Theorem bj-mptval 37870

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 6669	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
3		fnbrfvb 6929	. . 3 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑋) = 𝑌 ↔ 𝑋(𝑥 ∈ 𝐴 ↦ 𝐵)𝑌))
4	3	ex 418	. 2 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 → (𝑋 ∈ 𝐴 → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑋) = 𝑌 ↔ 𝑋(𝑥 ∈ 𝐴 ↦ 𝐵)𝑌)))
5	2, 4	syl 18	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝑋 ∈ 𝐴 → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑋) = 𝑌 ↔ 𝑋(𝑥 ∈ 𝐴 ↦ 𝐵)𝑌)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2145  Ⅎwnfc 2907  ∀wral 3076   class class class wbr 5103   ↦ cmpt 5186   Fn wfn 6528  ‘cfv 6533

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6489  df-fun 6535  df-fn 6536  df-fv 6541

This theorem is used by: (None)