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Theorem fndmin 7064
Description: Two ways to express the locus of equality between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)
Assertion
Ref Expression
fndmin ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) = {𝑥𝐴 ∣ (𝐹𝑥) = (𝐺𝑥)})
Distinct variable groups:   𝑥,𝐹   𝑥,𝐺   𝑥,𝐴

Proof of Theorem fndmin
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dffn5 6966 . . . . . . 7 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
21biimpi 216 . . . . . 6 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
3 df-mpt 5231 . . . . . 6 (𝑥𝐴 ↦ (𝐹𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))}
42, 3eqtrdi 2790 . . . . 5 (𝐹 Fn 𝐴𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))})
5 dffn5 6966 . . . . . . 7 (𝐺 Fn 𝐴𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
65biimpi 216 . . . . . 6 (𝐺 Fn 𝐴𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
7 df-mpt 5231 . . . . . 6 (𝑥𝐴 ↦ (𝐺𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐺𝑥))}
86, 7eqtrdi 2790 . . . . 5 (𝐺 Fn 𝐴𝐺 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐺𝑥))})
94, 8ineqan12d 4229 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹𝐺) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐺𝑥))}))
10 inopab 5841 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐺𝑥))}) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥)))}
119, 10eqtrdi 2790 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹𝐺) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥)))})
1211dmeqd 5918 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) = dom {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥)))})
13 19.42v 1950 . . . . 5 (∃𝑦(𝑥𝐴 ∧ (𝑦 = (𝐹𝑥) ∧ 𝑦 = (𝐺𝑥))) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑦 = (𝐹𝑥) ∧ 𝑦 = (𝐺𝑥))))
14 anandi 676 . . . . . 6 ((𝑥𝐴 ∧ (𝑦 = (𝐹𝑥) ∧ 𝑦 = (𝐺𝑥))) ↔ ((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥))))
1514exbii 1844 . . . . 5 (∃𝑦(𝑥𝐴 ∧ (𝑦 = (𝐹𝑥) ∧ 𝑦 = (𝐺𝑥))) ↔ ∃𝑦((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥))))
16 fvex 6919 . . . . . . 7 (𝐹𝑥) ∈ V
17 eqeq1 2738 . . . . . . 7 (𝑦 = (𝐹𝑥) → (𝑦 = (𝐺𝑥) ↔ (𝐹𝑥) = (𝐺𝑥)))
1816, 17ceqsexv 3529 . . . . . 6 (∃𝑦(𝑦 = (𝐹𝑥) ∧ 𝑦 = (𝐺𝑥)) ↔ (𝐹𝑥) = (𝐺𝑥))
1918anbi2i 623 . . . . 5 ((𝑥𝐴 ∧ ∃𝑦(𝑦 = (𝐹𝑥) ∧ 𝑦 = (𝐺𝑥))) ↔ (𝑥𝐴 ∧ (𝐹𝑥) = (𝐺𝑥)))
2013, 15, 193bitr3i 301 . . . 4 (∃𝑦((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥))) ↔ (𝑥𝐴 ∧ (𝐹𝑥) = (𝐺𝑥)))
2120abbii 2806 . . 3 {𝑥 ∣ ∃𝑦((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥)))} = {𝑥 ∣ (𝑥𝐴 ∧ (𝐹𝑥) = (𝐺𝑥))}
22 dmopab 5928 . . 3 dom {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥)))} = {𝑥 ∣ ∃𝑦((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥)))}
23 df-rab 3433 . . 3 {𝑥𝐴 ∣ (𝐹𝑥) = (𝐺𝑥)} = {𝑥 ∣ (𝑥𝐴 ∧ (𝐹𝑥) = (𝐺𝑥))}
2421, 22, 233eqtr4i 2772 . 2 dom {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥)))} = {𝑥𝐴 ∣ (𝐹𝑥) = (𝐺𝑥)}
2512, 24eqtrdi 2790 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) = {𝑥𝐴 ∣ (𝐹𝑥) = (𝐺𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wex 1775  wcel 2105  {cab 2711  {crab 3432  cin 3961  {copab 5209  cmpt 5230  dom cdm 5688   Fn wfn 6557  cfv 6562
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-pr 5437
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-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  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-iota 6515  df-fun 6564  df-fn 6565  df-fv 6570
This theorem is referenced by:  fneqeql  7065  fninfp  7193  mgmhmeql  18741  mhmeql  18851  ghmeql  19269  lmhmeql  21071  hauseqlcld  23669  cvmliftmolem1  35265  cvmliftmolem2  35266  hausgraph  43193
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