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Theorem fndmdif 5197
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 3047 . . . . 5 (𝐹𝐺) ⊆ 𝐹
2 dmss 4461 . . . . 5 ((𝐹𝐺) ⊆ 𝐹 → dom (𝐹𝐺) ⊆ dom 𝐹)
31, 2ax-mp 7 . . . 4 dom (𝐹𝐺) ⊆ dom 𝐹
4 fndm 4924 . . . . 5 (𝐹 Fn A → dom 𝐹 = A)
54adantr 261 . . . 4 ((𝐹 Fn A 𝐺 Fn A) → dom 𝐹 = A)
63, 5syl5sseq 2970 . . 3 ((𝐹 Fn A 𝐺 Fn A) → dom (𝐹𝐺) ⊆ A)
7 dfss1 3118 . . 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 2538 . . . . 5 x V
109eldm 4459 . . . 4 (x dom (𝐹𝐺) ↔ y x(𝐹𝐺)y)
11 eqcom 2024 . . . . . . . 8 ((𝐹x) = (𝐺x) ↔ (𝐺x) = (𝐹x))
12 fnbrfvb 5139 . . . . . . . 8 ((𝐺 Fn A x A) → ((𝐺x) = (𝐹x) ↔ x𝐺(𝐹x)))
1311, 12syl5bb 181 . . . . . . 7 ((𝐺 Fn A x A) → ((𝐹x) = (𝐺x) ↔ x𝐺(𝐹x)))
1413adantll 448 . . . . . 6 (((𝐹 Fn A 𝐺 Fn A) x A) → ((𝐹x) = (𝐺x) ↔ x𝐺(𝐹x)))
1514necon3abid 2222 . . . . 5 (((𝐹 Fn A 𝐺 Fn A) x A) → ((𝐹x) ≠ (𝐺x) ↔ ¬ x𝐺(𝐹x)))
16 funfvex 5117 . . . . . . . 8 ((Fun 𝐹 x dom 𝐹) → (𝐹x) V)
1716funfni 4925 . . . . . . 7 ((𝐹 Fn A x A) → (𝐹x) V)
1817adantlr 449 . . . . . 6 (((𝐹 Fn A 𝐺 Fn A) x A) → (𝐹x) V)
19 breq2 3742 . . . . . . . 8 (y = (𝐹x) → (x𝐺yx𝐺(𝐹x)))
2019notbid 579 . . . . . . 7 (y = (𝐹x) → (¬ x𝐺y ↔ ¬ x𝐺(𝐹x)))
2120ceqsexgv 2650 . . . . . 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 2024 . . . . . . . . . 10 (y = (𝐹x) ↔ (𝐹x) = y)
24 fnbrfvb 5139 . . . . . . . . . 10 ((𝐹 Fn A x A) → ((𝐹x) = yx𝐹y))
2523, 24syl5bb 181 . . . . . . . . 9 ((𝐹 Fn A x A) → (y = (𝐹x) ↔ x𝐹y))
2625adantlr 449 . . . . . . . 8 (((𝐹 Fn A 𝐺 Fn A) x A) → (y = (𝐹x) ↔ x𝐹y))
2726anbi1d 441 . . . . . . 7 (((𝐹 Fn A 𝐺 Fn A) x A) → ((y = (𝐹x) ¬ x𝐺y) ↔ (x𝐹y ¬ x𝐺y)))
28 brdif 3786 . . . . . . 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 1688 . . . . 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 3122 . 2 ((𝐹 Fn A 𝐺 Fn A) → (A ∩ dom (𝐹𝐺)) = {x A ∣ (𝐹x) ≠ (𝐺x)})
348, 33eqtr3d 2056 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 1228  wex 1362   wcel 1374  wne 2186  {crab 2288  Vcvv 2535  cdif 2891  cin 2893  wss 2894   class class class wbr 3738  dom cdm 4272   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-in1 532  ax-in2 533  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-ne 2188  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-sbc 2742  df-dif 2897  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-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:  fndmdifcom  5198
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