Theorem fndmdif 6321

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 3737	. . . . 5 ⊢ (𝐹 ∖ 𝐺) ⊆ 𝐹
2		dmss 5323	. . . . 5 ⊢ ((𝐹 ∖ 𝐺) ⊆ 𝐹 → dom (𝐹 ∖ 𝐺) ⊆ dom 𝐹)
3	1, 2	ax-mp 5	. . . 4 ⊢ dom (𝐹 ∖ 𝐺) ⊆ dom 𝐹
4		fndm 5990	. . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
5	4	adantr 481	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom 𝐹 = 𝐴)
6	3, 5	syl5sseq 3653	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∖ 𝐺) ⊆ 𝐴)
7		sseqin2 3817	. . 3 ⊢ (dom (𝐹 ∖ 𝐺) ⊆ 𝐴 ↔ (𝐴 ∩ dom (𝐹 ∖ 𝐺)) = dom (𝐹 ∖ 𝐺))
8	6, 7	sylib 208	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐴 ∩ dom (𝐹 ∖ 𝐺)) = dom (𝐹 ∖ 𝐺))
9		vex 3203	. . . . 5 ⊢ 𝑥 ∈ V
10	9	eldm 5321	. . . 4 ⊢ (𝑥 ∈ dom (𝐹 ∖ 𝐺) ↔ ∃𝑦 𝑥(𝐹 ∖ 𝐺)𝑦)
11		eqcom 2629	. . . . . . . . 9 ⊢ ((𝐹‘𝑥) = (𝐺‘𝑥) ↔ (𝐺‘𝑥) = (𝐹‘𝑥))
12		fnbrfvb 6236	. . . . . . . . 9 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥) = (𝐹‘𝑥) ↔ 𝑥𝐺(𝐹‘𝑥)))
13	11, 12	syl5bb 272	. . . . . . . 8 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐺‘𝑥) ↔ 𝑥𝐺(𝐹‘𝑥)))
14	13	adantll 750	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐺‘𝑥) ↔ 𝑥𝐺(𝐹‘𝑥)))
15	14	necon3abid 2830	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) ≠ (𝐺‘𝑥) ↔ ¬ 𝑥𝐺(𝐹‘𝑥)))
16		fvex 6201	. . . . . . 7 ⊢ (𝐹‘𝑥) ∈ V
17		breq2 4657	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑥𝐺𝑦 ↔ 𝑥𝐺(𝐹‘𝑥)))
18	17	notbid 308	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑥) → (¬ 𝑥𝐺𝑦 ↔ ¬ 𝑥𝐺(𝐹‘𝑥)))
19	16, 18	ceqsexv 3242	. . . . . 6 ⊢ (∃𝑦(𝑦 = (𝐹‘𝑥) ∧ ¬ 𝑥𝐺𝑦) ↔ ¬ 𝑥𝐺(𝐹‘𝑥))
20	15, 19	syl6bbr 278	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) ≠ (𝐺‘𝑥) ↔ ∃𝑦(𝑦 = (𝐹‘𝑥) ∧ ¬ 𝑥𝐺𝑦)))
21		eqcom 2629	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑦)
22		fnbrfvb 6236	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦))
23	21, 22	syl5bb 272	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑦))
24	23	adantlr 751	. . . . . . . 8 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑦))
25	24	anbi1d 741	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝑦 = (𝐹‘𝑥) ∧ ¬ 𝑥𝐺𝑦) ↔ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐺𝑦)))
26		brdif 4705	. . . . . . 7 ⊢ (𝑥(𝐹 ∖ 𝐺)𝑦 ↔ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐺𝑦))
27	25, 26	syl6bbr 278	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝑦 = (𝐹‘𝑥) ∧ ¬ 𝑥𝐺𝑦) ↔ 𝑥(𝐹 ∖ 𝐺)𝑦))
28	27	exbidv 1850	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → (∃𝑦(𝑦 = (𝐹‘𝑥) ∧ ¬ 𝑥𝐺𝑦) ↔ ∃𝑦 𝑥(𝐹 ∖ 𝐺)𝑦))
29	20, 28	bitr2d 269	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → (∃𝑦 𝑥(𝐹 ∖ 𝐺)𝑦 ↔ (𝐹‘𝑥) ≠ (𝐺‘𝑥)))
30	10, 29	syl5bb 272	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ dom (𝐹 ∖ 𝐺) ↔ (𝐹‘𝑥) ≠ (𝐺‘𝑥)))
31	30	rabbi2dva 3821	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐴 ∩ dom (𝐹 ∖ 𝐺)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐺‘𝑥)})
32	8, 31	eqtr3d 2658	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∖ 𝐺) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐺‘𝑥)})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  {crab 2916   ∖ cdif 3571   ∩ cin 3573   ⊆ wss 3574   class class class wbr 4653  dom cdm 5114   Fn wfn 5883  ‘cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  fndmdifcom  6322  fndmdifeq0  6323  fndifnfp  6442  wemapsolem  8455  wemapso2lem  8457  dsmmbas2  20081  frlmbas  20099  ptcmplem2  21857