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Theorem bj-fvmptunsn2 33743
 Description: Value of a function expressed as a union of a mapsto expression and a singleton on a couple (with disjoint domain) at a point in the domain of the mapsto construction. (Contributed by BJ, 18-Mar-2023.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-fvmptunsn.un (𝜑𝐹 = ((𝑥𝐴𝐵) ∪ {⟨𝐶, 𝐷⟩}))
bj-fvmptunsn.nel (𝜑 → ¬ 𝐶𝐴)
bj-fvmptunsn2.el (𝜑𝐸𝐴)
bj-fvmptunsn2.ex (𝜑𝐺𝑉)
bj-fvmptunsn2.es ((𝜑𝑥 = 𝐸) → 𝐵 = 𝐺)
Assertion
Ref Expression
bj-fvmptunsn2 (𝜑 → (𝐹𝐸) = 𝐺)
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝑥,𝐸   𝑥,𝐺
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝐷(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem bj-fvmptunsn2
StepHypRef Expression
1 bj-fvmptunsn.un . . 3 (𝜑𝐹 = ((𝑥𝐴𝐵) ∪ {⟨𝐶, 𝐷⟩}))
2 bj-fvmptunsn2.el . . . 4 (𝜑𝐸𝐴)
3 bj-fvmptunsn.nel . . . 4 (𝜑 → ¬ 𝐶𝐴)
4 nelneq 2883 . . . 4 ((𝐸𝐴 ∧ ¬ 𝐶𝐴) → ¬ 𝐸 = 𝐶)
52, 3, 4syl2anc 579 . . 3 (𝜑 → ¬ 𝐸 = 𝐶)
61, 5bj-fununsn1 33738 . 2 (𝜑 → (𝐹𝐸) = ((𝑥𝐴𝐵)‘𝐸))
7 eqidd 2779 . . 3 (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐵))
8 bj-fvmptunsn2.es . . 3 ((𝜑𝑥 = 𝐸) → 𝐵 = 𝐺)
9 bj-fvmptunsn2.ex . . 3 (𝜑𝐺𝑉)
107, 8, 2, 9fvmptd 6550 . 2 (𝜑 → ((𝑥𝐴𝐵)‘𝐸) = 𝐺)
116, 10eqtrd 2814 1 (𝜑 → (𝐹𝐸) = 𝐺)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107   ∪ cun 3790  {csn 4398  ⟨cop 4404   ↦ cmpt 4967  ‘cfv 6137 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fv 6145 This theorem is referenced by: (None)
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