Theorem fndmdif 7061

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 4145	. . . . 5 ⊢ (𝐹 ∖ 𝐺) ⊆ 𝐹
2		dmss 5915	. . . . 5 ⊢ ((𝐹 ∖ 𝐺) ⊆ 𝐹 → dom (𝐹 ∖ 𝐺) ⊆ dom 𝐹)
3	1, 2	ax-mp 5	. . . 4 ⊢ dom (𝐹 ∖ 𝐺) ⊆ dom 𝐹
4		fndm 6671	. . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
5	4	adantr 480	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom 𝐹 = 𝐴)
6	3, 5	sseqtrid 4047	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∖ 𝐺) ⊆ 𝐴)
7		sseqin2 4230	. . 3 ⊢ (dom (𝐹 ∖ 𝐺) ⊆ 𝐴 ↔ (𝐴 ∩ dom (𝐹 ∖ 𝐺)) = dom (𝐹 ∖ 𝐺))
8	6, 7	sylib 218	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐴 ∩ dom (𝐹 ∖ 𝐺)) = dom (𝐹 ∖ 𝐺))
9		vex 3481	. . . . 5 ⊢ 𝑥 ∈ V
10	9	eldm 5913	. . . 4 ⊢ (𝑥 ∈ dom (𝐹 ∖ 𝐺) ↔ ∃𝑦 𝑥(𝐹 ∖ 𝐺)𝑦)
11		eqcom 2741	. . . . . . . . 9 ⊢ ((𝐹‘𝑥) = (𝐺‘𝑥) ↔ (𝐺‘𝑥) = (𝐹‘𝑥))
12		fnbrfvb 6959	. . . . . . . . 9 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥) = (𝐹‘𝑥) ↔ 𝑥𝐺(𝐹‘𝑥)))
13	11, 12	bitrid 283	. . . . . . . 8 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐺‘𝑥) ↔ 𝑥𝐺(𝐹‘𝑥)))
14	13	adantll 714	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐺‘𝑥) ↔ 𝑥𝐺(𝐹‘𝑥)))
15	14	necon3abid 2974	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) ≠ (𝐺‘𝑥) ↔ ¬ 𝑥𝐺(𝐹‘𝑥)))
16		fvex 6919	. . . . . . 7 ⊢ (𝐹‘𝑥) ∈ V
17		breq2 5151	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑥𝐺𝑦 ↔ 𝑥𝐺(𝐹‘𝑥)))
18	17	notbid 318	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑥) → (¬ 𝑥𝐺𝑦 ↔ ¬ 𝑥𝐺(𝐹‘𝑥)))
19	16, 18	ceqsexv 3529	. . . . . 6 ⊢ (∃𝑦(𝑦 = (𝐹‘𝑥) ∧ ¬ 𝑥𝐺𝑦) ↔ ¬ 𝑥𝐺(𝐹‘𝑥))
20	15, 19	bitr4di 289	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) ≠ (𝐺‘𝑥) ↔ ∃𝑦(𝑦 = (𝐹‘𝑥) ∧ ¬ 𝑥𝐺𝑦)))
21		eqcom 2741	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑦)
22		fnbrfvb 6959	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦))
23	21, 22	bitrid 283	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑦))
24	23	adantlr 715	. . . . . . . 8 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑦))
25	24	anbi1d 631	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝑦 = (𝐹‘𝑥) ∧ ¬ 𝑥𝐺𝑦) ↔ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐺𝑦)))
26		brdif 5200	. . . . . . 7 ⊢ (𝑥(𝐹 ∖ 𝐺)𝑦 ↔ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐺𝑦))
27	25, 26	bitr4di 289	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝑦 = (𝐹‘𝑥) ∧ ¬ 𝑥𝐺𝑦) ↔ 𝑥(𝐹 ∖ 𝐺)𝑦))
28	27	exbidv 1918	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → (∃𝑦(𝑦 = (𝐹‘𝑥) ∧ ¬ 𝑥𝐺𝑦) ↔ ∃𝑦 𝑥(𝐹 ∖ 𝐺)𝑦))
29	20, 28	bitr2d 280	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → (∃𝑦 𝑥(𝐹 ∖ 𝐺)𝑦 ↔ (𝐹‘𝑥) ≠ (𝐺‘𝑥)))
30	10, 29	bitrid 283	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ dom (𝐹 ∖ 𝐺) ↔ (𝐹‘𝑥) ≠ (𝐺‘𝑥)))
31	30	rabbi2dva 4233	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐴 ∩ dom (𝐹 ∖ 𝐺)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐺‘𝑥)})
32	8, 31	eqtr3d 2776	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∖ 𝐺) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐺‘𝑥)})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1536  ∃wex 1775   ∈ wcel 2105   ≠ wne 2937  {crab 3432   ∖ cdif 3959   ∩ cin 3961   ⊆ wss 3962   class class class wbr 5147  dom cdm 5688   Fn wfn 6557  ‘cfv 6562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fn 6565  df-fv 6570

This theorem is referenced by:  fndmdifcom  7062  fndmdifeq0  7063  fndifnfp  7195  wemapsolem  9587  wemapso2lem  9589  dsmmbas2  21774  frlmbas  21792  ptcmplem2  24076