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Theorem numiunnum 36438
Description: An indexed union of sets is numerable if its index set is numerable and there exists a collection of well-orderings on its members. (Contributed by Matthew House, 23-Aug-2025.)
Assertion
Ref Expression
numiunnum ((𝐴 ∈ dom card ∧ ∀𝑥𝐴 (𝐵𝑉𝑆 We 𝐵)) → 𝑥𝐴 𝐵 ∈ dom card)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝑆(𝑥)   𝑉(𝑥)

Proof of Theorem numiunnum
Dummy variables 𝑠 𝑡 𝑢 𝑣 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfac8b 10102 . . . 4 (𝐴 ∈ dom card → ∃𝑠 𝑠 We 𝐴)
21adantr 480 . . 3 ((𝐴 ∈ dom card ∧ ∀𝑥𝐴 (𝐵𝑉𝑆 We 𝐵)) → ∃𝑠 𝑠 We 𝐴)
3 simpll 766 . . . . . . 7 (((𝐴 ∈ dom card ∧ ∀𝑥𝐴 (𝐵𝑉𝑆 We 𝐵)) ∧ 𝑠 We 𝐴) → 𝐴 ∈ dom card)
4 simplr 768 . . . . . . . . 9 (((𝐴 ∈ dom card ∧ ∀𝑥𝐴 (𝐵𝑉𝑆 We 𝐵)) ∧ 𝑠 We 𝐴) → ∀𝑥𝐴 (𝐵𝑉𝑆 We 𝐵))
5 r19.26 3117 . . . . . . . . 9 (∀𝑥𝐴 (𝐵𝑉𝑆 We 𝐵) ↔ (∀𝑥𝐴 𝐵𝑉 ∧ ∀𝑥𝐴 𝑆 We 𝐵))
64, 5sylib 218 . . . . . . . 8 (((𝐴 ∈ dom card ∧ ∀𝑥𝐴 (𝐵𝑉𝑆 We 𝐵)) ∧ 𝑠 We 𝐴) → (∀𝑥𝐴 𝐵𝑉 ∧ ∀𝑥𝐴 𝑆 We 𝐵))
76simpld 494 . . . . . . 7 (((𝐴 ∈ dom card ∧ ∀𝑥𝐴 (𝐵𝑉𝑆 We 𝐵)) ∧ 𝑠 We 𝐴) → ∀𝑥𝐴 𝐵𝑉)
8 iunexg 8006 . . . . . . 7 ((𝐴 ∈ dom card ∧ ∀𝑥𝐴 𝐵𝑉) → 𝑥𝐴 𝐵 ∈ V)
93, 7, 8syl2anc 583 . . . . . 6 (((𝐴 ∈ dom card ∧ ∀𝑥𝐴 (𝐵𝑉𝑆 We 𝐵)) ∧ 𝑠 We 𝐴) → 𝑥𝐴 𝐵 ∈ V)
109, 9xpexd 7788 . . . . 5 (((𝐴 ∈ dom card ∧ ∀𝑥𝐴 (𝐵𝑉𝑆 We 𝐵)) ∧ 𝑠 We 𝐴) → ( 𝑥𝐴 𝐵 × 𝑥𝐴 𝐵) ∈ V)
11 opabssxp 5792 . . . . . 6 {⟨𝑦, 𝑧⟩ ∣ ((𝑦 𝑥𝐴 𝐵𝑧 𝑥𝐴 𝐵) ∧ (((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑦)𝑠((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑧) ∨ (((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑦) = ((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑧) ∧ 𝑦((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑦) / 𝑥𝑆𝑧)))} ⊆ ( 𝑥𝐴 𝐵 × 𝑥𝐴 𝐵)
1211a1i 11 . . . . 5 (((𝐴 ∈ dom card ∧ ∀𝑥𝐴 (𝐵𝑉𝑆 We 𝐵)) ∧ 𝑠 We 𝐴) → {⟨𝑦, 𝑧⟩ ∣ ((𝑦 𝑥𝐴 𝐵𝑧 𝑥𝐴 𝐵) ∧ (((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑦)𝑠((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑧) ∨ (((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑦) = ((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑧) ∧ 𝑦((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑦) / 𝑥𝑆𝑧)))} ⊆ ( 𝑥𝐴 𝐵 × 𝑥𝐴 𝐵))
1310, 12ssexd 5342 . . . 4 (((𝐴 ∈ dom card ∧ ∀𝑥𝐴 (𝐵𝑉𝑆 We 𝐵)) ∧ 𝑠 We 𝐴) → {⟨𝑦, 𝑧⟩ ∣ ((𝑦 𝑥𝐴 𝐵𝑧 𝑥𝐴 𝐵) ∧ (((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑦)𝑠((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑧) ∨ (((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑦) = ((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑧) ∧ 𝑦((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑦) / 𝑥𝑆𝑧)))} ∈ V)
14 simpr 484 . . . . 5 (((𝐴 ∈ dom card ∧ ∀𝑥𝐴 (𝐵𝑉𝑆 We 𝐵)) ∧ 𝑠 We 𝐴) → 𝑠 We 𝐴)
15 exse 5660 . . . . . 6 (𝐴 ∈ dom card → 𝑠 Se 𝐴)
1615ad2antrr 725 . . . . 5 (((𝐴 ∈ dom card ∧ ∀𝑥𝐴 (𝐵𝑉𝑆 We 𝐵)) ∧ 𝑠 We 𝐴) → 𝑠 Se 𝐴)
176simprd 495 . . . . 5 (((𝐴 ∈ dom card ∧ ∀𝑥𝐴 (𝐵𝑉𝑆 We 𝐵)) ∧ 𝑠 We 𝐴) → ∀𝑥𝐴 𝑆 We 𝐵)
18 eqid 2740 . . . . . 6 (𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢)) = (𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))
19 eqid 2740 . . . . . 6 {⟨𝑦, 𝑧⟩ ∣ ((𝑦 𝑥𝐴 𝐵𝑧 𝑥𝐴 𝐵) ∧ (((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑦)𝑠((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑧) ∨ (((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑦) = ((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑧) ∧ 𝑦((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑦) / 𝑥𝑆𝑧)))} = {⟨𝑦, 𝑧⟩ ∣ ((𝑦 𝑥𝐴 𝐵𝑧 𝑥𝐴 𝐵) ∧ (((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑦)𝑠((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑧) ∨ (((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑦) = ((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑧) ∧ 𝑦((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑦) / 𝑥𝑆𝑧)))}
2018, 19weiunwe 36437 . . . . 5 ((𝑠 We 𝐴𝑠 Se 𝐴 ∧ ∀𝑥𝐴 𝑆 We 𝐵) → {⟨𝑦, 𝑧⟩ ∣ ((𝑦 𝑥𝐴 𝐵𝑧 𝑥𝐴 𝐵) ∧ (((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑦)𝑠((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑧) ∨ (((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑦) = ((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑧) ∧ 𝑦((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑦) / 𝑥𝑆𝑧)))} We 𝑥𝐴 𝐵)
2114, 16, 17, 20syl3anc 1371 . . . 4 (((𝐴 ∈ dom card ∧ ∀𝑥𝐴 (𝐵𝑉𝑆 We 𝐵)) ∧ 𝑠 We 𝐴) → {⟨𝑦, 𝑧⟩ ∣ ((𝑦 𝑥𝐴 𝐵𝑧 𝑥𝐴 𝐵) ∧ (((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑦)𝑠((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑧) ∨ (((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑦) = ((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑧) ∧ 𝑦((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑦) / 𝑥𝑆𝑧)))} We 𝑥𝐴 𝐵)
22 weeq1 5687 . . . 4 (𝑡 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦 𝑥𝐴 𝐵𝑧 𝑥𝐴 𝐵) ∧ (((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑦)𝑠((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑧) ∨ (((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑦) = ((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑧) ∧ 𝑦((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑦) / 𝑥𝑆𝑧)))} → (𝑡 We 𝑥𝐴 𝐵 ↔ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 𝑥𝐴 𝐵𝑧 𝑥𝐴 𝐵) ∧ (((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑦)𝑠((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑧) ∨ (((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑦) = ((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑧) ∧ 𝑦((𝑤 𝑥𝐴 𝐵 ↦ (𝑢 ∈ {𝑥𝐴𝑤𝐵}∀𝑣 ∈ {𝑥𝐴𝑤𝐵} ¬ 𝑣𝑠𝑢))‘𝑦) / 𝑥𝑆𝑧)))} We 𝑥𝐴 𝐵))
2313, 21, 22spcedv 3611 . . 3 (((𝐴 ∈ dom card ∧ ∀𝑥𝐴 (𝐵𝑉𝑆 We 𝐵)) ∧ 𝑠 We 𝐴) → ∃𝑡 𝑡 We 𝑥𝐴 𝐵)
242, 23exlimddv 1934 . 2 ((𝐴 ∈ dom card ∧ ∀𝑥𝐴 (𝐵𝑉𝑆 We 𝐵)) → ∃𝑡 𝑡 We 𝑥𝐴 𝐵)
25 ween 10106 . 2 ( 𝑥𝐴 𝐵 ∈ dom card ↔ ∃𝑡 𝑡 We 𝑥𝐴 𝐵)
2624, 25sylibr 234 1 ((𝐴 ∈ dom card ∧ ∀𝑥𝐴 (𝐵𝑉𝑆 We 𝐵)) → 𝑥𝐴 𝐵 ∈ dom card)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 846   = wceq 1537  wex 1777  wcel 2108  wral 3067  {crab 3443  Vcvv 3488  csb 3921  wss 3976   ciun 5015   class class class wbr 5166  {copab 5228  cmpt 5249   Se wse 5650   We wwe 5651   × cxp 5698  dom cdm 5700  cfv 6575  crio 7405  cardccrd 10006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7772
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6334  df-ord 6400  df-on 6401  df-suc 6403  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-f1 6580  df-fo 6581  df-f1o 6582  df-fv 6583  df-isom 6584  df-riota 7406  df-ov 7453  df-2nd 8033  df-frecs 8324  df-wrecs 8355  df-recs 8429  df-en 9006  df-card 10010
This theorem is referenced by: (None)
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