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Theorem mptmpt2opabbrd 7475
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.)
Hypotheses
Ref Expression
mptmpt2opabbrd.g (𝜑𝐺𝑊)
mptmpt2opabbrd.x (𝜑𝑋 ∈ (𝐴𝐺))
mptmpt2opabbrd.y (𝜑𝑌 ∈ (𝐵𝐺))
mptmpt2opabbrd.v (𝜑 → {⟨𝑓, ⟩ ∣ 𝜓} ∈ 𝑉)
mptmpt2opabbrd.r ((𝜑𝑓(𝐷𝐺)) → 𝜓)
mptmpt2opabbrd.1 ((𝑎 = 𝑋𝑏 = 𝑌) → (𝜏𝜃))
mptmpt2opabbrd.2 (𝑔 = 𝐺 → (𝜒𝜏))
mptmpt2opabbrd.m 𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴𝑔), 𝑏 ∈ (𝐵𝑔) ↦ {⟨𝑓, ⟩ ∣ (𝜒𝑓(𝐷𝑔))}))
Assertion
Ref Expression
mptmpt2opabbrd (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 mptmpt2opabbrd
StepHypRef Expression
1 mptmpt2opabbrd.g . . . 4 (𝜑𝐺𝑊)
2 mptmpt2opabbrd.m . . . . . 6 𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴𝑔), 𝑏 ∈ (𝐵𝑔) ↦ {⟨𝑓, ⟩ ∣ (𝜒𝑓(𝐷𝑔))}))
32a1i 11 . . . . 5 ((𝐺𝑊𝐺𝑊) → 𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴𝑔), 𝑏 ∈ (𝐵𝑔) ↦ {⟨𝑓, ⟩ ∣ (𝜒𝑓(𝐷𝑔))})))
4 fveq2 6402 . . . . . . 7 (𝑔 = 𝐺 → (𝐴𝑔) = (𝐴𝐺))
5 fveq2 6402 . . . . . . 7 (𝑔 = 𝐺 → (𝐵𝑔) = (𝐵𝐺))
6 mptmpt2opabbrd.2 . . . . . . . . 9 (𝑔 = 𝐺 → (𝜒𝜏))
7 fveq2 6402 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝐷𝑔) = (𝐷𝐺))
87breqd 4848 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑓(𝐷𝑔)𝑓(𝐷𝐺)))
96, 8anbi12d 618 . . . . . . . 8 (𝑔 = 𝐺 → ((𝜒𝑓(𝐷𝑔)) ↔ (𝜏𝑓(𝐷𝐺))))
109opabbidv 4903 . . . . . . 7 (𝑔 = 𝐺 → {⟨𝑓, ⟩ ∣ (𝜒𝑓(𝐷𝑔))} = {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))})
114, 5, 10mpt2eq123dv 6941 . . . . . 6 (𝑔 = 𝐺 → (𝑎 ∈ (𝐴𝑔), 𝑏 ∈ (𝐵𝑔) ↦ {⟨𝑓, ⟩ ∣ (𝜒𝑓(𝐷𝑔))}) = (𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))}))
1211adantl 469 . . . . 5 (((𝐺𝑊𝐺𝑊) ∧ 𝑔 = 𝐺) → (𝑎 ∈ (𝐴𝑔), 𝑏 ∈ (𝐵𝑔) ↦ {⟨𝑓, ⟩ ∣ (𝜒𝑓(𝐷𝑔))}) = (𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))}))
13 elex 3402 . . . . . 6 (𝐺𝑊𝐺 ∈ V)
1413adantr 468 . . . . 5 ((𝐺𝑊𝐺𝑊) → 𝐺 ∈ V)
15 fvex 6415 . . . . . . 7 (𝐴𝐺) ∈ V
16 fvex 6415 . . . . . . 7 (𝐵𝐺) ∈ V
1715, 16pm3.2i 458 . . . . . 6 ((𝐴𝐺) ∈ V ∧ (𝐵𝐺) ∈ V)
18 mpt2exga 7473 . . . . . 6 (((𝐴𝐺) ∈ V ∧ (𝐵𝐺) ∈ V) → (𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))}) ∈ V)
1917, 18mp1i 13 . . . . 5 ((𝐺𝑊𝐺𝑊) → (𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))}) ∈ V)
203, 12, 14, 19fvmptd 6503 . . . 4 ((𝐺𝑊𝐺𝑊) → (𝑀𝐺) = (𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))}))
211, 1, 20syl2anc 575 . . 3 (𝜑 → (𝑀𝐺) = (𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))}))
2221oveqd 6885 . 2 (𝜑 → (𝑋(𝑀𝐺)𝑌) = (𝑋(𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))})𝑌))
23 mptmpt2opabbrd.x . . 3 (𝜑𝑋 ∈ (𝐴𝐺))
24 mptmpt2opabbrd.y . . 3 (𝜑𝑌 ∈ (𝐵𝐺))
25 ancom 450 . . . . 5 ((𝜃𝑓(𝐷𝐺)) ↔ (𝑓(𝐷𝐺)𝜃))
2625opabbii 4904 . . . 4 {⟨𝑓, ⟩ ∣ (𝜃𝑓(𝐷𝐺))} = {⟨𝑓, ⟩ ∣ (𝑓(𝐷𝐺)𝜃)}
27 mptmpt2opabbrd.r . . . . 5 ((𝜑𝑓(𝐷𝐺)) → 𝜓)
28 mptmpt2opabbrd.v . . . . 5 (𝜑 → {⟨𝑓, ⟩ ∣ 𝜓} ∈ 𝑉)
2927, 28opabresex2d 6920 . . . 4 (𝜑 → {⟨𝑓, ⟩ ∣ (𝑓(𝐷𝐺)𝜃)} ∈ V)
3026, 29syl5eqel 2885 . . 3 (𝜑 → {⟨𝑓, ⟩ ∣ (𝜃𝑓(𝐷𝐺))} ∈ V)
31 mptmpt2opabbrd.1 . . . . . 6 ((𝑎 = 𝑋𝑏 = 𝑌) → (𝜏𝜃))
3231anbi1d 617 . . . . 5 ((𝑎 = 𝑋𝑏 = 𝑌) → ((𝜏𝑓(𝐷𝐺)) ↔ (𝜃𝑓(𝐷𝐺))))
3332opabbidv 4903 . . . 4 ((𝑎 = 𝑋𝑏 = 𝑌) → {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))} = {⟨𝑓, ⟩ ∣ (𝜃𝑓(𝐷𝐺))})
34 eqid 2802 . . . 4 (𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))}) = (𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))})
3533, 34ovmpt2ga 7014 . . 3 ((𝑋 ∈ (𝐴𝐺) ∧ 𝑌 ∈ (𝐵𝐺) ∧ {⟨𝑓, ⟩ ∣ (𝜃𝑓(𝐷𝐺))} ∈ V) → (𝑋(𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))})𝑌) = {⟨𝑓, ⟩ ∣ (𝜃𝑓(𝐷𝐺))})
3623, 24, 30, 35syl3anc 1483 . 2 (𝜑 → (𝑋(𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))})𝑌) = {⟨𝑓, ⟩ ∣ (𝜃𝑓(𝐷𝐺))})
3722, 36eqtrd 2836 1 (𝜑 → (𝑋(𝑀𝐺)𝑌) = {⟨𝑓, ⟩ ∣ (𝜃𝑓(𝐷𝐺))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wcel 2155  Vcvv 3387   class class class wbr 4837  {copab 4899  cmpt 4916  cfv 6095  (class class class)co 6868  cmpt2 6870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-8 2157  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-rep 4957  ax-sep 4968  ax-nul 4977  ax-pow 5029  ax-pr 5090  ax-un 7173
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ne 2975  df-ral 3097  df-rex 3098  df-reu 3099  df-rab 3101  df-v 3389  df-sbc 3628  df-csb 3723  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-nul 4111  df-if 4274  df-pw 4347  df-sn 4365  df-pr 4367  df-op 4371  df-uni 4624  df-iun 4707  df-br 4838  df-opab 4900  df-mpt 4917  df-id 5213  df-xp 5311  df-rel 5312  df-cnv 5313  df-co 5314  df-dm 5315  df-rn 5316  df-res 5317  df-ima 5318  df-iota 6058  df-fun 6097  df-fn 6098  df-f 6099  df-f1 6100  df-fo 6101  df-f1o 6102  df-fv 6103  df-ov 6871  df-oprab 6872  df-mpt2 6873  df-1st 7392  df-2nd 7393
This theorem is referenced by:  mptmpt2opabovd  7476  wlkson  26774
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