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Mirrors > Home > MPE Home > Th. List > Mathboxes > iunsnima2 | Structured version Visualization version GIF version |
Description: Version of iunsnima 30649 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 5945 | . . . . 5 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑧 ∈ V) → (𝑧 ∈ (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) “ {𝑌}) ↔ 〈𝑌, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) | |
2 | 1 | elvd 3408 | . . . 4 ⊢ (𝑌 ∈ 𝐴 → (𝑧 ∈ (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) “ {𝑌}) ↔ 〈𝑌, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) |
3 | 2 | adantl 485 | . . 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 7941 | . . . . 5 ⊢ (〈𝑌, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝑌 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)) |
7 | 6 | baib 539 | . . . 4 ⊢ (𝑌 ∈ 𝐴 → (〈𝑌, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ 𝑧 ∈ 𝐶)) |
8 | 7 | adantl 485 | . . 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 2732 | 1 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐴) → (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) “ {𝑌}) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 Ⅎwnfc 2880 Vcvv 3401 {csn 4531 〈cop 4537 ∪ ciun 4894 × cxp 5538 “ cima 5543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pr 5311 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-sn 4532 df-pr 4534 df-op 4538 df-iun 4896 df-br 5044 df-opab 5106 df-xp 5546 df-rel 5547 df-cnv 5548 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 |
This theorem is referenced by: gsumpart 31006 |
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