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 (𝐹 ∖ 𝐺) = {x ∈ A ∣ (𝐹‘x) ≠ (𝐺‘x)})

Distinct variable groups:   x,𝐹   x,𝐺   x,A

Proof of Theorem fndmdif

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		difss 3047	. . . . 5 ⊢ (𝐹 ∖ 𝐺) ⊆ 𝐹
2		dmss 4461	. . . . 5 ⊢ ((𝐹 ∖ 𝐺) ⊆ 𝐹 → dom (𝐹 ∖ 𝐺) ⊆ dom 𝐹)
3	1, 2	ax-mp 7	. . . 4 ⊢ dom (𝐹 ∖ 𝐺) ⊆ dom 𝐹
4		fndm 4924	. . . . 5 ⊢ (𝐹 Fn A → dom 𝐹 = A)
5	4	adantr 261	. . . 4 ⊢ ((𝐹 Fn A ∧ 𝐺 Fn A) → dom 𝐹 = A)
6	3, 5	syl5sseq 2970	. . 3 ⊢ ((𝐹 Fn A ∧ 𝐺 Fn A) → dom (𝐹 ∖ 𝐺) ⊆ A)
7		dfss1 3118	. . 3 ⊢ (dom (𝐹 ∖ 𝐺) ⊆ A ↔ (A ∩ dom (𝐹 ∖ 𝐺)) = dom (𝐹 ∖ 𝐺))
8	6, 7	sylib 127	. 2 ⊢ ((𝐹 Fn A ∧ 𝐺 Fn A) → (A ∩ dom (𝐹 ∖ 𝐺)) = dom (𝐹 ∖ 𝐺))
9		vex 2538	. . . . 5 ⊢ x ∈ V
10	9	eldm 4459	. . . 4 ⊢ (x ∈ dom (𝐹 ∖ 𝐺) ↔ ∃y x(𝐹 ∖ 𝐺)y)
11		eqcom 2024	. . . . . . . 8 ⊢ ((𝐹‘x) = (𝐺‘x) ↔ (𝐺‘x) = (𝐹‘x))
12		fnbrfvb 5139	. . . . . . . 8 ⊢ ((𝐺 Fn A ∧ x ∈ A) → ((𝐺‘x) = (𝐹‘x) ↔ x𝐺(𝐹‘x)))
13	11, 12	syl5bb 181	. . . . . . 7 ⊢ ((𝐺 Fn A ∧ x ∈ A) → ((𝐹‘x) = (𝐺‘x) ↔ x𝐺(𝐹‘x)))
14	13	adantll 448	. . . . . 6 ⊢ (((𝐹 Fn A ∧ 𝐺 Fn A) ∧ x ∈ A) → ((𝐹‘x) = (𝐺‘x) ↔ x𝐺(𝐹‘x)))
15	14	necon3abid 2222	. . . . 5 ⊢ (((𝐹 Fn A ∧ 𝐺 Fn A) ∧ x ∈ A) → ((𝐹‘x) ≠ (𝐺‘x) ↔ ¬ x𝐺(𝐹‘x)))
16		funfvex 5117	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ x ∈ dom 𝐹) → (𝐹‘x) ∈ V)
17	16	funfni 4925	. . . . . . 7 ⊢ ((𝐹 Fn A ∧ x ∈ A) → (𝐹‘x) ∈ V)
18	17	adantlr 449	. . . . . 6 ⊢ (((𝐹 Fn A ∧ 𝐺 Fn A) ∧ x ∈ A) → (𝐹‘x) ∈ V)
19		breq2 3742	. . . . . . . 8 ⊢ (y = (𝐹‘x) → (x𝐺y ↔ x𝐺(𝐹‘x)))
20	19	notbid 579	. . . . . . 7 ⊢ (y = (𝐹‘x) → (¬ x𝐺y ↔ ¬ x𝐺(𝐹‘x)))
21	20	ceqsexgv 2650	. . . . . 6 ⊢ ((𝐹‘x) ∈ V → (∃y(y = (𝐹‘x) ∧ ¬ x𝐺y) ↔ ¬ x𝐺(𝐹‘x)))
22	18, 21	syl 14	. . . . 5 ⊢ (((𝐹 Fn A ∧ 𝐺 Fn A) ∧ x ∈ A) → (∃y(y = (𝐹‘x) ∧ ¬ x𝐺y) ↔ ¬ x𝐺(𝐹‘x)))
23		eqcom 2024	. . . . . . . . . 10 ⊢ (y = (𝐹‘x) ↔ (𝐹‘x) = y)
24		fnbrfvb 5139	. . . . . . . . . 10 ⊢ ((𝐹 Fn A ∧ x ∈ A) → ((𝐹‘x) = y ↔ x𝐹y))
25	23, 24	syl5bb 181	. . . . . . . . 9 ⊢ ((𝐹 Fn A ∧ x ∈ A) → (y = (𝐹‘x) ↔ x𝐹y))
26	25	adantlr 449	. . . . . . . 8 ⊢ (((𝐹 Fn A ∧ 𝐺 Fn A) ∧ x ∈ A) → (y = (𝐹‘x) ↔ x𝐹y))
27	26	anbi1d 441	. . . . . . 7 ⊢ (((𝐹 Fn A ∧ 𝐺 Fn A) ∧ x ∈ A) → ((y = (𝐹‘x) ∧ ¬ x𝐺y) ↔ (x𝐹y ∧ ¬ x𝐺y)))
28		brdif 3786	. . . . . . 7 ⊢ (x(𝐹 ∖ 𝐺)y ↔ (x𝐹y ∧ ¬ x𝐺y))
29	27, 28	syl6bbr 187	. . . . . 6 ⊢ (((𝐹 Fn A ∧ 𝐺 Fn A) ∧ x ∈ A) → ((y = (𝐹‘x) ∧ ¬ x𝐺y) ↔ x(𝐹 ∖ 𝐺)y))
30	29	exbidv 1688	. . . . 5 ⊢ (((𝐹 Fn A ∧ 𝐺 Fn A) ∧ x ∈ A) → (∃y(y = (𝐹‘x) ∧ ¬ x𝐺y) ↔ ∃y x(𝐹 ∖ 𝐺)y))
31	15, 22, 30	3bitr2rd 206	. . . 4 ⊢ (((𝐹 Fn A ∧ 𝐺 Fn A) ∧ x ∈ A) → (∃y x(𝐹 ∖ 𝐺)y ↔ (𝐹‘x) ≠ (𝐺‘x)))
32	10, 31	syl5bb 181	. . 3 ⊢ (((𝐹 Fn A ∧ 𝐺 Fn A) ∧ x ∈ A) → (x ∈ dom (𝐹 ∖ 𝐺) ↔ (𝐹‘x) ≠ (𝐺‘x)))
33	32	rabbi2dva 3122	. 2 ⊢ ((𝐹 Fn A ∧ 𝐺 Fn A) → (A ∩ dom (𝐹 ∖ 𝐺)) = {x ∈ A ∣ (𝐹‘x) ≠ (𝐺‘x)})
34	8, 33	eqtr3d 2056	1 ⊢ ((𝐹 Fn A ∧ 𝐺 Fn A) → dom (𝐹 ∖ 𝐺) = {x ∈ A ∣ (𝐹‘x) ≠ (𝐺‘x)})

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