Theorem fnmptfvd 5804

Description: A function with a given domain is a mapping defined by its function values. (Contributed by AV, 1-Mar-2019.)

Hypotheses

Ref	Expression
fnmptfvd.m	⊢ (𝜑 → 𝑀 Fn 𝐴)
fnmptfvd.s	⊢ (𝑖 = 𝑎 → 𝐷 = 𝐶)
fnmptfvd.d	⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → 𝐷 ∈ 𝑈)
fnmptfvd.c	⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝐶 ∈ 𝑉)

Assertion

Ref	Expression
fnmptfvd	⊢ (𝜑 → (𝑀 = (𝑎 ∈ 𝐴 ↦ 𝐶) ↔ ∀𝑖 ∈ 𝐴 (𝑀‘𝑖) = 𝐷))

Distinct variable groups:   𝐴,𝑎,𝑖   𝐶,𝑖   𝐷,𝑎   𝑀,𝑎,𝑖   𝑈,𝑎,𝑖   𝑉,𝑎,𝑖   𝜑,𝑎,𝑖

Allowed substitution hints:   𝐶(𝑎)   𝐷(𝑖)

Proof of Theorem fnmptfvd

Step	Hyp	Ref	Expression
1		fnmptfvd.m	. . 3 ⊢ (𝜑 → 𝑀 Fn 𝐴)
2		fnmptfvd.c	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝐶 ∈ 𝑉)
3	2	ralrimiva 2623	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 𝐶 ∈ 𝑉)
4		eqid 2238	. . . . 5 ⊢ (𝑎 ∈ 𝐴 ↦ 𝐶) = (𝑎 ∈ 𝐴 ↦ 𝐶)
5	4	fnmpt 5505	. . . 4 ⊢ (∀𝑎 ∈ 𝐴 𝐶 ∈ 𝑉 → (𝑎 ∈ 𝐴 ↦ 𝐶) Fn 𝐴)
6	3, 5	syl 14	. . 3 ⊢ (𝜑 → (𝑎 ∈ 𝐴 ↦ 𝐶) Fn 𝐴)
7		eqfnfv 5797	. . 3 ⊢ ((𝑀 Fn 𝐴 ∧ (𝑎 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) → (𝑀 = (𝑎 ∈ 𝐴 ↦ 𝐶) ↔ ∀𝑖 ∈ 𝐴 (𝑀‘𝑖) = ((𝑎 ∈ 𝐴 ↦ 𝐶)‘𝑖)))
8	1, 6, 7	syl2anc 415	. 2 ⊢ (𝜑 → (𝑀 = (𝑎 ∈ 𝐴 ↦ 𝐶) ↔ ∀𝑖 ∈ 𝐴 (𝑀‘𝑖) = ((𝑎 ∈ 𝐴 ↦ 𝐶)‘𝑖)))
9		fnmptfvd.s	. . . . . . . 8 ⊢ (𝑖 = 𝑎 → 𝐷 = 𝐶)
10	9	cbvmptv 4222	. . . . . . 7 ⊢ (𝑖 ∈ 𝐴 ↦ 𝐷) = (𝑎 ∈ 𝐴 ↦ 𝐶)
11	10	eqcomi 2242	. . . . . 6 ⊢ (𝑎 ∈ 𝐴 ↦ 𝐶) = (𝑖 ∈ 𝐴 ↦ 𝐷)
12	11	a1i 9	. . . . 5 ⊢ (𝜑 → (𝑎 ∈ 𝐴 ↦ 𝐶) = (𝑖 ∈ 𝐴 ↦ 𝐷))
13	12	fveq1d 5692	. . . 4 ⊢ (𝜑 → ((𝑎 ∈ 𝐴 ↦ 𝐶)‘𝑖) = ((𝑖 ∈ 𝐴 ↦ 𝐷)‘𝑖))
14	13	eqeq2d 2250	. . 3 ⊢ (𝜑 → ((𝑀‘𝑖) = ((𝑎 ∈ 𝐴 ↦ 𝐶)‘𝑖) ↔ (𝑀‘𝑖) = ((𝑖 ∈ 𝐴 ↦ 𝐷)‘𝑖)))
15	14	ralbidv 2550	. 2 ⊢ (𝜑 → (∀𝑖 ∈ 𝐴 (𝑀‘𝑖) = ((𝑎 ∈ 𝐴 ↦ 𝐶)‘𝑖) ↔ ∀𝑖 ∈ 𝐴 (𝑀‘𝑖) = ((𝑖 ∈ 𝐴 ↦ 𝐷)‘𝑖)))
16		simpr 110	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → 𝑖 ∈ 𝐴)
17		fnmptfvd.d	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → 𝐷 ∈ 𝑈)
18		eqid 2238	. . . . . 6 ⊢ (𝑖 ∈ 𝐴 ↦ 𝐷) = (𝑖 ∈ 𝐴 ↦ 𝐷)
19	18	fvmpt2 5783	. . . . 5 ⊢ ((𝑖 ∈ 𝐴 ∧ 𝐷 ∈ 𝑈) → ((𝑖 ∈ 𝐴 ↦ 𝐷)‘𝑖) = 𝐷)
20	16, 17, 19	syl2anc 415	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ((𝑖 ∈ 𝐴 ↦ 𝐷)‘𝑖) = 𝐷)
21	20	eqeq2d 2250	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ((𝑀‘𝑖) = ((𝑖 ∈ 𝐴 ↦ 𝐷)‘𝑖) ↔ (𝑀‘𝑖) = 𝐷))
22	21	ralbidva 2546	. 2 ⊢ (𝜑 → (∀𝑖 ∈ 𝐴 (𝑀‘𝑖) = ((𝑖 ∈ 𝐴 ↦ 𝐷)‘𝑖) ↔ ∀𝑖 ∈ 𝐴 (𝑀‘𝑖) = 𝐷))
23	8, 15, 22	3bitrd 214	1 ⊢ (𝜑 → (𝑀 = (𝑎 ∈ 𝐴 ↦ 𝐶) ↔ ∀𝑖 ∈ 𝐴 (𝑀‘𝑖) = 𝐷))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ∀wral 2528   ↦ cmpt 4187   Fn wfn 5367  ‘cfv 5372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380

This theorem is referenced by:  nninfdcinf  7501  nninfwlporlemd  7502  nninfwlporlem  7503