . . 6 (𝐺𝑊𝐺 ∈ V)
1211adantr 480 . . . . 5 ((𝐺𝑊𝐺𝑊) → 𝐺 ∈ V)
13 fvex 6904 . . . . . . 7 (𝐴𝐺) ∈ V
14 fvex 6904 . . . . . . 7 (𝐵𝐺) ∈ V
1513, 14pm3.2i 470 . . . . . 6 ((𝐴𝐺) ∈ V ∧ (𝐵𝐺) ∈ V)
16 mpoexga 8076 . . . . . 6 (((𝐴𝐺) ∈ V ∧ (𝐵𝐺) ∈ V) → (𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))}) ∈ V)
1715, 16mp1i 13 . . . . 5 ((𝐺𝑊𝐺𝑊) → (𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))}) ∈ V)
182, 10, 12, 17fvmptd3 7022 . . . 4 ((𝐺𝑊𝐺𝑊) → (𝑀𝐺) = (𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))}))
191, 1, 18syl2anc 583 . . 3 (𝜑 → (𝑀𝐺) = (𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))}))
2019oveqd 7431 . 2 (𝜑 → (𝑋(𝑀𝐺)𝑌) = (𝑋(𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))})𝑌))
21 mptmpoopabbrdOLD.x . . 3 (𝜑𝑋 ∈ (𝐴𝐺))
22 mptmpoopabbrdOLD.y . . 3 (𝜑𝑌 ∈ (𝐵𝐺))
23 ancom 460 . . . . 5 ((𝜃𝑓(𝐷𝐺)) ↔ (𝑓(𝐷𝐺)𝜃))
2423opabbii 5209 . . . 4 {⟨𝑓, ⟩ ∣ (𝜃𝑓(𝐷𝐺))} = {⟨𝑓, ⟩ ∣ (𝑓(𝐷𝐺)𝜃)}
25 mptmpoopabbrdOLD.r . . . . 5 ((𝜑𝑓(𝐷𝐺)) → 𝜓)
26 mptmpoopabbrdOLD.v . . . . 5 (𝜑 → {⟨𝑓, ⟩ ∣ 𝜓} ∈ 𝑉)
2725, 26opabresex2d 7467 . . . 4 (𝜑 → {⟨𝑓, ⟩ ∣ (𝑓(𝐷𝐺)𝜃)} ∈ V)
2824, 27eqeltrid 2832 . . 3 (𝜑 → {⟨𝑓, ⟩ ∣ (𝜃𝑓(𝐷𝐺))} ∈ V)
29 mptmpoopabbrdOLD.1 . . . . . 6 ((𝑎 = 𝑋𝑏 = 𝑌) → (𝜏𝜃))
3029anbi1d 629 . . . . 5 ((𝑎 = 𝑋𝑏 = 𝑌) → ((𝜏𝑓(𝐷𝐺)) ↔ (𝜃𝑓(𝐷𝐺))))
3130opabbidv 5208 . . . 4 ((𝑎 = 𝑋𝑏 = 𝑌) → {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))} = {⟨𝑓, ⟩ ∣ (𝜃𝑓(𝐷𝐺))})
32 eqid 2727 . . . 4 (𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))}) = (𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))})
3331, 32ovmpoga 7569 . . 3 ((𝑋 ∈ (𝐴𝐺) ∧ 𝑌 ∈ (𝐵𝐺) ∧ {⟨𝑓, ⟩ ∣ (𝜃𝑓(𝐷𝐺))} ∈ V) → (𝑋(𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))})𝑌) = {⟨𝑓, ⟩ ∣ (𝜃𝑓(𝐷𝐺))})
3421, 22, 28, 33syl3anc 1369 . 2 (𝜑 → (𝑋(𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))})𝑌) = {⟨𝑓, ⟩ ∣ (𝜃𝑓(𝐷𝐺))})
3520, 34eqtrd 2767 1 (𝜑 → (𝑋(𝑀𝐺)𝑌) = {⟨𝑓, ⟩ ∣ (𝜃𝑓(𝐷𝐺))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  wcel 2099  Vcvv 3469   class class class wbr 5142  {copab 5204  cmpt 5225  cfv 6542  (class class class)co 7414  cmpo 7416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988
This theorem is referenced by:  mptmpoopabovdOLD  8083
  Copyright terms: Public domain W3C validator