Theorem fnmptfvd 6320

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 2966	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 𝐶 ∈ 𝑉)
4		eqid 2622	. . . . 5 ⊢ (𝑎 ∈ 𝐴 ↦ 𝐶) = (𝑎 ∈ 𝐴 ↦ 𝐶)
5	4	fnmpt 6020	. . . 4 ⊢ (∀𝑎 ∈ 𝐴 𝐶 ∈ 𝑉 → (𝑎 ∈ 𝐴 ↦ 𝐶) Fn 𝐴)
6	3, 5	syl 17	. . 3 ⊢ (𝜑 → (𝑎 ∈ 𝐴 ↦ 𝐶) Fn 𝐴)
7		eqfnfv 6311	. . 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 693	. 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 4750	. . . . . . 7 ⊢ (𝑖 ∈ 𝐴 ↦ 𝐷) = (𝑎 ∈ 𝐴 ↦ 𝐶)
11	10	eqcomi 2631	. . . . . 6 ⊢ (𝑎 ∈ 𝐴 ↦ 𝐶) = (𝑖 ∈ 𝐴 ↦ 𝐷)
12	11	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑎 ∈ 𝐴 ↦ 𝐶) = (𝑖 ∈ 𝐴 ↦ 𝐷))
13	12	fveq1d 6193	. . . 4 ⊢ (𝜑 → ((𝑎 ∈ 𝐴 ↦ 𝐶)‘𝑖) = ((𝑖 ∈ 𝐴 ↦ 𝐷)‘𝑖))
14	13	eqeq2d 2632	. . 3 ⊢ (𝜑 → ((𝑀‘𝑖) = ((𝑎 ∈ 𝐴 ↦ 𝐶)‘𝑖) ↔ (𝑀‘𝑖) = ((𝑖 ∈ 𝐴 ↦ 𝐷)‘𝑖)))
15	14	ralbidv 2986	. 2 ⊢ (𝜑 → (∀𝑖 ∈ 𝐴 (𝑀‘𝑖) = ((𝑎 ∈ 𝐴 ↦ 𝐶)‘𝑖) ↔ ∀𝑖 ∈ 𝐴 (𝑀‘𝑖) = ((𝑖 ∈ 𝐴 ↦ 𝐷)‘𝑖)))
16		simpr 477	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → 𝑖 ∈ 𝐴)
17		fnmptfvd.d	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → 𝐷 ∈ 𝑈)
18		eqid 2622	. . . . . 6 ⊢ (𝑖 ∈ 𝐴 ↦ 𝐷) = (𝑖 ∈ 𝐴 ↦ 𝐷)
19	18	fvmpt2 6291	. . . . 5 ⊢ ((𝑖 ∈ 𝐴 ∧ 𝐷 ∈ 𝑈) → ((𝑖 ∈ 𝐴 ↦ 𝐷)‘𝑖) = 𝐷)
20	16, 17, 19	syl2anc 693	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ((𝑖 ∈ 𝐴 ↦ 𝐷)‘𝑖) = 𝐷)
21	20	eqeq2d 2632	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ((𝑀‘𝑖) = ((𝑖 ∈ 𝐴 ↦ 𝐷)‘𝑖) ↔ (𝑀‘𝑖) = 𝐷))
22	21	ralbidva 2985	. 2 ⊢ (𝜑 → (∀𝑖 ∈ 𝐴 (𝑀‘𝑖) = ((𝑖 ∈ 𝐴 ↦ 𝐷)‘𝑖) ↔ ∀𝑖 ∈ 𝐴 (𝑀‘𝑖) = 𝐷))
23	8, 15, 22	3bitrd 294	1 ⊢ (𝜑 → (𝑀 = (𝑎 ∈ 𝐴 ↦ 𝐶) ↔ ∀𝑖 ∈ 𝐴 (𝑀‘𝑖) = 𝐷))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912   ↦ cmpt 4729   Fn wfn 5883  ‘cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  cramerlem1  20493  dssmapnvod  38314