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Theorem mptmpoopabbrd 8103
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 5284. (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 6906 . . . . 5 (𝑔 = 𝐺 → (𝐴𝑔) = (𝐴𝐺))
3 fveq2 6906 . . . . 5 (𝑔 = 𝐺 → (𝐵𝑔) = (𝐵𝐺))
4 mptmpoopabbrd.2 . . . . . . 7 (𝑔 = 𝐺 → (𝜒𝜏))
5 fveq2 6906 . . . . . . . 8 (𝑔 = 𝐺 → (𝐷𝑔) = (𝐷𝐺))
65breqd 5158 . . . . . . 7 (𝑔 = 𝐺 → (𝑓(𝐷𝑔)𝑓(𝐷𝐺)))
74, 6anbi12d 632 . . . . . 6 (𝑔 = 𝐺 → ((𝜒𝑓(𝐷𝑔)) ↔ (𝜏𝑓(𝐷𝐺))))
87opabbidv 5213 . . . . 5 (𝑔 = 𝐺 → {⟨𝑓, ⟩ ∣ (𝜒𝑓(𝐷𝑔))} = {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))})
92, 3, 8mpoeq123dv 7507 . . . 4 (𝑔 = 𝐺 → (𝑎 ∈ (𝐴𝑔), 𝑏 ∈ (𝐵𝑔) ↦ {⟨𝑓, ⟩ ∣ (𝜒𝑓(𝐷𝑔))}) = (𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))}))
10 mptmpoopabbrd.g . . . . 5 (𝜑𝐺𝑊)
1110elexd 3501 . . . 4 (𝜑𝐺 ∈ V)
12 fvex 6919 . . . . . 6 (𝐴𝐺) ∈ V
13 fvex 6919 . . . . . 6 (𝐵𝐺) ∈ V
14 fvex 6919 . . . . . . 7 (𝐷𝐺) ∈ V
1514pwex 5385 . . . . . 6 𝒫 (𝐷𝐺) ∈ V
16 simpr 484 . . . . . . . . . 10 ((𝜏𝑓(𝐷𝐺)) → 𝑓(𝐷𝐺))
1716ssopab2i 5559 . . . . . . . . 9 {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))} ⊆ {⟨𝑓, ⟩ ∣ 𝑓(𝐷𝐺)}
18 opabss 5211 . . . . . . . . 9 {⟨𝑓, ⟩ ∣ 𝑓(𝐷𝐺)} ⊆ (𝐷𝐺)
1917, 18sstri 4004 . . . . . . . 8 {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))} ⊆ (𝐷𝐺)
2014, 19elpwi2 5340 . . . . . . 7 {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))} ∈ 𝒫 (𝐷𝐺)
2120rgen2w 3063 . . . . . 6 𝑎 ∈ (𝐴𝐺)∀𝑏 ∈ (𝐵𝐺){⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))} ∈ 𝒫 (𝐷𝐺)
2212, 13, 15, 21mpoexw 8101 . . . . 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 7038 . . 3 (𝜑 → (𝑀𝐺) = (𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))}))
2524oveqd 7447 . 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 631 . . . . 5 ((𝑎 = 𝑋𝑏 = 𝑌) → ((𝜏𝑓(𝐷𝐺)) ↔ (𝜃𝑓(𝐷𝐺))))
3029opabbidv 5213 . . . 4 ((𝑎 = 𝑋𝑏 = 𝑌) → {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))} = {⟨𝑓, ⟩ ∣ (𝜃𝑓(𝐷𝐺))})
31 eqid 2734 . . . 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 5214 . . . . 5 {⟨𝑓, ⟩ ∣ (𝜃𝑓(𝐷𝐺))} = {⟨𝑓, ⟩ ∣ (𝑓(𝐷𝐺)𝜃)}
34 opabresex2 7484 . . . . 5 {⟨𝑓, ⟩ ∣ (𝑓(𝐷𝐺)𝜃)} ∈ V
3533, 34eqeltri 2834 . . . 4 {⟨𝑓, ⟩ ∣ (𝜃𝑓(𝐷𝐺))} ∈ V
3630, 31, 35ovmpoa 7587 . . 3 ((𝑋 ∈ (𝐴𝐺) ∧ 𝑌 ∈ (𝐵𝐺)) → (𝑋(𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))})𝑌) = {⟨𝑓, ⟩ ∣ (𝜃𝑓(𝐷𝐺))})
3726, 27, 36syl2anc 584 . 2 (𝜑 → (𝑋(𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))})𝑌) = {⟨𝑓, ⟩ ∣ (𝜃𝑓(𝐷𝐺))})
3825, 37eqtrd 2774 1 (𝜑 → (𝑋(𝑀𝐺)𝑌) = {⟨𝑓, ⟩ ∣ (𝜃𝑓(𝐷𝐺))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  Vcvv 3477  𝒫 cpw 4604   class class class wbr 5147  {copab 5209  cmpt 5230  cfv 6562  (class class class)co 7430  cmpo 7432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013
This theorem is referenced by:  mptmpoopabovd  8105  wlkson  29688
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