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

Theorem mptmpoopabbrd 8079
Description: The operation value of a function value of a collection of ordered pairs of elements related in two ways. (Contributed by Alexander van Vekens, 8-Nov-2017.) (Revised by AV, 15-Jan-2021.) Add disjoint variable condition on 𝐷, 𝑓, to remove hypotheses; avoid ax-rep 5279. (Revised by SN, 7-Apr-2025.)
Hypotheses
Ref Expression
mptmpoopabbrd.g (𝜑𝐺𝑊)
mptmpoopabbrd.x (𝜑𝑋 ∈ (𝐴𝐺))
mptmpoopabbrd.y (𝜑𝑌 ∈ (𝐵𝐺))
mptmpoopabbrd.1 ((𝑎 = 𝑋𝑏 = 𝑌) → (𝜏𝜃))
mptmpoopabbrd.2 (𝑔 = 𝐺 → (𝜒𝜏))
mptmpoopabbrd.m 𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴𝑔), 𝑏 ∈ (𝐵𝑔) ↦ {⟨𝑓, ⟩ ∣ (𝜒𝑓(𝐷𝑔))}))
Assertion
Ref Expression
mptmpoopabbrd (𝜑 → (𝑋(𝑀𝐺)𝑌) = {⟨𝑓, ⟩ ∣ (𝜃𝑓(𝐷𝐺))})
Distinct variable groups:   𝐴,𝑎,𝑏,𝑔   𝐵,𝑎,𝑏,𝑔   𝐷,𝑎,𝑏,𝑓,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)   𝜒(𝑓,𝑔,,𝑎,𝑏)   𝜃(𝑓,𝑔,)   𝜏(𝑓,,𝑎,𝑏)   𝐴(𝑓,)   𝐵(𝑓,)   𝑀(𝑓,𝑔,,𝑎,𝑏)   𝑊(𝑓,,𝑎,𝑏)

Proof of Theorem mptmpoopabbrd
StepHypRef Expression
1 mptmpoopabbrd.m . . . 4 𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴𝑔), 𝑏 ∈ (𝐵𝑔) ↦ {⟨𝑓, ⟩ ∣ (𝜒𝑓(𝐷𝑔))}))
2 fveq2 6891 . . . . 5 (𝑔 = 𝐺 → (𝐴𝑔) = (𝐴𝐺))
3 fveq2 6891 . . . . 5 (𝑔 = 𝐺 → (𝐵𝑔) = (𝐵𝐺))
4 mptmpoopabbrd.2 . . . . . . 7 (𝑔 = 𝐺 → (𝜒𝜏))
5 fveq2 6891 . . . . . . . 8 (𝑔 = 𝐺 → (𝐷𝑔) = (𝐷𝐺))
65breqd 5153 . . . . . . 7 (𝑔 = 𝐺 → (𝑓(𝐷𝑔)𝑓(𝐷𝐺)))
74, 6anbi12d 630 . . . . . 6 (𝑔 = 𝐺 → ((𝜒𝑓(𝐷𝑔)) ↔ (𝜏𝑓(𝐷𝐺))))
87opabbidv 5208 . . . . 5 (𝑔 = 𝐺 → {⟨𝑓, ⟩ ∣ (𝜒𝑓(𝐷𝑔))} = {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))})
92, 3, 8mpoeq123dv 7489 . . . 4 (𝑔 = 𝐺 → (𝑎 ∈ (𝐴𝑔), 𝑏 ∈ (𝐵𝑔) ↦ {⟨𝑓, ⟩ ∣ (𝜒𝑓(𝐷𝑔))}) = (𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))}))
10 mptmpoopabbrd.g . . . . 5 (𝜑𝐺𝑊)
1110elexd 3490 . . . 4 (𝜑𝐺 ∈ V)
12 fvex 6904 . . . . . 6 (𝐴𝐺) ∈ V
13 fvex 6904 . . . . . 6 (𝐵𝐺) ∈ V
14 fvex 6904 . . . . . . 7 (𝐷𝐺) ∈ V
1514pwex 5374 . . . . . 6 𝒫 (𝐷𝐺) ∈ V
16 simpr 484 . . . . . . . . . 10 ((𝜏𝑓(𝐷𝐺)) → 𝑓(𝐷𝐺))
1716ssopab2i 5546 . . . . . . . . 9 {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))} ⊆ {⟨𝑓, ⟩ ∣ 𝑓(𝐷𝐺)}
18 opabss 5206 . . . . . . . . 9 {⟨𝑓, ⟩ ∣ 𝑓(𝐷𝐺)} ⊆ (𝐷𝐺)
1917, 18sstri 3987 . . . . . . . 8 {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))} ⊆ (𝐷𝐺)
2014, 19elpwi2 5342 . . . . . . 7 {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))} ∈ 𝒫 (𝐷𝐺)
2120rgen2w 3061 . . . . . 6 𝑎 ∈ (𝐴𝐺)∀𝑏 ∈ (𝐵𝐺){⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))} ∈ 𝒫 (𝐷𝐺)
2212, 13, 15, 21mpoexw 8077 . . . . 5 (𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))}) ∈ V
2322a1i 11 . . . 4 (𝜑 → (𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))}) ∈ V)
241, 9, 11, 23fvmptd3 7022 . . 3 (𝜑 → (𝑀𝐺) = (𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))}))
2524oveqd 7431 . 2 (𝜑 → (𝑋(𝑀𝐺)𝑌) = (𝑋(𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))})𝑌))
26 mptmpoopabbrd.x . . 3 (𝜑𝑋 ∈ (𝐴𝐺))
27 mptmpoopabbrd.y . . 3 (𝜑𝑌 ∈ (𝐵𝐺))
28 mptmpoopabbrd.1 . . . . . 6 ((𝑎 = 𝑋𝑏 = 𝑌) → (𝜏𝜃))
2928anbi1d 629 . . . . 5 ((𝑎 = 𝑋𝑏 = 𝑌) → ((𝜏𝑓(𝐷𝐺)) ↔ (𝜃𝑓(𝐷𝐺))))
3029opabbidv 5208 . . . 4 ((𝑎 = 𝑋𝑏 = 𝑌) → {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))} = {⟨𝑓, ⟩ ∣ (𝜃𝑓(𝐷𝐺))})
31 eqid 2727 . . . 4 (𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))}) = (𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))})
32 ancom 460 . . . . . 6 ((𝜃𝑓(𝐷𝐺)) ↔ (𝑓(𝐷𝐺)𝜃))
3332opabbii 5209 . . . . 5 {⟨𝑓, ⟩ ∣ (𝜃𝑓(𝐷𝐺))} = {⟨𝑓, ⟩ ∣ (𝑓(𝐷𝐺)𝜃)}
34 opabresex2 7466 . . . . 5 {⟨𝑓, ⟩ ∣ (𝑓(𝐷𝐺)𝜃)} ∈ V
3533, 34eqeltri 2824 . . . 4 {⟨𝑓, ⟩ ∣ (𝜃𝑓(𝐷𝐺))} ∈ V
3630, 31, 35ovmpoa 7570 . . 3 ((𝑋 ∈ (𝐴𝐺) ∧ 𝑌 ∈ (𝐵𝐺)) → (𝑋(𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))})𝑌) = {⟨𝑓, ⟩ ∣ (𝜃𝑓(𝐷𝐺))})
3726, 27, 36syl2anc 583 . 2 (𝜑 → (𝑋(𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))})𝑌) = {⟨𝑓, ⟩ ∣ (𝜃𝑓(𝐷𝐺))})
3825, 37eqtrd 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  𝒫 cpw 4598   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-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-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-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988
This theorem is referenced by:  mptmpoopabovd  8081  wlkson  29457
  Copyright terms: Public domain W3C validator