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Theorem fndmdif 5215
Description: Two ways to express the locus of differences between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)
Assertion
Ref Expression
fndmdif ((𝐹 Fn A 𝐺 Fn A) → dom (𝐹𝐺) = {x A ∣ (𝐹x) ≠ (𝐺x)})
Distinct variable groups:   x,𝐹   x,𝐺   x,A

Proof of Theorem fndmdif
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 difss 3064 . . . . 5 (𝐹𝐺) ⊆ 𝐹
2 dmss 4477 . . . . 5 ((𝐹𝐺) ⊆ 𝐹 → dom (𝐹𝐺) ⊆ dom 𝐹)
31, 2ax-mp 7 . . . 4 dom (𝐹𝐺) ⊆ dom 𝐹
4 fndm 4941 . . . . 5 (𝐹 Fn A → dom 𝐹 = A)
54adantr 261 . . . 4 ((𝐹 Fn A 𝐺 Fn A) → dom 𝐹 = A)
63, 5syl5sseq 2987 . . 3 ((𝐹 Fn A 𝐺 Fn A) → dom (𝐹𝐺) ⊆ A)
7 dfss1 3135 . . 3 (dom (𝐹𝐺) ⊆ A ↔ (A ∩ dom (𝐹𝐺)) = dom (𝐹𝐺))
86, 7sylib 127 . 2 ((𝐹 Fn A 𝐺 Fn A) → (A ∩ dom (𝐹𝐺)) = dom (𝐹𝐺))
9 vex 2554 . . . . 5 x V
109eldm 4475 . . . 4 (x dom (𝐹𝐺) ↔ y x(𝐹𝐺)y)
11 eqcom 2039 . . . . . . . 8 ((𝐹x) = (𝐺x) ↔ (𝐺x) = (𝐹x))
12 fnbrfvb 5157 . . . . . . . 8 ((𝐺 Fn A x A) → ((𝐺x) = (𝐹x) ↔ x𝐺(𝐹x)))
1311, 12syl5bb 181 . . . . . . 7 ((𝐺 Fn A x A) → ((𝐹x) = (𝐺x) ↔ x𝐺(𝐹x)))
1413adantll 445 . . . . . 6 (((𝐹 Fn A 𝐺 Fn A) x A) → ((𝐹x) = (𝐺x) ↔ x𝐺(𝐹x)))
1514necon3abid 2238 . . . . 5 (((𝐹 Fn A 𝐺 Fn A) x A) → ((𝐹x) ≠ (𝐺x) ↔ ¬ x𝐺(𝐹x)))
16 funfvex 5135 . . . . . . . 8 ((Fun 𝐹 x dom 𝐹) → (𝐹x) V)
1716funfni 4942 . . . . . . 7 ((𝐹 Fn A x A) → (𝐹x) V)
1817adantlr 446 . . . . . 6 (((𝐹 Fn A 𝐺 Fn A) x A) → (𝐹x) V)
19 breq2 3759 . . . . . . . 8 (y = (𝐹x) → (x𝐺yx𝐺(𝐹x)))
2019notbid 591 . . . . . . 7 (y = (𝐹x) → (¬ x𝐺y ↔ ¬ x𝐺(𝐹x)))
2120ceqsexgv 2667 . . . . . 6 ((𝐹x) V → (y(y = (𝐹x) ¬ x𝐺y) ↔ ¬ x𝐺(𝐹x)))
2218, 21syl 14 . . . . 5 (((𝐹 Fn A 𝐺 Fn A) x A) → (y(y = (𝐹x) ¬ x𝐺y) ↔ ¬ x𝐺(𝐹x)))
23 eqcom 2039 . . . . . . . . . 10 (y = (𝐹x) ↔ (𝐹x) = y)
24 fnbrfvb 5157 . . . . . . . . . 10 ((𝐹 Fn A x A) → ((𝐹x) = yx𝐹y))
2523, 24syl5bb 181 . . . . . . . . 9 ((𝐹 Fn A x A) → (y = (𝐹x) ↔ x𝐹y))
2625adantlr 446 . . . . . . . 8 (((𝐹 Fn A 𝐺 Fn A) x A) → (y = (𝐹x) ↔ x𝐹y))
2726anbi1d 438 . . . . . . 7 (((𝐹 Fn A 𝐺 Fn A) x A) → ((y = (𝐹x) ¬ x𝐺y) ↔ (x𝐹y ¬ x𝐺y)))
28 brdif 3803 . . . . . . 7 (x(𝐹𝐺)y ↔ (x𝐹y ¬ x𝐺y))
2927, 28syl6bbr 187 . . . . . 6 (((𝐹 Fn A 𝐺 Fn A) x A) → ((y = (𝐹x) ¬ x𝐺y) ↔ x(𝐹𝐺)y))
3029exbidv 1703 . . . . 5 (((𝐹 Fn A 𝐺 Fn A) x A) → (y(y = (𝐹x) ¬ x𝐺y) ↔ y x(𝐹𝐺)y))
3115, 22, 303bitr2rd 206 . . . 4 (((𝐹 Fn A 𝐺 Fn A) x A) → (y x(𝐹𝐺)y ↔ (𝐹x) ≠ (𝐺x)))
3210, 31syl5bb 181 . . 3 (((𝐹 Fn A 𝐺 Fn A) x A) → (x dom (𝐹𝐺) ↔ (𝐹x) ≠ (𝐺x)))
3332rabbi2dva 3139 . 2 ((𝐹 Fn A 𝐺 Fn A) → (A ∩ dom (𝐹𝐺)) = {x A ∣ (𝐹x) ≠ (𝐺x)})
348, 33eqtr3d 2071 1 ((𝐹 Fn A 𝐺 Fn A) → dom (𝐹𝐺) = {x A ∣ (𝐹x) ≠ (𝐺x)})
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   = wceq 1242  wex 1378   wcel 1390  wne 2201  {crab 2304  Vcvv 2551  cdif 2908  cin 2910  wss 2911   class class class wbr 3755  dom cdm 4288   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-in1 544  ax-in2 545  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-ne 2203  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-dif 2914  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-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:  fndmdifcom  5216
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