MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mptmpt2opabbrd Structured version   Visualization version   GIF version

Theorem mptmpt2opabbrd 7416
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 6352 . . . . . . 7 (𝑔 = 𝐺 → (𝐴𝑔) = (𝐴𝐺))
5 fveq2 6352 . . . . . . 7 (𝑔 = 𝐺 → (𝐵𝑔) = (𝐵𝐺))
6 mptmpt2opabbrd.2 . . . . . . . . 9 (𝑔 = 𝐺 → (𝜒𝜏))
7 fveq2 6352 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝐷𝑔) = (𝐷𝐺))
87breqd 4815 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑓(𝐷𝑔)𝑓(𝐷𝐺)))
96, 8anbi12d 749 . . . . . . . 8 (𝑔 = 𝐺 → ((𝜒𝑓(𝐷𝑔)) ↔ (𝜏𝑓(𝐷𝐺))))
109opabbidv 4868 . . . . . . 7 (𝑔 = 𝐺 → {⟨𝑓, ⟩ ∣ (𝜒𝑓(𝐷𝑔))} = {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))})
114, 5, 10mpt2eq123dv 6882 . . . . . 6 (𝑔 = 𝐺 → (𝑎 ∈ (𝐴𝑔), 𝑏 ∈ (𝐵𝑔) ↦ {⟨𝑓, ⟩ ∣ (𝜒𝑓(𝐷𝑔))}) = (𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))}))
1211adantl 473 . . . . 5 (((𝐺𝑊𝐺𝑊) ∧ 𝑔 = 𝐺) → (𝑎 ∈ (𝐴𝑔), 𝑏 ∈ (𝐵𝑔) ↦ {⟨𝑓, ⟩ ∣ (𝜒𝑓(𝐷𝑔))}) = (𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))}))
13 elex 3352 . . . . . 6 (𝐺𝑊𝐺 ∈ V)
1413adantr 472 . . . . 5 ((𝐺𝑊𝐺𝑊) → 𝐺 ∈ V)
15 fvex 6362 . . . . . . 7 (𝐴𝐺) ∈ V
16 fvex 6362 . . . . . . 7 (𝐵𝐺) ∈ V
1715, 16pm3.2i 470 . . . . . 6 ((𝐴𝐺) ∈ V ∧ (𝐵𝐺) ∈ V)
18 mpt2exga 7414 . . . . . 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 6450 . . . 4 ((𝐺𝑊𝐺𝑊) → (𝑀𝐺) = (𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))}))
211, 1, 20syl2anc 696 . . 3 (𝜑 → (𝑀𝐺) = (𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))}))
2221oveqd 6830 . 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 465 . . . . 5 ((𝜃𝑓(𝐷𝐺)) ↔ (𝑓(𝐷𝐺)𝜃))
2625opabbii 4869 . . . 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 6861 . . . 4 (𝜑 → {⟨𝑓, ⟩ ∣ (𝑓(𝐷𝐺)𝜃)} ∈ V)
3026, 29syl5eqel 2843 . . 3 (𝜑 → {⟨𝑓, ⟩ ∣ (𝜃𝑓(𝐷𝐺))} ∈ V)
31 mptmpt2opabbrd.1 . . . . . 6 ((𝑎 = 𝑋𝑏 = 𝑌) → (𝜏𝜃))
3231anbi1d 743 . . . . 5 ((𝑎 = 𝑋𝑏 = 𝑌) → ((𝜏𝑓(𝐷𝐺)) ↔ (𝜃𝑓(𝐷𝐺))))
3332opabbidv 4868 . . . 4 ((𝑎 = 𝑋𝑏 = 𝑌) → {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))} = {⟨𝑓, ⟩ ∣ (𝜃𝑓(𝐷𝐺))})
34 eqid 2760 . . . 4 (𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))}) = (𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))})
3533, 34ovmpt2ga 6955 . . 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 1477 . 2 (𝜑 → (𝑋(𝑎 ∈ (𝐴𝐺), 𝑏 ∈ (𝐵𝐺) ↦ {⟨𝑓, ⟩ ∣ (𝜏𝑓(𝐷𝐺))})𝑌) = {⟨𝑓, ⟩ ∣ (𝜃𝑓(𝐷𝐺))})
3722, 36eqtrd 2794 1 (𝜑 → (𝑋(𝑀𝐺)𝑌) = {⟨𝑓, ⟩ ∣ (𝜃𝑓(𝐷𝐺))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  Vcvv 3340   class class class wbr 4804  {copab 4864  cmpt 4881  cfv 6049  (class class class)co 6813  cmpt2 6815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334
This theorem is referenced by:  mptmpt2opabovd  7417  wlkson  26762
  Copyright terms: Public domain W3C validator