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Theorem iunsnima2 30386
 Description: Version of iunsnima 30385 with different variables. (Contributed by Thierry Arnoux, 22-Jun-2024.)
Hypotheses
Ref Expression
iunsnima.1 (𝜑𝐴𝑉)
iunsnima.2 ((𝜑𝑥𝐴) → 𝐵𝑊)
iunsnima2.1 𝑥𝐶
iunsnima2.2 (𝑥 = 𝑌𝐵 = 𝐶)
Assertion
Ref Expression
iunsnima2 ((𝜑𝑌𝐴) → ( 𝑥𝐴 ({𝑥} × 𝐵) “ {𝑌}) = 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑌
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem iunsnima2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 elimasng 5926 . . . . 5 ((𝑌𝐴𝑧 ∈ V) → (𝑧 ∈ ( 𝑥𝐴 ({𝑥} × 𝐵) “ {𝑌}) ↔ ⟨𝑌, 𝑧⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵)))
21elvd 3450 . . . 4 (𝑌𝐴 → (𝑧 ∈ ( 𝑥𝐴 ({𝑥} × 𝐵) “ {𝑌}) ↔ ⟨𝑌, 𝑧⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵)))
32adantl 485 . . 3 ((𝜑𝑌𝐴) → (𝑧 ∈ ( 𝑥𝐴 ({𝑥} × 𝐵) “ {𝑌}) ↔ ⟨𝑌, 𝑧⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵)))
4 iunsnima2.1 . . . . . 6 𝑥𝐶
5 iunsnima2.2 . . . . . 6 (𝑥 = 𝑌𝐵 = 𝐶)
64, 5opeliunxp2f 7863 . . . . 5 (⟨𝑌, 𝑧⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝑌𝐴𝑧𝐶))
76baib 539 . . . 4 (𝑌𝐴 → (⟨𝑌, 𝑧⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ 𝑧𝐶))
87adantl 485 . . 3 ((𝜑𝑌𝐴) → (⟨𝑌, 𝑧⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ 𝑧𝐶))
93, 8bitrd 282 . 2 ((𝜑𝑌𝐴) → (𝑧 ∈ ( 𝑥𝐴 ({𝑥} × 𝐵) “ {𝑌}) ↔ 𝑧𝐶))
109eqrdv 2799 1 ((𝜑𝑌𝐴) → ( 𝑥𝐴 ({𝑥} × 𝐵) “ {𝑌}) = 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2112  Ⅎwnfc 2939  Vcvv 3444  {csn 4528  ⟨cop 4534  ∪ ciun 4884   × cxp 5521   “ cima 5526 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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298 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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-iun 4886  df-br 5034  df-opab 5096  df-xp 5529  df-rel 5530  df-cnv 5531  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536 This theorem is referenced by:  gsumpart  30743
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