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 (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 3064	. . . . 5 ⊢ (𝐹 ∖ 𝐺) ⊆ 𝐹
2		dmss 4477	. . . . 5 ⊢ ((𝐹 ∖ 𝐺) ⊆ 𝐹 → dom (𝐹 ∖ 𝐺) ⊆ dom 𝐹)
3	1, 2	ax-mp 7	. . . 4 ⊢ dom (𝐹 ∖ 𝐺) ⊆ dom 𝐹
4		fndm 4941	. . . . 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 2987	. . 3 ⊢ ((𝐹 Fn A ∧ 𝐺 Fn A) → dom (𝐹 ∖ 𝐺) ⊆ A)
7		dfss1 3135	. . 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 2554	. . . . 5 ⊢ x ∈ V
10	9	eldm 4475	. . . 4 ⊢ (x ∈ dom (𝐹 ∖ 𝐺) ↔ ∃y x(𝐹 ∖ 𝐺)y)
11		eqcom 2039	. . . . . . . 8 ⊢ ((𝐹‘x) = (𝐺‘x) ↔ (𝐺‘x) = (𝐹‘x))
12		fnbrfvb 5157	. . . . . . . 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 445	. . . . . 6 ⊢ (((𝐹 Fn A ∧ 𝐺 Fn A) ∧ x ∈ A) → ((𝐹‘x) = (𝐺‘x) ↔ x𝐺(𝐹‘x)))
15	14	necon3abid 2238	. . . . 5 ⊢ (((𝐹 Fn A ∧ 𝐺 Fn A) ∧ x ∈ A) → ((𝐹‘x) ≠ (𝐺‘x) ↔ ¬ x𝐺(𝐹‘x)))
16		funfvex 5135	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ x ∈ dom 𝐹) → (𝐹‘x) ∈ V)
17	16	funfni 4942	. . . . . . 7 ⊢ ((𝐹 Fn A ∧ x ∈ A) → (𝐹‘x) ∈ V)
18	17	adantlr 446	. . . . . 6 ⊢ (((𝐹 Fn A ∧ 𝐺 Fn A) ∧ x ∈ A) → (𝐹‘x) ∈ V)
19		breq2 3759	. . . . . . . 8 ⊢ (y = (𝐹‘x) → (x𝐺y ↔ x𝐺(𝐹‘x)))
20	19	notbid 591	. . . . . . 7 ⊢ (y = (𝐹‘x) → (¬ x𝐺y ↔ ¬ x𝐺(𝐹‘x)))
21	20	ceqsexgv 2667	. . . . . 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 2039	. . . . . . . . . 10 ⊢ (y = (𝐹‘x) ↔ (𝐹‘x) = y)
24		fnbrfvb 5157	. . . . . . . . . 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 446	. . . . . . . 8 ⊢ (((𝐹 Fn A ∧ 𝐺 Fn A) ∧ x ∈ A) → (y = (𝐹‘x) ↔ x𝐹y))
27	26	anbi1d 438	. . . . . . 7 ⊢ (((𝐹 Fn A ∧ 𝐺 Fn A) ∧ x ∈ A) → ((y = (𝐹‘x) ∧ ¬ x𝐺y) ↔ (x𝐹y ∧ ¬ x𝐺y)))
28		brdif 3803	. . . . . . 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 1703	. . . . 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 3139	. 2 ⊢ ((𝐹 Fn A ∧ 𝐺 Fn A) → (A ∩ dom (𝐹 ∖ 𝐺)) = {x ∈ A ∣ (𝐹‘x) ≠ (𝐺‘x)})
34	8, 33	eqtr3d 2071	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 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-bndl 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