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Theorem fvmptopab 6739
Description: The function value of a mapping 𝑀 to a restricted binary relation expressed as an ordered-pair class abstraction: The restricted binary relation is a binary relation given as value of a function 𝐹 restricted by the condition 𝜓. (Contributed by AV, 31-Jan-2021.) (Revised by AV, 29-Oct-2021.)
Hypotheses
Ref Expression
fvmptopab.1 ((𝜑𝑧 = 𝑍) → (𝜒𝜓))
fvmptopab.2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐹𝑍)𝑦} ∈ V)
fvmptopab.3 𝑀 = (𝑧 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑧)𝑦𝜒)})
Assertion
Ref Expression
fvmptopab (𝜑 → (𝑀𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)})
Distinct variable groups:   𝑧,𝐹   𝑥,𝑍,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝜓,𝑧
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦,𝑧)   𝐹(𝑥,𝑦)   𝑀(𝑥,𝑦,𝑧)

Proof of Theorem fvmptopab
StepHypRef Expression
1 fvmptopab.3 . . . . 5 𝑀 = (𝑧 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑧)𝑦𝜒)})
21a1i 11 . . . 4 ((𝑍 ∈ V ∧ 𝜑) → 𝑀 = (𝑧 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑧)𝑦𝜒)}))
3 fveq2 6229 . . . . . . . 8 (𝑧 = 𝑍 → (𝐹𝑧) = (𝐹𝑍))
43breqd 4696 . . . . . . 7 (𝑧 = 𝑍 → (𝑥(𝐹𝑧)𝑦𝑥(𝐹𝑍)𝑦))
54adantl 481 . . . . . 6 (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑧 = 𝑍) → (𝑥(𝐹𝑧)𝑦𝑥(𝐹𝑍)𝑦))
6 fvmptopab.1 . . . . . . 7 ((𝜑𝑧 = 𝑍) → (𝜒𝜓))
76adantll 750 . . . . . 6 (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑧 = 𝑍) → (𝜒𝜓))
85, 7anbi12d 747 . . . . 5 (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑧 = 𝑍) → ((𝑥(𝐹𝑧)𝑦𝜒) ↔ (𝑥(𝐹𝑍)𝑦𝜓)))
98opabbidv 4749 . . . 4 (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑧 = 𝑍) → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑧)𝑦𝜒)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)})
10 simpl 472 . . . 4 ((𝑍 ∈ V ∧ 𝜑) → 𝑍 ∈ V)
11 id 22 . . . . . 6 (𝑥(𝐹𝑍)𝑦𝑥(𝐹𝑍)𝑦)
1211gen2 1763 . . . . 5 𝑥𝑦(𝑥(𝐹𝑍)𝑦𝑥(𝐹𝑍)𝑦)
13 fvmptopab.2 . . . . . 6 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐹𝑍)𝑦} ∈ V)
1413adantl 481 . . . . 5 ((𝑍 ∈ V ∧ 𝜑) → {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐹𝑍)𝑦} ∈ V)
15 opabbrex 6737 . . . . 5 ((∀𝑥𝑦(𝑥(𝐹𝑍)𝑦𝑥(𝐹𝑍)𝑦) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐹𝑍)𝑦} ∈ V) → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)} ∈ V)
1612, 14, 15sylancr 696 . . . 4 ((𝑍 ∈ V ∧ 𝜑) → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)} ∈ V)
172, 9, 10, 16fvmptd 6327 . . 3 ((𝑍 ∈ V ∧ 𝜑) → (𝑀𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)})
1817ex 449 . 2 (𝑍 ∈ V → (𝜑 → (𝑀𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)}))
19 fvprc 6223 . . . 4 𝑍 ∈ V → (𝑀𝑍) = ∅)
20 br0 4734 . . . . . . . 8 ¬ 𝑥𝑦
21 fvprc 6223 . . . . . . . . 9 𝑍 ∈ V → (𝐹𝑍) = ∅)
2221breqd 4696 . . . . . . . 8 𝑍 ∈ V → (𝑥(𝐹𝑍)𝑦𝑥𝑦))
2320, 22mtbiri 316 . . . . . . 7 𝑍 ∈ V → ¬ 𝑥(𝐹𝑍)𝑦)
2423intnanrd 983 . . . . . 6 𝑍 ∈ V → ¬ (𝑥(𝐹𝑍)𝑦𝜓))
2524alrimivv 1896 . . . . 5 𝑍 ∈ V → ∀𝑥𝑦 ¬ (𝑥(𝐹𝑍)𝑦𝜓))
26 opab0 5036 . . . . 5 ({⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)} = ∅ ↔ ∀𝑥𝑦 ¬ (𝑥(𝐹𝑍)𝑦𝜓))
2725, 26sylibr 224 . . . 4 𝑍 ∈ V → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)} = ∅)
2819, 27eqtr4d 2688 . . 3 𝑍 ∈ V → (𝑀𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)})
2928a1d 25 . 2 𝑍 ∈ V → (𝜑 → (𝑀𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)}))
3018, 29pm2.61i 176 1 (𝜑 → (𝑀𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wal 1521   = wceq 1523  wcel 2030  Vcvv 3231  c0 3948   class class class wbr 4685  {copab 4745  cmpt 4762  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934
This theorem is referenced by:  trlsfval  26648  pthsfval  26673  spthsfval  26674  clwlks  26724  crcts  26739  cycls  26740  eupths  27178
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