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Theorem mptmpoopabbrd 8005
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. (Revised by SN, 13-Dec-2024.)
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 6839 . . . . 5 (𝑔 = 𝐺 → (𝐴𝑔) = (𝐴𝐺))
3 fveq2 6839 . . . . 5 (𝑔 = 𝐺 → (𝐵𝑔) = (𝐵𝐺))
4 mptmpoopabbrd.2 . . . . . . 7 (𝑔 = 𝐺 → (𝜒𝜏))
5 fveq2 6839 . . . . . . . 8 (𝑔 = 𝐺 → (𝐷𝑔) = (𝐷𝐺))
65breqd 5114 . . . . . . 7 (𝑔 = 𝐺 → (𝑓(𝐷𝑔)𝑓(𝐷𝐺)))
74, 6anbi12d 631 . . . . . 6 (𝑔 = 𝐺 → ((𝜒𝑓(𝐷𝑔)) ↔ (𝜏𝑓(𝐷𝐺))))
87opabbidv 5169 . . . . 5 (𝑔 = 𝐺 → {⟨𝑓, ⟩ ∣ (𝜒𝑓(𝐷𝑔))} = {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))})
92, 3, 8mpoeq123dv 7426 . . . 4 (𝑔 = 𝐺 → (𝑎 ∈ (𝐴𝑔), 𝑏 ∈ (𝐵𝑔) ↦ {⟨𝑓, ⟩ ∣ (𝜒𝑓(𝐷𝑔))}) = (𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))}))
10 mptmpoopabbrd.g . . . . 5 (𝜑𝐺𝑊)
1110elexd 3463 . . . 4 (𝜑𝐺 ∈ V)
12 fvex 6852 . . . . . 6 (𝐴𝐺) ∈ V
13 fvex 6852 . . . . . 6 (𝐵𝐺) ∈ V
1412, 13mpoex 8004 . . . . 5 (𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))}) ∈ V
1514a1i 11 . . . 4 (𝜑 → (𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))}) ∈ V)
161, 9, 11, 15fvmptd3 6968 . . 3 (𝜑 → (𝑀𝐺) = (𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))}))
1716oveqd 7368 . 2 (𝜑 → (𝑋(𝑀𝐺)𝑌) = (𝑋(𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))})𝑌))
18 mptmpoopabbrd.x . . 3 (𝜑𝑋 ∈ (𝐴𝐺))
19 mptmpoopabbrd.y . . 3 (𝜑𝑌 ∈ (𝐵𝐺))
20 mptmpoopabbrd.1 . . . . . 6 ((𝑎 = 𝑋𝑏 = 𝑌) → (𝜏𝜃))
2120anbi1d 630 . . . . 5 ((𝑎 = 𝑋𝑏 = 𝑌) → ((𝜏𝑓(𝐷𝐺)) ↔ (𝜃𝑓(𝐷𝐺))))
2221opabbidv 5169 . . . 4 ((𝑎 = 𝑋𝑏 = 𝑌) → {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))} = {⟨𝑓, ⟩ ∣ (𝜃𝑓(𝐷𝐺))})
23 eqid 2737 . . . 4 (𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))}) = (𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))})
24 ancom 461 . . . . . 6 ((𝜃𝑓(𝐷𝐺)) ↔ (𝑓(𝐷𝐺)𝜃))
2524opabbii 5170 . . . . 5 {⟨𝑓, ⟩ ∣ (𝜃𝑓(𝐷𝐺))} = {⟨𝑓, ⟩ ∣ (𝑓(𝐷𝐺)𝜃)}
26 opabresex2 7403 . . . . 5 {⟨𝑓, ⟩ ∣ (𝑓(𝐷𝐺)𝜃)} ∈ V
2725, 26eqeltri 2834 . . . 4 {⟨𝑓, ⟩ ∣ (𝜃𝑓(𝐷𝐺))} ∈ V
2822, 23, 27ovmpoa 7504 . . 3 ((𝑋 ∈ (𝐴𝐺) ∧ 𝑌 ∈ (𝐵𝐺)) → (𝑋(𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))})𝑌) = {⟨𝑓, ⟩ ∣ (𝜃𝑓(𝐷𝐺))})
2918, 19, 28syl2anc 584 . 2 (𝜑 → (𝑋(𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))})𝑌) = {⟨𝑓, ⟩ ∣ (𝜃𝑓(𝐷𝐺))})
3017, 29eqtrd 2777 1 (𝜑 → (𝑋(𝑀𝐺)𝑌) = {⟨𝑓, ⟩ ∣ (𝜃𝑓(𝐷𝐺))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  Vcvv 3443   class class class wbr 5103  {copab 5165  cmpt 5186  cfv 6493  (class class class)co 7351  cmpo 7353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-1st 7913  df-2nd 7914
This theorem is referenced by:  mptmpoopabovd  8006  wlkson  28433
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