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Theorem fndmin 5495
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 dffn5im 5435 . . . . . 6 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
2 df-mpt 3961 . . . . . 6 (𝑥𝐴 ↦ (𝐹𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))}
31, 2syl6eq 2166 . . . . 5 (𝐹 Fn 𝐴𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))})
4 dffn5im 5435 . . . . . 6 (𝐺 Fn 𝐴𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
5 df-mpt 3961 . . . . . 6 (𝑥𝐴 ↦ (𝐺𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐺𝑥))}
64, 5syl6eq 2166 . . . . 5 (𝐺 Fn 𝐴𝐺 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐺𝑥))})
73, 6ineqan12d 3249 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹𝐺) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐺𝑥))}))
8 inopab 4641 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐺𝑥))}) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥)))}
97, 8syl6eq 2166 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹𝐺) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥)))})
109dmeqd 4711 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) = dom {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥)))})
11 anandi 564 . . . . . . . 8 ((𝑥𝐴 ∧ (𝑦 = (𝐹𝑥) ∧ 𝑦 = (𝐺𝑥))) ↔ ((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥))))
1211exbii 1569 . . . . . . 7 (∃𝑦(𝑥𝐴 ∧ (𝑦 = (𝐹𝑥) ∧ 𝑦 = (𝐺𝑥))) ↔ ∃𝑦((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥))))
13 19.42v 1862 . . . . . . 7 (∃𝑦(𝑥𝐴 ∧ (𝑦 = (𝐹𝑥) ∧ 𝑦 = (𝐺𝑥))) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑦 = (𝐹𝑥) ∧ 𝑦 = (𝐺𝑥))))
1412, 13bitr3i 185 . . . . . 6 (∃𝑦((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥))) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑦 = (𝐹𝑥) ∧ 𝑦 = (𝐺𝑥))))
15 funfvex 5406 . . . . . . . . 9 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ V)
16 eqeq1 2124 . . . . . . . . . 10 (𝑦 = (𝐹𝑥) → (𝑦 = (𝐺𝑥) ↔ (𝐹𝑥) = (𝐺𝑥)))
1716ceqsexgv 2788 . . . . . . . . 9 ((𝐹𝑥) ∈ V → (∃𝑦(𝑦 = (𝐹𝑥) ∧ 𝑦 = (𝐺𝑥)) ↔ (𝐹𝑥) = (𝐺𝑥)))
1815, 17syl 14 . . . . . . . 8 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (∃𝑦(𝑦 = (𝐹𝑥) ∧ 𝑦 = (𝐺𝑥)) ↔ (𝐹𝑥) = (𝐺𝑥)))
1918funfni 5193 . . . . . . 7 ((𝐹 Fn 𝐴𝑥𝐴) → (∃𝑦(𝑦 = (𝐹𝑥) ∧ 𝑦 = (𝐺𝑥)) ↔ (𝐹𝑥) = (𝐺𝑥)))
2019pm5.32da 447 . . . . . 6 (𝐹 Fn 𝐴 → ((𝑥𝐴 ∧ ∃𝑦(𝑦 = (𝐹𝑥) ∧ 𝑦 = (𝐺𝑥))) ↔ (𝑥𝐴 ∧ (𝐹𝑥) = (𝐺𝑥))))
2114, 20syl5bb 191 . . . . 5 (𝐹 Fn 𝐴 → (∃𝑦((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥))) ↔ (𝑥𝐴 ∧ (𝐹𝑥) = (𝐺𝑥))))
2221abbidv 2235 . . . 4 (𝐹 Fn 𝐴 → {𝑥 ∣ ∃𝑦((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥)))} = {𝑥 ∣ (𝑥𝐴 ∧ (𝐹𝑥) = (𝐺𝑥))})
23 dmopab 4720 . . . 4 dom {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥)))} = {𝑥 ∣ ∃𝑦((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥)))}
24 df-rab 2402 . . . 4 {𝑥𝐴 ∣ (𝐹𝑥) = (𝐺𝑥)} = {𝑥 ∣ (𝑥𝐴 ∧ (𝐹𝑥) = (𝐺𝑥))}
2522, 23, 243eqtr4g 2175 . . 3 (𝐹 Fn 𝐴 → dom {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥)))} = {𝑥𝐴 ∣ (𝐹𝑥) = (𝐺𝑥)})
2625adantr 274 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦 = (𝐹𝑥)) ∧ (𝑥𝐴𝑦 = (𝐺𝑥)))} = {𝑥𝐴 ∣ (𝐹𝑥) = (𝐺𝑥)})
2710, 26eqtrd 2150 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) = {𝑥𝐴 ∣ (𝐹𝑥) = (𝐺𝑥)})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1316  wex 1453  wcel 1465  {cab 2103  {crab 2397  Vcvv 2660  cin 3040  {copab 3958  cmpt 3959  dom cdm 4509  Fun wfun 5087   Fn wfn 5088  cfv 5093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fn 5096  df-fv 5101
This theorem is referenced by:  fneqeql  5496
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