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Theorem fndmin 6587
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 6501 . . . . . . 7 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
21biimpi 208 . . . . . 6 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
3 df-mpt 4966 . . . . . 6 (𝑥𝐴 ↦ (𝐹𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))}
42, 3syl6eq 2830 . . . . 5 (𝐹 Fn 𝐴𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))})
5 dffn5 6501 . . . . . . 7 (𝐺 Fn 𝐴𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
65biimpi 208 . . . . . 6 (𝐺 Fn 𝐴𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
7 df-mpt 4966 . . . . . 6 (𝑥𝐴 ↦ (𝐺𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐺𝑥))}
86, 7syl6eq 2830 . . . . 5 (𝐺 Fn 𝐴𝐺 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐺𝑥))})
94, 8ineqan12d 4039 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹𝐺) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐺𝑥))}))
10 inopab 5498 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐺𝑥))}) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥)))}
119, 10syl6eq 2830 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹𝐺) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥)))})
1211dmeqd 5571 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) = dom {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥)))})
13 19.42v 1996 . . . . 5 (∃𝑦(𝑥𝐴 ∧ (𝑦 = (𝐹𝑥) ∧ 𝑦 = (𝐺𝑥))) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑦 = (𝐹𝑥) ∧ 𝑦 = (𝐺𝑥))))
14 anandi 666 . . . . . 6 ((𝑥𝐴 ∧ (𝑦 = (𝐹𝑥) ∧ 𝑦 = (𝐺𝑥))) ↔ ((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥))))
1514exbii 1892 . . . . 5 (∃𝑦(𝑥𝐴 ∧ (𝑦 = (𝐹𝑥) ∧ 𝑦 = (𝐺𝑥))) ↔ ∃𝑦((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥))))
16 fvex 6459 . . . . . . 7 (𝐹𝑥) ∈ V
17 eqeq1 2782 . . . . . . 7 (𝑦 = (𝐹𝑥) → (𝑦 = (𝐺𝑥) ↔ (𝐹𝑥) = (𝐺𝑥)))
1816, 17ceqsexv 3444 . . . . . 6 (∃𝑦(𝑦 = (𝐹𝑥) ∧ 𝑦 = (𝐺𝑥)) ↔ (𝐹𝑥) = (𝐺𝑥))
1918anbi2i 616 . . . . 5 ((𝑥𝐴 ∧ ∃𝑦(𝑦 = (𝐹𝑥) ∧ 𝑦 = (𝐺𝑥))) ↔ (𝑥𝐴 ∧ (𝐹𝑥) = (𝐺𝑥)))
2013, 15, 193bitr3i 293 . . . 4 (∃𝑦((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥))) ↔ (𝑥𝐴 ∧ (𝐹𝑥) = (𝐺𝑥)))
2120abbii 2908 . . 3 {𝑥 ∣ ∃𝑦((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥)))} = {𝑥 ∣ (𝑥𝐴 ∧ (𝐹𝑥) = (𝐺𝑥))}
22 dmopab 5580 . . 3 dom {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥)))} = {𝑥 ∣ ∃𝑦((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥)))}
23 df-rab 3099 . . 3 {𝑥𝐴 ∣ (𝐹𝑥) = (𝐺𝑥)} = {𝑥 ∣ (𝑥𝐴 ∧ (𝐹𝑥) = (𝐺𝑥))}
2421, 22, 233eqtr4i 2812 . 2 dom {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥)))} = {𝑥𝐴 ∣ (𝐹𝑥) = (𝐺𝑥)}
2512, 24syl6eq 2830 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) = {𝑥𝐴 ∣ (𝐹𝑥) = (𝐺𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wex 1823  wcel 2107  {cab 2763  {crab 3094  cin 3791  {copab 4948  cmpt 4965  dom cdm 5355   Fn wfn 6130  cfv 6135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fn 6138  df-fv 6143
This theorem is referenced by:  fneqeql  6588  fninfp  6707  mhmeql  17750  ghmeql  18067  lmhmeql  19450  hauseqlcld  21858  cvmliftmolem1  31862  cvmliftmolem2  31863  hausgraph  38749  mgmhmeql  42818
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