Theorem fndmin 5672

Description: Two ways to express the locus of equality between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Assertion

Ref	Expression
fndmin	⊢ ((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 fndmin

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffn5im 5609	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
2		df-mpt 4097	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))}
3	1, 2	eqtrdi 2245	. . . . 5 ⊢ (𝐹 Fn 𝐴 → 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))})
4		dffn5im 5609	. . . . . 6 ⊢ (𝐺 Fn 𝐴 → 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥)))
5		df-mpt 4097	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))}
6	4, 5	eqtrdi 2245	. . . . 5 ⊢ (𝐺 Fn 𝐴 → 𝐺 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))})
7	3, 6	ineqan12d 3367	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 ∩ 𝐺) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))}))
8		inopab 4799	. . . 4 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))}) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)))}
9	7, 8	eqtrdi 2245	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 ∩ 𝐺) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)))})
10	9	dmeqd 4869	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∩ 𝐺) = dom {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)))})
11		anandi 590	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑦 = (𝐹‘𝑥) ∧ 𝑦 = (𝐺‘𝑥))) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))))
12	11	exbii 1619	. . . . . . 7 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 = (𝐹‘𝑥) ∧ 𝑦 = (𝐺‘𝑥))) ↔ ∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))))
13		19.42v 1921	. . . . . . 7 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 = (𝐹‘𝑥) ∧ 𝑦 = (𝐺‘𝑥))) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 = (𝐹‘𝑥) ∧ 𝑦 = (𝐺‘𝑥))))
14	12, 13	bitr3i 186	. . . . . 6 ⊢ (∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 = (𝐹‘𝑥) ∧ 𝑦 = (𝐺‘𝑥))))
15		funfvex 5578	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ V)
16		eqeq1 2203	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑦 = (𝐺‘𝑥) ↔ (𝐹‘𝑥) = (𝐺‘𝑥)))
17	16	ceqsexgv 2893	. . . . . . . . 9 ⊢ ((𝐹‘𝑥) ∈ V → (∃𝑦(𝑦 = (𝐹‘𝑥) ∧ 𝑦 = (𝐺‘𝑥)) ↔ (𝐹‘𝑥) = (𝐺‘𝑥)))
18	15, 17	syl 14	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (∃𝑦(𝑦 = (𝐹‘𝑥) ∧ 𝑦 = (𝐺‘𝑥)) ↔ (𝐹‘𝑥) = (𝐺‘𝑥)))
19	18	funfni 5361	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (∃𝑦(𝑦 = (𝐹‘𝑥) ∧ 𝑦 = (𝐺‘𝑥)) ↔ (𝐹‘𝑥) = (𝐺‘𝑥)))
20	19	pm5.32da 452	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → ((𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 = (𝐹‘𝑥) ∧ 𝑦 = (𝐺‘𝑥))) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = (𝐺‘𝑥))))
21	14, 20	bitrid 192	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = (𝐺‘𝑥))))
22	21	abbidv 2314	. . . 4 ⊢ (𝐹 Fn 𝐴 → {𝑥 ∣ ∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)))} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = (𝐺‘𝑥))})
23		dmopab 4878	. . . 4 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)))} = {𝑥 ∣ ∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)))}
24		df-rab 2484	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = (𝐺‘𝑥))}
25	22, 23, 24	3eqtr4g 2254	. . 3 ⊢ (𝐹 Fn 𝐴 → dom {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)))} = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)})
26	25	adantr 276	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)))} = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)})
27	10, 26	eqtrd 2229	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∩ 𝐺) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)})

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364  ∃wex 1506   ∈ wcel 2167  {cab 2182  {crab 2479  Vcvv 2763   ∩ cin 3156  {copab 4094   ↦ cmpt 4095  dom cdm 4664  Fun wfun 5253   Fn wfn 5254  ‘cfv 5259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-iota 5220  df-fun 5261  df-fn 5262  df-fv 5267

This theorem is referenced by:  fneqeql  5673  mhmeql  13194  ghmeql  13473