Theorem fndmin 6801

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		dffn5 6710	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
2	1	biimpi 218	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
3		df-mpt 5133	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))}
4	2, 3	syl6eq 2872	. . . . 5 ⊢ (𝐹 Fn 𝐴 → 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))})
5		dffn5 6710	. . . . . . 7 ⊢ (𝐺 Fn 𝐴 ↔ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥)))
6	5	biimpi 218	. . . . . 6 ⊢ (𝐺 Fn 𝐴 → 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥)))
7		df-mpt 5133	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))}
8	6, 7	syl6eq 2872	. . . . 5 ⊢ (𝐺 Fn 𝐴 → 𝐺 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))})
9	4, 8	ineqan12d 4179	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 ∩ 𝐺) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))}))
10		inopab 5687	. . . 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)))}
11	9, 10	syl6eq 2872	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 ∩ 𝐺) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)))})
12	11	dmeqd 5760	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∩ 𝐺) = dom {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)))})
13		19.42v 1954	. . . . 5 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 = (𝐹‘𝑥) ∧ 𝑦 = (𝐺‘𝑥))) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 = (𝐹‘𝑥) ∧ 𝑦 = (𝐺‘𝑥))))
14		anandi 674	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑦 = (𝐹‘𝑥) ∧ 𝑦 = (𝐺‘𝑥))) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))))
15	14	exbii 1848	. . . . 5 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 = (𝐹‘𝑥) ∧ 𝑦 = (𝐺‘𝑥))) ↔ ∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))))
16		fvex 6669	. . . . . . 7 ⊢ (𝐹‘𝑥) ∈ V
17		eqeq1 2825	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑦 = (𝐺‘𝑥) ↔ (𝐹‘𝑥) = (𝐺‘𝑥)))
18	16, 17	ceqsexv 3533	. . . . . 6 ⊢ (∃𝑦(𝑦 = (𝐹‘𝑥) ∧ 𝑦 = (𝐺‘𝑥)) ↔ (𝐹‘𝑥) = (𝐺‘𝑥))
19	18	anbi2i 624	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 = (𝐹‘𝑥) ∧ 𝑦 = (𝐺‘𝑥))) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = (𝐺‘𝑥)))
20	13, 15, 19	3bitr3i 303	. . . 4 ⊢ (∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = (𝐺‘𝑥)))
21	20	abbii 2886	. . 3 ⊢ {𝑥 ∣ ∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)))} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = (𝐺‘𝑥))}
22		dmopab 5770	. . 3 ⊢ 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)))}
23		df-rab 3147	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = (𝐺‘𝑥))}
24	21, 22, 23	3eqtr4i 2854	. 2 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)))} = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)}
25	12, 24	syl6eq 2872	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∩ 𝐺) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)})

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114  {cab 2799  {crab 3142   ∩ cin 3923  {copab 5114   ↦ cmpt 5132  dom cdm 5541   Fn wfn 6336  ‘cfv 6341

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5189  ax-nul 5196  ax-pr 5316

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3488  df-sbc 3764  df-dif 3927  df-un 3929  df-in 3931  df-ss 3940  df-nul 4280  df-if 4454  df-sn 4554  df-pr 4556  df-op 4560  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5446  df-xp 5547  df-rel 5548  df-cnv 5549  df-co 5550  df-dm 5551  df-iota 6300  df-fun 6343  df-fn 6344  df-fv 6349

This theorem is referenced by:  fneqeql  6802  fninfp  6922  mhmeql  17973  ghmeql  18364  lmhmeql  19810  hauseqlcld  22237  cvmliftmolem1  32535  cvmliftmolem2  32536  hausgraph  39904  mgmhmeql  44155