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Theorem fnmptfvd 6546
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 (𝜑𝑀 Fn 𝐴)
fnmptfvd.s (𝑖 = 𝑎𝐷 = 𝐶)
fnmptfvd.d ((𝜑𝑖𝐴) → 𝐷𝑈)
fnmptfvd.c ((𝜑𝑎𝐴) → 𝐶𝑉)
Assertion
Ref Expression
fnmptfvd (𝜑 → (𝑀 = (𝑎𝐴𝐶) ↔ ∀𝑖𝐴 (𝑀𝑖) = 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
StepHypRef Expression
1 fnmptfvd.m . . 3 (𝜑𝑀 Fn 𝐴)
2 fnmptfvd.c . . . . 5 ((𝜑𝑎𝐴) → 𝐶𝑉)
32ralrimiva 3147 . . . 4 (𝜑 → ∀𝑎𝐴 𝐶𝑉)
4 eqid 2799 . . . . 5 (𝑎𝐴𝐶) = (𝑎𝐴𝐶)
54fnmpt 6231 . . . 4 (∀𝑎𝐴 𝐶𝑉 → (𝑎𝐴𝐶) Fn 𝐴)
63, 5syl 17 . . 3 (𝜑 → (𝑎𝐴𝐶) Fn 𝐴)
7 eqfnfv 6537 . . 3 ((𝑀 Fn 𝐴 ∧ (𝑎𝐴𝐶) Fn 𝐴) → (𝑀 = (𝑎𝐴𝐶) ↔ ∀𝑖𝐴 (𝑀𝑖) = ((𝑎𝐴𝐶)‘𝑖)))
81, 6, 7syl2anc 580 . 2 (𝜑 → (𝑀 = (𝑎𝐴𝐶) ↔ ∀𝑖𝐴 (𝑀𝑖) = ((𝑎𝐴𝐶)‘𝑖)))
9 fnmptfvd.s . . . . . . . 8 (𝑖 = 𝑎𝐷 = 𝐶)
109cbvmptv 4943 . . . . . . 7 (𝑖𝐴𝐷) = (𝑎𝐴𝐶)
1110eqcomi 2808 . . . . . 6 (𝑎𝐴𝐶) = (𝑖𝐴𝐷)
1211a1i 11 . . . . 5 (𝜑 → (𝑎𝐴𝐶) = (𝑖𝐴𝐷))
1312fveq1d 6413 . . . 4 (𝜑 → ((𝑎𝐴𝐶)‘𝑖) = ((𝑖𝐴𝐷)‘𝑖))
1413eqeq2d 2809 . . 3 (𝜑 → ((𝑀𝑖) = ((𝑎𝐴𝐶)‘𝑖) ↔ (𝑀𝑖) = ((𝑖𝐴𝐷)‘𝑖)))
1514ralbidv 3167 . 2 (𝜑 → (∀𝑖𝐴 (𝑀𝑖) = ((𝑎𝐴𝐶)‘𝑖) ↔ ∀𝑖𝐴 (𝑀𝑖) = ((𝑖𝐴𝐷)‘𝑖)))
16 simpr 478 . . . . 5 ((𝜑𝑖𝐴) → 𝑖𝐴)
17 fnmptfvd.d . . . . 5 ((𝜑𝑖𝐴) → 𝐷𝑈)
18 eqid 2799 . . . . . 6 (𝑖𝐴𝐷) = (𝑖𝐴𝐷)
1918fvmpt2 6516 . . . . 5 ((𝑖𝐴𝐷𝑈) → ((𝑖𝐴𝐷)‘𝑖) = 𝐷)
2016, 17, 19syl2anc 580 . . . 4 ((𝜑𝑖𝐴) → ((𝑖𝐴𝐷)‘𝑖) = 𝐷)
2120eqeq2d 2809 . . 3 ((𝜑𝑖𝐴) → ((𝑀𝑖) = ((𝑖𝐴𝐷)‘𝑖) ↔ (𝑀𝑖) = 𝐷))
2221ralbidva 3166 . 2 (𝜑 → (∀𝑖𝐴 (𝑀𝑖) = ((𝑖𝐴𝐷)‘𝑖) ↔ ∀𝑖𝐴 (𝑀𝑖) = 𝐷))
238, 15, 223bitrd 297 1 (𝜑 → (𝑀 = (𝑎𝐴𝐶) ↔ ∀𝑖𝐴 (𝑀𝑖) = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wral 3089  cmpt 4922   Fn wfn 6096  cfv 6101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-fv 6109
This theorem is referenced by:  cramerlem1  20821  dssmapnvod  39096
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