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Theorem fndmin 5199
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 5144 . . . . . 6 (𝐹 Fn A𝐹 = (x A ↦ (𝐹x)))
2 df-mpt 3794 . . . . . 6 (x A ↦ (𝐹x)) = {⟨x, y⟩ ∣ (x A y = (𝐹x))}
31, 2syl6eq 2070 . . . . 5 (𝐹 Fn A𝐹 = {⟨x, y⟩ ∣ (x A y = (𝐹x))})
4 dffn5im 5144 . . . . . 6 (𝐺 Fn A𝐺 = (x A ↦ (𝐺x)))
5 df-mpt 3794 . . . . . 6 (x A ↦ (𝐺x)) = {⟨x, y⟩ ∣ (x A y = (𝐺x))}
64, 5syl6eq 2070 . . . . 5 (𝐺 Fn A𝐺 = {⟨x, y⟩ ∣ (x A y = (𝐺x))})
73, 6ineqan12d 3117 . . . 4 ((𝐹 Fn A 𝐺 Fn A) → (𝐹𝐺) = ({⟨x, y⟩ ∣ (x A y = (𝐹x))} ∩ {⟨x, y⟩ ∣ (x A y = (𝐺x))}))
8 inopab 4395 . . . 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 2070 . . 3 ((𝐹 Fn A 𝐺 Fn A) → (𝐹𝐺) = {⟨x, y⟩ ∣ ((x A y = (𝐹x)) (x A y = (𝐺x)))})
109dmeqd 4464 . 2 ((𝐹 Fn A 𝐺 Fn A) → dom (𝐹𝐺) = dom {⟨x, y⟩ ∣ ((x A y = (𝐹x)) (x A y = (𝐺x)))})
11 anandi 511 . . . . . . . 8 ((x A (y = (𝐹x) y = (𝐺x))) ↔ ((x A y = (𝐹x)) (x A y = (𝐺x))))
1211exbii 1478 . . . . . . 7 (y(x A (y = (𝐹x) y = (𝐺x))) ↔ y((x A y = (𝐹x)) (x A y = (𝐺x))))
13 19.42v 1768 . . . . . . 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 5117 . . . . . . . . 9 ((Fun 𝐹 x dom 𝐹) → (𝐹x) V)
16 eqeq1 2028 . . . . . . . . . 10 (y = (𝐹x) → (y = (𝐺x) ↔ (𝐹x) = (𝐺x)))
1716ceqsexgv 2650 . . . . . . . . 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 4925 . . . . . . 7 ((𝐹 Fn A x A) → (y(y = (𝐹x) y = (𝐺x)) ↔ (𝐹x) = (𝐺x)))
2019pm5.32da 428 . . . . . 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 2137 . . . 4 (𝐹 Fn A → {xy((x A y = (𝐹x)) (x A y = (𝐺x)))} = {x ∣ (x A (𝐹x) = (𝐺x))})
23 dmopab 4473 . . . 4 dom {⟨x, y⟩ ∣ ((x A y = (𝐹x)) (x A y = (𝐺x)))} = {xy((x A y = (𝐹x)) (x A y = (𝐺x)))}
24 df-rab 2293 . . . 4 {x A ∣ (𝐹x) = (𝐺x)} = {x ∣ (x A (𝐹x) = (𝐺x))}
2522, 23, 243eqtr4g 2079 . . 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 2054 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 1228  wex 1362   wcel 1374  {cab 2008  {crab 2288  Vcvv 2535  cin 2893  {copab 3791  cmpt 3792  dom cdm 4272  Fun wfun 4823   Fn wfn 4824  cfv 4829
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-iota 4794  df-fun 4831  df-fn 4832  df-fv 4837
This theorem is referenced by:  fneqeql  5200
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