Theorem fndmdif 7033

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 4083	. . . . 5 ⊢ (𝐹 ∖ 𝐺) ⊆ 𝐹
2		dmss 5884	. . . . 5 ⊢ ((𝐹 ∖ 𝐺) ⊆ 𝐹 → dom (𝐹 ∖ 𝐺) ⊆ dom 𝐹)
3	1, 2	ax-mp 5	. . . 4 ⊢ dom (𝐹 ∖ 𝐺) ⊆ dom 𝐹
4		fndm 6634	. . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
5	4	adantr 486	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom 𝐹 = 𝐴)
6	3, 5	sseqtrid 3973	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∖ 𝐺) ⊆ 𝐴)
7		sseqin2 4169	. . 3 ⊢ (dom (𝐹 ∖ 𝐺) ⊆ 𝐴 ↔ (𝐴 ∩ dom (𝐹 ∖ 𝐺)) = dom (𝐹 ∖ 𝐺))
8	6, 7	sylib 221	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐴 ∩ dom (𝐹 ∖ 𝐺)) = dom (𝐹 ∖ 𝐺))
9		vex 3455	. . . . 5 ⊢ 𝑥 ∈ V
10	9	eldm 5882	. . . 4 ⊢ (𝑥 ∈ dom (𝐹 ∖ 𝐺) ↔ ∃𝑦 𝑥(𝐹 ∖ 𝐺)𝑦)
11		eqcom 2768	. . . . . . . . 9 ⊢ ((𝐹‘𝑥) = (𝐺‘𝑥) ↔ (𝐺‘𝑥) = (𝐹‘𝑥))
12		fnbrfvb 6927	. . . . . . . . 9 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥) = (𝐹‘𝑥) ↔ 𝑥𝐺(𝐹‘𝑥)))
13	11, 12	bitrid 286	. . . . . . . 8 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐺‘𝑥) ↔ 𝑥𝐺(𝐹‘𝑥)))
14	13	adantll 727	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐺‘𝑥) ↔ 𝑥𝐺(𝐹‘𝑥)))
15	14	necon3abid 2992	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) ≠ (𝐺‘𝑥) ↔ ¬ 𝑥𝐺(𝐹‘𝑥)))
16		fvex 6890	. . . . . . 7 ⊢ (𝐹‘𝑥) ∈ V
17		breq2 5107	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑥𝐺𝑦 ↔ 𝑥𝐺(𝐹‘𝑥)))
18	17	notbid 321	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑥) → (¬ 𝑥𝐺𝑦 ↔ ¬ 𝑥𝐺(𝐹‘𝑥)))
19	16, 18	ceqsexv 3499	. . . . . 6 ⊢ (∃𝑦(𝑦 = (𝐹‘𝑥) ∧ ¬ 𝑥𝐺𝑦) ↔ ¬ 𝑥𝐺(𝐹‘𝑥))
20	15, 19	bitr4di 292	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) ≠ (𝐺‘𝑥) ↔ ∃𝑦(𝑦 = (𝐹‘𝑥) ∧ ¬ 𝑥𝐺𝑦)))
21		eqcom 2768	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑦)
22		fnbrfvb 6927	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦))
23	21, 22	bitrid 286	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑦))
24	23	adantlr 728	. . . . . . . 8 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑦))
25	24	anbi1d 643	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝑦 = (𝐹‘𝑥) ∧ ¬ 𝑥𝐺𝑦) ↔ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐺𝑦)))
26		brdif 5158	. . . . . . 7 ⊢ (𝑥(𝐹 ∖ 𝐺)𝑦 ↔ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐺𝑦))
27	25, 26	bitr4di 292	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝑦 = (𝐹‘𝑥) ∧ ¬ 𝑥𝐺𝑦) ↔ 𝑥(𝐹 ∖ 𝐺)𝑦))
28	27	exbidv 1954	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → (∃𝑦(𝑦 = (𝐹‘𝑥) ∧ ¬ 𝑥𝐺𝑦) ↔ ∃𝑦 𝑥(𝐹 ∖ 𝐺)𝑦))
29	20, 28	bitr2d 283	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → (∃𝑦 𝑥(𝐹 ∖ 𝐺)𝑦 ↔ (𝐹‘𝑥) ≠ (𝐺‘𝑥)))
30	10, 29	bitrid 286	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ dom (𝐹 ∖ 𝐺) ↔ (𝐹‘𝑥) ≠ (𝐺‘𝑥)))
31	30	rabbi2dva 4171	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐴 ∩ dom (𝐹 ∖ 𝐺)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐺‘𝑥)})
32	8, 31	eqtr3d 2798	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∖ 𝐺) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐺‘𝑥)})

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145   ≠ wne 2956  {crab 3413   ∖ cdif 3896   ∩ cin 3898   ⊆ wss 3899   class class class wbr 5103  dom cdm 5651   Fn wfn 6526  ‘cfv 6531

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6487  df-fun 6533  df-fn 6534  df-fv 6539

This theorem is used by:  fndmdifcom  7034  fndmdifeq0  7035  fndifnfp  7173  wemapsolem  9528  wemapso2lem  9530  dsmmbas2  22023  frlmbas  22041  ptcmplem2  24352