Theorem fndmdif 7043

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 4131	. . . . 5 ⊢ (𝐹 ∖ 𝐺) ⊆ 𝐹
2		dmss 5902	. . . . 5 ⊢ ((𝐹 ∖ 𝐺) ⊆ 𝐹 → dom (𝐹 ∖ 𝐺) ⊆ dom 𝐹)
3	1, 2	ax-mp 5	. . . 4 ⊢ dom (𝐹 ∖ 𝐺) ⊆ dom 𝐹
4		fndm 6652	. . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
5	4	adantr 481	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom 𝐹 = 𝐴)
6	3, 5	sseqtrid 4034	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∖ 𝐺) ⊆ 𝐴)
7		sseqin2 4215	. . 3 ⊢ (dom (𝐹 ∖ 𝐺) ⊆ 𝐴 ↔ (𝐴 ∩ dom (𝐹 ∖ 𝐺)) = dom (𝐹 ∖ 𝐺))
8	6, 7	sylib 217	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐴 ∩ dom (𝐹 ∖ 𝐺)) = dom (𝐹 ∖ 𝐺))
9		vex 3478	. . . . 5 ⊢ 𝑥 ∈ V
10	9	eldm 5900	. . . 4 ⊢ (𝑥 ∈ dom (𝐹 ∖ 𝐺) ↔ ∃𝑦 𝑥(𝐹 ∖ 𝐺)𝑦)
11		eqcom 2739	. . . . . . . . 9 ⊢ ((𝐹‘𝑥) = (𝐺‘𝑥) ↔ (𝐺‘𝑥) = (𝐹‘𝑥))
12		fnbrfvb 6944	. . . . . . . . 9 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥) = (𝐹‘𝑥) ↔ 𝑥𝐺(𝐹‘𝑥)))
13	11, 12	bitrid 282	. . . . . . . 8 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐺‘𝑥) ↔ 𝑥𝐺(𝐹‘𝑥)))
14	13	adantll 712	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐺‘𝑥) ↔ 𝑥𝐺(𝐹‘𝑥)))
15	14	necon3abid 2977	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) ≠ (𝐺‘𝑥) ↔ ¬ 𝑥𝐺(𝐹‘𝑥)))
16		fvex 6904	. . . . . . 7 ⊢ (𝐹‘𝑥) ∈ V
17		breq2 5152	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑥𝐺𝑦 ↔ 𝑥𝐺(𝐹‘𝑥)))
18	17	notbid 317	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑥) → (¬ 𝑥𝐺𝑦 ↔ ¬ 𝑥𝐺(𝐹‘𝑥)))
19	16, 18	ceqsexv 3525	. . . . . 6 ⊢ (∃𝑦(𝑦 = (𝐹‘𝑥) ∧ ¬ 𝑥𝐺𝑦) ↔ ¬ 𝑥𝐺(𝐹‘𝑥))
20	15, 19	bitr4di 288	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) ≠ (𝐺‘𝑥) ↔ ∃𝑦(𝑦 = (𝐹‘𝑥) ∧ ¬ 𝑥𝐺𝑦)))
21		eqcom 2739	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑦)
22		fnbrfvb 6944	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦))
23	21, 22	bitrid 282	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑦))
24	23	adantlr 713	. . . . . . . 8 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑦))
25	24	anbi1d 630	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝑦 = (𝐹‘𝑥) ∧ ¬ 𝑥𝐺𝑦) ↔ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐺𝑦)))
26		brdif 5201	. . . . . . 7 ⊢ (𝑥(𝐹 ∖ 𝐺)𝑦 ↔ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐺𝑦))
27	25, 26	bitr4di 288	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝑦 = (𝐹‘𝑥) ∧ ¬ 𝑥𝐺𝑦) ↔ 𝑥(𝐹 ∖ 𝐺)𝑦))
28	27	exbidv 1924	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → (∃𝑦(𝑦 = (𝐹‘𝑥) ∧ ¬ 𝑥𝐺𝑦) ↔ ∃𝑦 𝑥(𝐹 ∖ 𝐺)𝑦))
29	20, 28	bitr2d 279	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → (∃𝑦 𝑥(𝐹 ∖ 𝐺)𝑦 ↔ (𝐹‘𝑥) ≠ (𝐺‘𝑥)))
30	10, 29	bitrid 282	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ dom (𝐹 ∖ 𝐺) ↔ (𝐹‘𝑥) ≠ (𝐺‘𝑥)))
31	30	rabbi2dva 4217	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐴 ∩ dom (𝐹 ∖ 𝐺)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐺‘𝑥)})
32	8, 31	eqtr3d 2774	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∖ 𝐺) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐺‘𝑥)})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106   ≠ wne 2940  {crab 3432   ∖ cdif 3945   ∩ cin 3947   ⊆ wss 3948   class class class wbr 5148  dom cdm 5676   Fn wfn 6538  ‘cfv 6543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551

This theorem is referenced by:  fndmdifcom  7044  fndmdifeq0  7045  fndifnfp  7176  wemapsolem  9547  wemapso2lem  9549  dsmmbas2  21511  frlmbas  21529  ptcmplem2  23777