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Theorem fndmin 5217
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 A 𝐺 Fn A) → dom (𝐹𝐺) = {x A ∣ (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 5162 . . . . . 6 (𝐹 Fn A𝐹 = (x A ↦ (𝐹x)))
2 df-mpt 3811 . . . . . 6 (x A ↦ (𝐹x)) = {⟨x, y⟩ ∣ (x A y = (𝐹x))}
31, 2syl6eq 2085 . . . . 5 (𝐹 Fn A𝐹 = {⟨x, y⟩ ∣ (x A y = (𝐹x))})
4 dffn5im 5162 . . . . . 6 (𝐺 Fn A𝐺 = (x A ↦ (𝐺x)))
5 df-mpt 3811 . . . . . 6 (x A ↦ (𝐺x)) = {⟨x, y⟩ ∣ (x A y = (𝐺x))}
64, 5syl6eq 2085 . . . . 5 (𝐺 Fn A𝐺 = {⟨x, y⟩ ∣ (x A y = (𝐺x))})
73, 6ineqan12d 3134 . . . 4 ((𝐹 Fn A 𝐺 Fn A) → (𝐹𝐺) = ({⟨x, y⟩ ∣ (x A y = (𝐹x))} ∩ {⟨x, y⟩ ∣ (x A y = (𝐺x))}))
8 inopab 4411 . . . 4 ({⟨x, y⟩ ∣ (x A y = (𝐹x))} ∩ {⟨x, y⟩ ∣ (x A y = (𝐺x))}) = {⟨x, y⟩ ∣ ((x A y = (𝐹x)) (x A y = (𝐺x)))}
97, 8syl6eq 2085 . . 3 ((𝐹 Fn A 𝐺 Fn A) → (𝐹𝐺) = {⟨x, y⟩ ∣ ((x A y = (𝐹x)) (x A y = (𝐺x)))})
109dmeqd 4480 . 2 ((𝐹 Fn A 𝐺 Fn A) → dom (𝐹𝐺) = dom {⟨x, y⟩ ∣ ((x A y = (𝐹x)) (x A y = (𝐺x)))})
11 anandi 524 . . . . . . . 8 ((x A (y = (𝐹x) y = (𝐺x))) ↔ ((x A y = (𝐹x)) (x A y = (𝐺x))))
1211exbii 1493 . . . . . . 7 (y(x A (y = (𝐹x) y = (𝐺x))) ↔ y((x A y = (𝐹x)) (x A y = (𝐺x))))
13 19.42v 1783 . . . . . . 7 (y(x A (y = (𝐹x) y = (𝐺x))) ↔ (x A y(y = (𝐹x) y = (𝐺x))))
1412, 13bitr3i 175 . . . . . 6 (y((x A y = (𝐹x)) (x A y = (𝐺x))) ↔ (x A y(y = (𝐹x) y = (𝐺x))))
15 funfvex 5135 . . . . . . . . 9 ((Fun 𝐹 x dom 𝐹) → (𝐹x) V)
16 eqeq1 2043 . . . . . . . . . 10 (y = (𝐹x) → (y = (𝐺x) ↔ (𝐹x) = (𝐺x)))
1716ceqsexgv 2667 . . . . . . . . 9 ((𝐹x) V → (y(y = (𝐹x) y = (𝐺x)) ↔ (𝐹x) = (𝐺x)))
1815, 17syl 14 . . . . . . . 8 ((Fun 𝐹 x dom 𝐹) → (y(y = (𝐹x) y = (𝐺x)) ↔ (𝐹x) = (𝐺x)))
1918funfni 4942 . . . . . . 7 ((𝐹 Fn A x A) → (y(y = (𝐹x) y = (𝐺x)) ↔ (𝐹x) = (𝐺x)))
2019pm5.32da 425 . . . . . 6 (𝐹 Fn A → ((x A y(y = (𝐹x) y = (𝐺x))) ↔ (x A (𝐹x) = (𝐺x))))
2114, 20syl5bb 181 . . . . 5 (𝐹 Fn A → (y((x A y = (𝐹x)) (x A y = (𝐺x))) ↔ (x A (𝐹x) = (𝐺x))))
2221abbidv 2152 . . . 4 (𝐹 Fn A → {xy((x A y = (𝐹x)) (x A y = (𝐺x)))} = {x ∣ (x A (𝐹x) = (𝐺x))})
23 dmopab 4489 . . . 4 dom {⟨x, y⟩ ∣ ((x A y = (𝐹x)) (x A y = (𝐺x)))} = {xy((x A y = (𝐹x)) (x A y = (𝐺x)))}
24 df-rab 2309 . . . 4 {x A ∣ (𝐹x) = (𝐺x)} = {x ∣ (x A (𝐹x) = (𝐺x))}
2522, 23, 243eqtr4g 2094 . . 3 (𝐹 Fn A → dom {⟨x, y⟩ ∣ ((x A y = (𝐹x)) (x A y = (𝐺x)))} = {x A ∣ (𝐹x) = (𝐺x)})
2625adantr 261 . 2 ((𝐹 Fn A 𝐺 Fn A) → dom {⟨x, y⟩ ∣ ((x A y = (𝐹x)) (x A y = (𝐺x)))} = {x A ∣ (𝐹x) = (𝐺x)})
2710, 26eqtrd 2069 1 ((𝐹 Fn A 𝐺 Fn A) → dom (𝐹𝐺) = {x A ∣ (𝐹x) = (𝐺x)})
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wex 1378   wcel 1390  {cab 2023  {crab 2304  Vcvv 2551  cin 2910  {copab 3808  cmpt 3809  dom cdm 4288  Fun wfun 4839   Fn wfn 4840  cfv 4845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853
This theorem is referenced by:  fneqeql  5218
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