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Theorem fndmin 7065
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 6967 . . . . . . 7 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
21biimpi 216 . . . . . 6 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
3 df-mpt 5226 . . . . . 6 (𝑥𝐴 ↦ (𝐹𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))}
42, 3eqtrdi 2793 . . . . 5 (𝐹 Fn 𝐴𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))})
5 dffn5 6967 . . . . . . 7 (𝐺 Fn 𝐴𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
65biimpi 216 . . . . . 6 (𝐺 Fn 𝐴𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
7 df-mpt 5226 . . . . . 6 (𝑥𝐴 ↦ (𝐺𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐺𝑥))}
86, 7eqtrdi 2793 . . . . 5 (𝐺 Fn 𝐴𝐺 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐺𝑥))})
94, 8ineqan12d 4222 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹𝐺) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐺𝑥))}))
10 inopab 5839 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐺𝑥))}) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥)))}
119, 10eqtrdi 2793 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹𝐺) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥)))})
1211dmeqd 5916 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) = dom {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥)))})
13 19.42v 1953 . . . . 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 1848 . . . . 5 (∃𝑦(𝑥𝐴 ∧ (𝑦 = (𝐹𝑥) ∧ 𝑦 = (𝐺𝑥))) ↔ ∃𝑦((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥))))
16 fvex 6919 . . . . . . 7 (𝐹𝑥) ∈ V
17 eqeq1 2741 . . . . . . 7 (𝑦 = (𝐹𝑥) → (𝑦 = (𝐺𝑥) ↔ (𝐹𝑥) = (𝐺𝑥)))
1816, 17ceqsexv 3532 . . . . . 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 2809 . . 3 {𝑥 ∣ ∃𝑦((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥)))} = {𝑥 ∣ (𝑥𝐴 ∧ (𝐹𝑥) = (𝐺𝑥))}
22 dmopab 5926 . . 3 dom {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥)))} = {𝑥 ∣ ∃𝑦((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥)))}
23 df-rab 3437 . . 3 {𝑥𝐴 ∣ (𝐹𝑥) = (𝐺𝑥)} = {𝑥 ∣ (𝑥𝐴 ∧ (𝐹𝑥) = (𝐺𝑥))}
2421, 22, 233eqtr4i 2775 . 2 dom {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥)))} = {𝑥𝐴 ∣ (𝐹𝑥) = (𝐺𝑥)}
2512, 24eqtrdi 2793 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) = {𝑥𝐴 ∣ (𝐹𝑥) = (𝐺𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wex 1779  wcel 2108  {cab 2714  {crab 3436  cin 3950  {copab 5205  cmpt 5225  dom cdm 5685   Fn wfn 6556  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569
This theorem is referenced by:  fneqeql  7066  fninfp  7194  mgmhmeql  18729  mhmeql  18839  ghmeql  19257  lmhmeql  21054  hauseqlcld  23654  cvmliftmolem1  35286  cvmliftmolem2  35287  hausgraph  43217
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