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Theorem iuneqconst 5003
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 4995 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
2 iuneqconst.p . . . . . . . 8 (𝑥 = 𝑋𝐵 = 𝐶)
32eleq2d 2827 . . . . . . 7 (𝑥 = 𝑋 → (𝑦𝐵𝑦𝐶))
43rspcev 3622 . . . . . 6 ((𝑋𝐴𝑦𝐶) → ∃𝑥𝐴 𝑦𝐵)
54adantlr 715 . . . . 5 (((𝑋𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶) ∧ 𝑦𝐶) → ∃𝑥𝐴 𝑦𝐵)
65ex 412 . . . 4 ((𝑋𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶) → (𝑦𝐶 → ∃𝑥𝐴 𝑦𝐵))
7 nfv 1914 . . . . . 6 𝑥 𝑋𝐴
8 nfra1 3284 . . . . . 6 𝑥𝑥𝐴 𝐵 = 𝐶
97, 8nfan 1899 . . . . 5 𝑥(𝑋𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶)
10 nfv 1914 . . . . 5 𝑥 𝑦𝐶
11 rsp 3247 . . . . . . 7 (∀𝑥𝐴 𝐵 = 𝐶 → (𝑥𝐴𝐵 = 𝐶))
12 eleq2 2830 . . . . . . . 8 (𝐵 = 𝐶 → (𝑦𝐵𝑦𝐶))
1312biimpd 229 . . . . . . 7 (𝐵 = 𝐶 → (𝑦𝐵𝑦𝐶))
1411, 13syl6 35 . . . . . 6 (∀𝑥𝐴 𝐵 = 𝐶 → (𝑥𝐴 → (𝑦𝐵𝑦𝐶)))
1514adantl 481 . . . . 5 ((𝑋𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶) → (𝑥𝐴 → (𝑦𝐵𝑦𝐶)))
169, 10, 15rexlimd 3266 . . . 4 ((𝑋𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶) → (∃𝑥𝐴 𝑦𝐵𝑦𝐶))
176, 16impbid 212 . . 3 ((𝑋𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶) → (𝑦𝐶 ↔ ∃𝑥𝐴 𝑦𝐵))
181, 17bitr4id 290 . 2 ((𝑋𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶) → (𝑦 𝑥𝐴 𝐵𝑦𝐶))
1918eqrdv 2735 1 ((𝑋𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶) → 𝑥𝐴 𝐵 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061  wrex 3070   ciun 4991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-v 3482  df-iun 4993
This theorem is referenced by:  uniimafveqt  47368  imasetpreimafvbijlemfv  47389
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