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Theorem bj-fvmptunsn2 34634
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 2938 . . . 4 ((𝐸𝐴 ∧ ¬ 𝐶𝐴) → ¬ 𝐸 = 𝐶)
52, 3, 4syl2anc 587 . . 3 (𝜑 → ¬ 𝐸 = 𝐶)
61, 5bj-fununsn1 34629 . 2 (𝜑 → (𝐹𝐸) = ((𝑥𝐴𝐵)‘𝐸))
7 eqidd 2823 . . 3 (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐵))
8 bj-fvmptunsn2.is . . 3 ((𝜑𝑥 = 𝐸) → 𝐵 = 𝐺)
9 bj-fvmptunsn2.ex . . 3 (𝜑𝐺𝑉)
107, 8, 2, 9fvmptd 6757 . 2 (𝜑 → ((𝑥𝐴𝐵)‘𝐸) = 𝐺)
116, 10eqtrd 2857 1 (𝜑 → (𝐹𝐸) = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2114  cun 3906  {csn 4539  cop 4545  cmpt 5122  cfv 6334
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307
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 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fv 6342
This theorem is referenced by: (None)
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