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Theorem bj-fvmptunsn2 36868
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.is ((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 2849 . . . 4 ((𝐸𝐴 ∧ ¬ 𝐶𝐴) → ¬ 𝐸 = 𝐶)
52, 3, 4syl2anc 582 . . 3 (𝜑 → ¬ 𝐸 = 𝐶)
61, 5bj-fununsn1 36863 . 2 (𝜑 → (𝐹𝐸) = ((𝑥𝐴𝐵)‘𝐸))
7 eqidd 2726 . . 3 (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐵))
8 bj-fvmptunsn2.is . . 3 ((𝜑𝑥 = 𝐸) → 𝐵 = 𝐺)
9 bj-fvmptunsn2.ex . . 3 (𝜑𝐺𝑉)
107, 8, 2, 9fvmptd 7011 . 2 (𝜑 → ((𝑥𝐴𝐵)‘𝐸) = 𝐺)
116, 10eqtrd 2765 1 (𝜑 → (𝐹𝐸) = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1533  wcel 2098  cun 3942  {csn 4630  cop 4636  cmpt 5232  cfv 6549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fv 6557
This theorem is referenced by: (None)
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