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Theorem fnmptfvd 6213
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 2948 . . . 4 (𝜑 → ∀𝑎𝐴 𝐶𝑉)
4 eqid 2609 . . . . 5 (𝑎𝐴𝐶) = (𝑎𝐴𝐶)
54fnmpt 5919 . . . 4 (∀𝑎𝐴 𝐶𝑉 → (𝑎𝐴𝐶) Fn 𝐴)
63, 5syl 17 . . 3 (𝜑 → (𝑎𝐴𝐶) Fn 𝐴)
7 eqfnfv 6204 . . 3 ((𝑀 Fn 𝐴 ∧ (𝑎𝐴𝐶) Fn 𝐴) → (𝑀 = (𝑎𝐴𝐶) ↔ ∀𝑖𝐴 (𝑀𝑖) = ((𝑎𝐴𝐶)‘𝑖)))
81, 6, 7syl2anc 690 . 2 (𝜑 → (𝑀 = (𝑎𝐴𝐶) ↔ ∀𝑖𝐴 (𝑀𝑖) = ((𝑎𝐴𝐶)‘𝑖)))
9 fnmptfvd.s . . . . . . . 8 (𝑖 = 𝑎𝐷 = 𝐶)
109cbvmptv 4672 . . . . . . 7 (𝑖𝐴𝐷) = (𝑎𝐴𝐶)
1110eqcomi 2618 . . . . . 6 (𝑎𝐴𝐶) = (𝑖𝐴𝐷)
1211a1i 11 . . . . 5 (𝜑 → (𝑎𝐴𝐶) = (𝑖𝐴𝐷))
1312fveq1d 6090 . . . 4 (𝜑 → ((𝑎𝐴𝐶)‘𝑖) = ((𝑖𝐴𝐷)‘𝑖))
1413eqeq2d 2619 . . 3 (𝜑 → ((𝑀𝑖) = ((𝑎𝐴𝐶)‘𝑖) ↔ (𝑀𝑖) = ((𝑖𝐴𝐷)‘𝑖)))
1514ralbidv 2968 . 2 (𝜑 → (∀𝑖𝐴 (𝑀𝑖) = ((𝑎𝐴𝐶)‘𝑖) ↔ ∀𝑖𝐴 (𝑀𝑖) = ((𝑖𝐴𝐷)‘𝑖)))
16 simpr 475 . . . . 5 ((𝜑𝑖𝐴) → 𝑖𝐴)
17 fnmptfvd.d . . . . 5 ((𝜑𝑖𝐴) → 𝐷𝑈)
18 eqid 2609 . . . . . 6 (𝑖𝐴𝐷) = (𝑖𝐴𝐷)
1918fvmpt2 6185 . . . . 5 ((𝑖𝐴𝐷𝑈) → ((𝑖𝐴𝐷)‘𝑖) = 𝐷)
2016, 17, 19syl2anc 690 . . . 4 ((𝜑𝑖𝐴) → ((𝑖𝐴𝐷)‘𝑖) = 𝐷)
2120eqeq2d 2619 . . 3 ((𝜑𝑖𝐴) → ((𝑀𝑖) = ((𝑖𝐴𝐷)‘𝑖) ↔ (𝑀𝑖) = 𝐷))
2221ralbidva 2967 . 2 (𝜑 → (∀𝑖𝐴 (𝑀𝑖) = ((𝑖𝐴𝐷)‘𝑖) ↔ ∀𝑖𝐴 (𝑀𝑖) = 𝐷))
238, 15, 223bitrd 292 1 (𝜑 → (𝑀 = (𝑎𝐴𝐶) ↔ ∀𝑖𝐴 (𝑀𝑖) = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wral 2895  cmpt 4637   Fn wfn 5785  cfv 5790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-fv 5798
This theorem is referenced by:  cramerlem1  20260  dssmapnvod  37158
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