Theorem fvmptf 7015

Description: Value of a function given by an ordered-pair class abstraction. This version of fvmptg 6991 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 2938	. . . 4 ⊢ Ⅎ𝑥 𝐶 ∈ V
4		fvmptf.4	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)
5		nfmpt1 5203	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐷 ↦ 𝐵)
6	4, 5	nfcxfr 2920	. . . . . 6 ⊢ Ⅎ𝑥𝐹
7	6, 1	nffv 6895	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝐴)
8	7, 2	nfeq 2935	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝐴) = 𝐶
9	3, 8	nfim 1928	. . 3 ⊢ Ⅎ𝑥(𝐶 ∈ V → (𝐹‘𝐴) = 𝐶)
10		fvmptf.3	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
11	10	eleq1d 2845	. . . 4 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ V ↔ 𝐶 ∈ V))
12		fveq2 6885	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
13	12, 10	eqeq12d 2776	. . . 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 7005	. . . 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 3535	. 2 ⊢ (𝐴 ∈ 𝐷 → (𝐶 ∈ V → (𝐹‘𝐴) = 𝐶))
18		elex 3471	. 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 1569   ∈ wcel 2144  Ⅎwnfc 2907  Vcvv 3450   ↦ cmpt 5185  ‘cfv 6538

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fv 6546

This theorem is used by:  fvmptnf  7016  elfvmptrab1w  7021  elfvmptrab1  7022  elovmpt3rab1  7681  rdgsucmptf  8439  frsucmpt  8449  fprodntriv  16114  prodss  16119  fprodefsum  16266  dvfsumabs  26348  dvfsumlem1  26351  dvfsumlem4  26354  dvfsum2  26359  dchrisumlem2  27822  dchrisumlem3  27823  rmfsupp2  33844  ptrest  38630  hlhilset  43084  orbitclmpt  46041  fsumsermpt  46684  mulc1cncfg  46694  expcnfg  46696  climsubmpt  46763  climeldmeqmpt  46771  climfveqmpt  46774  fnlimfvre  46777  climfveqmpt3  46785  climeldmeqmpt3  46792  climinf2mpt  46817  climinfmpt  46818  stoweidlem23  47126  stoweidlem34  47137  stoweidlem36  47139  wallispilem5  47172  stirlinglem4  47180  stirlinglem11  47187  stirlinglem12  47188  stirlinglem13  47189  stirlinglem14  47190  sge0lempt  47513  sge0isummpt2  47535  meadjiun  47569  hoimbl2  47768  vonhoire  47775