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Theorem iuneqconst 5008
Description: Indexed union of identical classes. (Contributed by AV, 5-Mar-2024.)
Hypothesis
Ref Expression
iuneqconst.p (𝑥 = 𝑋𝐵 = 𝐶)
Assertion
Ref Expression
iuneqconst ((𝑋𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶) → 𝑥𝐴 𝐵 = 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝑋
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem iuneqconst
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eliun 5001 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
2 iuneqconst.p . . . . . . . 8 (𝑥 = 𝑋𝐵 = 𝐶)
32eleq2d 2820 . . . . . . 7 (𝑥 = 𝑋 → (𝑦𝐵𝑦𝐶))
43rspcev 3613 . . . . . 6 ((𝑋𝐴𝑦𝐶) → ∃𝑥𝐴 𝑦𝐵)
54adantlr 714 . . . . 5 (((𝑋𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶) ∧ 𝑦𝐶) → ∃𝑥𝐴 𝑦𝐵)
65ex 414 . . . 4 ((𝑋𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶) → (𝑦𝐶 → ∃𝑥𝐴 𝑦𝐵))
7 nfv 1918 . . . . . 6 𝑥 𝑋𝐴
8 nfra1 3282 . . . . . 6 𝑥𝑥𝐴 𝐵 = 𝐶
97, 8nfan 1903 . . . . 5 𝑥(𝑋𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶)
10 nfv 1918 . . . . 5 𝑥 𝑦𝐶
11 rsp 3245 . . . . . . 7 (∀𝑥𝐴 𝐵 = 𝐶 → (𝑥𝐴𝐵 = 𝐶))
12 eleq2 2823 . . . . . . . 8 (𝐵 = 𝐶 → (𝑦𝐵𝑦𝐶))
1312biimpd 228 . . . . . . 7 (𝐵 = 𝐶 → (𝑦𝐵𝑦𝐶))
1411, 13syl6 35 . . . . . 6 (∀𝑥𝐴 𝐵 = 𝐶 → (𝑥𝐴 → (𝑦𝐵𝑦𝐶)))
1514adantl 483 . . . . 5 ((𝑋𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶) → (𝑥𝐴 → (𝑦𝐵𝑦𝐶)))
169, 10, 15rexlimd 3264 . . . 4 ((𝑋𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶) → (∃𝑥𝐴 𝑦𝐵𝑦𝐶))
176, 16impbid 211 . . 3 ((𝑋𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶) → (𝑦𝐶 ↔ ∃𝑥𝐴 𝑦𝐵))
181, 17bitr4id 290 . 2 ((𝑋𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶) → (𝑦 𝑥𝐴 𝐵𝑦𝐶))
1918eqrdv 2731 1 ((𝑋𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶) → 𝑥𝐴 𝐵 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3062  wrex 3071   ciun 4997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-v 3477  df-iun 4999
This theorem is referenced by:  uniimafveqt  46036  imasetpreimafvbijlemfv  46057
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