Theorem fvmptf 7012

Description: Value of a function given by an ordered-pair class abstraction. This version of fvmptg 6988 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
fvmptf.1	⊢ Ⅎ𝑥𝐴
fvmptf.2	⊢ Ⅎ𝑥𝐶
fvmptf.3	⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
fvmptf.4	⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)

Assertion

Ref	Expression
fvmptf	⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉) → (𝐹‘𝐴) = 𝐶)

Distinct variable group:   𝑥,𝐷

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem fvmptf

Step	Hyp	Ref	Expression
1		fvmptf.1	. . 3 ⊢ Ⅎ𝑥𝐴
2		fvmptf.2	. . . . 5 ⊢ Ⅎ𝑥𝐶
3	2	nfel1 2940	. . . 4 ⊢ Ⅎ𝑥 𝐶 ∈ V
4		fvmptf.4	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)
5		nfmpt1 5208	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐷 ↦ 𝐵)
6	4, 5	nfcxfr 2922	. . . . . 6 ⊢ Ⅎ𝑥𝐹
7	6, 1	nffv 6892	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝐴)
8	7, 2	nfeq 2937	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝐴) = 𝐶
9	3, 8	nfim 1929	. . 3 ⊢ Ⅎ𝑥(𝐶 ∈ V → (𝐹‘𝐴) = 𝐶)
10		fvmptf.3	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
11	10	eleq1d 2847	. . . 4 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ V ↔ 𝐶 ∈ V))
12		fveq2 6882	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
13	12, 10	eqeq12d 2778	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) = 𝐵 ↔ (𝐹‘𝐴) = 𝐶))
14	11, 13	imbi12d 347	. . 3 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ V → (𝐹‘𝑥) = 𝐵) ↔ (𝐶 ∈ V → (𝐹‘𝐴) = 𝐶)))
15	4	fvmpt2 7002	. . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝐵 ∈ V) → (𝐹‘𝑥) = 𝐵)
16	15	ex 418	. . 3 ⊢ (𝑥 ∈ 𝐷 → (𝐵 ∈ V → (𝐹‘𝑥) = 𝐵))
17	1, 9, 14, 16	vtoclgaf 3538	. 2 ⊢ (𝐴 ∈ 𝐷 → (𝐶 ∈ V → (𝐹‘𝐴) = 𝐶))
18		elex 3474	. 2 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V)
19	17, 18	impel 515	1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉) → (𝐹‘𝐴) = 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Ⅎwnfc 2909  Vcvv 3453   ↦ cmpt 5190  ‘cfv 6537

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-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fv 6545

This theorem is used by:  fvmptnf  7013  elfvmptrab1w  7018  elfvmptrab1  7019  elovmpt3rab1  7678  rdgsucmptf  8421  frsucmpt  8431  fprodntriv  16035  prodss  16040  fprodefsum  16187  dvfsumabs  26257  dvfsumlem1  26260  dvfsumlem4  26263  dvfsum2  26268  dchrisumlem2  27734  dchrisumlem3  27735  rmfsupp2  33685  ptrest  38376  hlhilset  42815  orbitclmpt  45789  fsumsermpt  46417  mulc1cncfg  46427  expcnfg  46429  climsubmpt  46496  climeldmeqmpt  46504  climfveqmpt  46507  fnlimfvre  46510  climfveqmpt3  46518  climeldmeqmpt3  46525  climinf2mpt  46550  climinfmpt  46551  stoweidlem23  46859  stoweidlem34  46870  stoweidlem36  46872  wallispilem5  46905  stirlinglem4  46913  stirlinglem11  46920  stirlinglem12  46921  stirlinglem13  46922  stirlinglem14  46923  sge0lempt  47246  sge0isummpt2  47268  meadjiun  47302  hoimbl2  47501  vonhoire  47508