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Theorem fvmpt4 39973
Description: Value of a function given by the "maps to" notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Assertion
Ref Expression
fvmpt4 ((𝑥𝐴𝐵𝐶) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem fvmpt4
StepHypRef Expression
1 simpl 469 . 2 ((𝑥𝐴𝐵𝐶) → 𝑥𝐴)
2 simpr 472 . 2 ((𝑥𝐴𝐵𝐶) → 𝐵𝐶)
3 eqid 2774 . . 3 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
43fvmpt2 6450 . 2 ((𝑥𝐴𝐵𝐶) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
51, 2, 4syl2anc 574 1 ((𝑥𝐴𝐵𝐶) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1634  wcel 2148  cmpt 4876  cfv 6042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-sep 4928  ax-nul 4936  ax-pow 4988  ax-pr 5048
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3357  df-sbc 3594  df-csb 3689  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-nul 4074  df-if 4236  df-sn 4327  df-pr 4329  df-op 4333  df-uni 4586  df-br 4798  df-opab 4860  df-mpt 4877  df-id 5171  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-rn 5274  df-res 5275  df-ima 5276  df-iota 6005  df-fun 6044  df-fv 6050
This theorem is referenced by:  climeldmeqmpt2  40451  liminfltlem  40560
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