Theorem bj-mptval 34532

Description: Value of a function given in maps-to notation. (Contributed by BJ, 30-Dec-2020.)

Hypothesis

Ref	Expression
bj-mptval.nf	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
bj-mptval	⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝑋 ∈ 𝐴 → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑋) = 𝑌 ↔ 𝑋(𝑥 ∈ 𝐴 ↦ 𝐵)𝑌)))

Proof of Theorem bj-mptval

Step	Hyp	Ref	Expression
1		bj-mptval.nf	. . 3 ⊢ Ⅎ𝑥𝐴
2	1	fnmptf 6456	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
3		fnbrfvb 6693	. . 3 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑋) = 𝑌 ↔ 𝑋(𝑥 ∈ 𝐴 ↦ 𝐵)𝑌))
4	3	ex 416	. 2 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 → (𝑋 ∈ 𝐴 → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑋) = 𝑌 ↔ 𝑋(𝑥 ∈ 𝐴 ↦ 𝐵)𝑌)))
5	2, 4	syl 17	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝑋 ∈ 𝐴 → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑋) = 𝑌 ↔ 𝑋(𝑥 ∈ 𝐴 ↦ 𝐵)𝑌)))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  Ⅎwnfc 2936  ∀wral 3106   class class class wbr 5030   ↦ cmpt 5110   Fn wfn 6319  ‘cfv 6324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fn 6327  df-fv 6332

This theorem is referenced by: (None)