Theorem fvmptg 5191

Description: Value of a function given in maps-to notation. (Contributed by NM, 2-Oct-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)

Hypotheses

Ref	Expression
fvmptg.1	⊢ (x = A → B = 𝐶)
fvmptg.2	⊢ 𝐹 = (x ∈ 𝐷 ↦ B)

Assertion

Ref	Expression
fvmptg	⊢ ((A ∈ 𝐷 ∧ 𝐶 ∈ 𝑅) → (𝐹‘A) = 𝐶)

Distinct variable groups:   x,A   x,𝐶   x,𝐷

Allowed substitution hints:   B(x)   𝑅(x)   𝐹(x)

Proof of Theorem fvmptg

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2037	. 2 ⊢ 𝐶 = 𝐶
2		fvmptg.1	. . . 4 ⊢ (x = A → B = 𝐶)
3	2	eqeq2d 2048	. . 3 ⊢ (x = A → (y = B ↔ y = 𝐶))
4		eqeq1 2043	. . 3 ⊢ (y = 𝐶 → (y = 𝐶 ↔ 𝐶 = 𝐶))
5		moeq 2710	. . . 4 ⊢ ∃*y y = B
6	5	a1i 9	. . 3 ⊢ (x ∈ 𝐷 → ∃*y y = B)
7		fvmptg.2	. . . 4 ⊢ 𝐹 = (x ∈ 𝐷 ↦ B)
8		df-mpt 3811	. . . 4 ⊢ (x ∈ 𝐷 ↦ B) = {⟨x, y⟩ ∣ (x ∈ 𝐷 ∧ y = B)}
9	7, 8	eqtri 2057	. . 3 ⊢ 𝐹 = {⟨x, y⟩ ∣ (x ∈ 𝐷 ∧ y = B)}
10	3, 4, 6, 9	fvopab3ig 5189	. 2 ⊢ ((A ∈ 𝐷 ∧ 𝐶 ∈ 𝑅) → (𝐶 = 𝐶 → (𝐹‘A) = 𝐶))
11	1, 10	mpi 15	1 ⊢ ((A ∈ 𝐷 ∧ 𝐶 ∈ 𝑅) → (𝐹‘A) = 𝐶)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390  ∃*wmo 1898  {copab 3808   ↦ cmpt 3809  ‘cfv 4845

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fv 4853

This theorem is referenced by:  fvmpt  5192  fvmpts  5193  fvmpt3  5194  fvmpt2  5197  f1mpt  5353  fnofval  5663  caofinvl  5675  1stvalg  5711  2ndvalg  5712  brtpos2  5807  frec0g  5922  frecsuclem3  5929  sucinc  5964  sucinc2  5965  omcl  5980  oeicl  5981  oav2  5982  omv2  5984  cjval  9053  reval  9057  imval  9058  absval  9190