Theorem fvmptg 5169

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 2018	. 2 ⊢ 𝐶 = 𝐶
2		fvmptg.1	. . . 4 ⊢ (x = A → B = 𝐶)
3	2	eqeq2d 2029	. . 3 ⊢ (x = A → (y = B ↔ y = 𝐶))
4		eqeq1 2024	. . 3 ⊢ (y = 𝐶 → (y = 𝐶 ↔ 𝐶 = 𝐶))
5		moeq 2689	. . . 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 3790	. . . 4 ⊢ (x ∈ 𝐷 ↦ B) = {⟨x, y⟩ ∣ (x ∈ 𝐷 ∧ y = B)}
9	7, 8	eqtri 2038	. . 3 ⊢ 𝐹 = {⟨x, y⟩ ∣ (x ∈ 𝐷 ∧ y = B)}
10	3, 4, 6, 9	fvopab3ig 5167	. 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 1226   ∈ wcel 1370  ∃*wmo 1879  {copab 3787   ↦ cmpt 3788  ‘cfv 4825

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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914

This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-sbc 2738  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-opab 3789  df-mpt 3790  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-iota 4790  df-fun 4827  df-fv 4833

This theorem is referenced by:  fvmpt  5170  fvmpts  5171  fvmpt3  5172  fvmpt2  5175  f1mpt  5331  fnofval  5640  caofinvl  5652  1stvalg  5688  2ndvalg  5689  brtpos2  5784  rdgi0g  5882  frec0g  5898  frecsuclem3  5902  sucinc  5936  sucinc2  5937  omcl  5952  oeicl  5953  oav2  5954  omv2  5956