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Theorem fnmptfvd 5612
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 2548 . . . 4 (𝜑 → ∀𝑎𝐴 𝐶𝑉)
4 eqid 2175 . . . . 5 (𝑎𝐴𝐶) = (𝑎𝐴𝐶)
54fnmpt 5334 . . . 4 (∀𝑎𝐴 𝐶𝑉 → (𝑎𝐴𝐶) Fn 𝐴)
63, 5syl 14 . . 3 (𝜑 → (𝑎𝐴𝐶) Fn 𝐴)
7 eqfnfv 5605 . . 3 ((𝑀 Fn 𝐴 ∧ (𝑎𝐴𝐶) Fn 𝐴) → (𝑀 = (𝑎𝐴𝐶) ↔ ∀𝑖𝐴 (𝑀𝑖) = ((𝑎𝐴𝐶)‘𝑖)))
81, 6, 7syl2anc 411 . 2 (𝜑 → (𝑀 = (𝑎𝐴𝐶) ↔ ∀𝑖𝐴 (𝑀𝑖) = ((𝑎𝐴𝐶)‘𝑖)))
9 fnmptfvd.s . . . . . . . 8 (𝑖 = 𝑎𝐷 = 𝐶)
109cbvmptv 4094 . . . . . . 7 (𝑖𝐴𝐷) = (𝑎𝐴𝐶)
1110eqcomi 2179 . . . . . 6 (𝑎𝐴𝐶) = (𝑖𝐴𝐷)
1211a1i 9 . . . . 5 (𝜑 → (𝑎𝐴𝐶) = (𝑖𝐴𝐷))
1312fveq1d 5509 . . . 4 (𝜑 → ((𝑎𝐴𝐶)‘𝑖) = ((𝑖𝐴𝐷)‘𝑖))
1413eqeq2d 2187 . . 3 (𝜑 → ((𝑀𝑖) = ((𝑎𝐴𝐶)‘𝑖) ↔ (𝑀𝑖) = ((𝑖𝐴𝐷)‘𝑖)))
1514ralbidv 2475 . 2 (𝜑 → (∀𝑖𝐴 (𝑀𝑖) = ((𝑎𝐴𝐶)‘𝑖) ↔ ∀𝑖𝐴 (𝑀𝑖) = ((𝑖𝐴𝐷)‘𝑖)))
16 simpr 110 . . . . 5 ((𝜑𝑖𝐴) → 𝑖𝐴)
17 fnmptfvd.d . . . . 5 ((𝜑𝑖𝐴) → 𝐷𝑈)
18 eqid 2175 . . . . . 6 (𝑖𝐴𝐷) = (𝑖𝐴𝐷)
1918fvmpt2 5591 . . . . 5 ((𝑖𝐴𝐷𝑈) → ((𝑖𝐴𝐷)‘𝑖) = 𝐷)
2016, 17, 19syl2anc 411 . . . 4 ((𝜑𝑖𝐴) → ((𝑖𝐴𝐷)‘𝑖) = 𝐷)
2120eqeq2d 2187 . . 3 ((𝜑𝑖𝐴) → ((𝑀𝑖) = ((𝑖𝐴𝐷)‘𝑖) ↔ (𝑀𝑖) = 𝐷))
2221ralbidva 2471 . 2 (𝜑 → (∀𝑖𝐴 (𝑀𝑖) = ((𝑖𝐴𝐷)‘𝑖) ↔ ∀𝑖𝐴 (𝑀𝑖) = 𝐷))
238, 15, 223bitrd 214 1 (𝜑 → (𝑀 = (𝑎𝐴𝐶) ↔ ∀𝑖𝐴 (𝑀𝑖) = 𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2146  wral 2453  cmpt 4059   Fn wfn 5203  cfv 5208
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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-sbc 2961  df-csb 3056  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-iota 5170  df-fun 5210  df-fn 5211  df-fv 5216
This theorem is referenced by:  nninfdcinf  7159  nninfwlporlemd  7160  nninfwlporlem  7161
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