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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iunsnima2 | Structured version Visualization version GIF version |
Description: Version of iunsnima 30465 with different variables. (Contributed by Thierry Arnoux, 22-Jun-2024.) |
Ref | Expression |
---|---|
iunsnima.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
iunsnima.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊) |
iunsnima2.1 | ⊢ Ⅎ𝑥𝐶 |
iunsnima2.2 | ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
iunsnima2 | ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐴) → (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) “ {𝑌}) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elimasng 5920 | . . . . 5 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑧 ∈ V) → (𝑧 ∈ (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) “ {𝑌}) ↔ 〈𝑌, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) | |
2 | 1 | elvd 3414 | . . . 4 ⊢ (𝑌 ∈ 𝐴 → (𝑧 ∈ (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) “ {𝑌}) ↔ 〈𝑌, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) |
3 | 2 | adantl 486 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐴) → (𝑧 ∈ (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) “ {𝑌}) ↔ 〈𝑌, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) |
4 | iunsnima2.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐶 | |
5 | iunsnima2.2 | . . . . . 6 ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐶) | |
6 | 4, 5 | opeliunxp2f 7879 | . . . . 5 ⊢ (〈𝑌, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝑌 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)) |
7 | 6 | baib 540 | . . . 4 ⊢ (𝑌 ∈ 𝐴 → (〈𝑌, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ 𝑧 ∈ 𝐶)) |
8 | 7 | adantl 486 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐴) → (〈𝑌, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ 𝑧 ∈ 𝐶)) |
9 | 3, 8 | bitrd 282 | . 2 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐴) → (𝑧 ∈ (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) “ {𝑌}) ↔ 𝑧 ∈ 𝐶)) |
10 | 9 | eqrdv 2757 | 1 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐴) → (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) “ {𝑌}) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1539 ∈ wcel 2112 Ⅎwnfc 2897 Vcvv 3407 {csn 4515 〈cop 4521 ∪ ciun 4876 × cxp 5515 “ cima 5520 |
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 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pr 5291 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3an 1087 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ral 3073 df-rex 3074 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-nul 4222 df-if 4414 df-sn 4516 df-pr 4518 df-op 4522 df-iun 4878 df-br 5026 df-opab 5088 df-xp 5523 df-rel 5524 df-cnv 5525 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 |
This theorem is referenced by: gsumpart 30826 |
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