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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2ndresdjuf1o | Structured version Visualization version GIF version |
Description: The 2nd function restricted to a disjoint union is a bijection. See also e.g. 2ndconst 8086. (Contributed by Thierry Arnoux, 23-Jun-2024.) |
Ref | Expression |
---|---|
2ndresdju.u | ⊢ 𝑈 = ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) |
2ndresdju.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
2ndresdju.x | ⊢ (𝜑 → 𝑋 ∈ 𝑊) |
2ndresdju.1 | ⊢ (𝜑 → Disj 𝑥 ∈ 𝑋 𝐶) |
2ndresdju.2 | ⊢ (𝜑 → ∪ 𝑥 ∈ 𝑋 𝐶 = 𝐴) |
Ref | Expression |
---|---|
2ndresdjuf1o | ⊢ (𝜑 → (2nd ↾ 𝑈):𝑈–1-1-onto→𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2ndresdju.u | . . 3 ⊢ 𝑈 = ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) | |
2 | 2ndresdju.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | 2ndresdju.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑊) | |
4 | 2ndresdju.1 | . . 3 ⊢ (𝜑 → Disj 𝑥 ∈ 𝑋 𝐶) | |
5 | 2ndresdju.2 | . . 3 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝑋 𝐶 = 𝐴) | |
6 | 1, 2, 3, 4, 5 | 2ndresdju 31869 | . 2 ⊢ (𝜑 → (2nd ↾ 𝑈):𝑈–1-1→𝐴) |
7 | 1 | iunfo 10533 | . . 3 ⊢ (2nd ↾ 𝑈):𝑈–onto→∪ 𝑥 ∈ 𝑋 𝐶 |
8 | foeq3 6803 | . . . 4 ⊢ (∪ 𝑥 ∈ 𝑋 𝐶 = 𝐴 → ((2nd ↾ 𝑈):𝑈–onto→∪ 𝑥 ∈ 𝑋 𝐶 ↔ (2nd ↾ 𝑈):𝑈–onto→𝐴)) | |
9 | 8 | biimpa 477 | . . 3 ⊢ ((∪ 𝑥 ∈ 𝑋 𝐶 = 𝐴 ∧ (2nd ↾ 𝑈):𝑈–onto→∪ 𝑥 ∈ 𝑋 𝐶) → (2nd ↾ 𝑈):𝑈–onto→𝐴) |
10 | 5, 7, 9 | sylancl 586 | . 2 ⊢ (𝜑 → (2nd ↾ 𝑈):𝑈–onto→𝐴) |
11 | df-f1o 6550 | . 2 ⊢ ((2nd ↾ 𝑈):𝑈–1-1-onto→𝐴 ↔ ((2nd ↾ 𝑈):𝑈–1-1→𝐴 ∧ (2nd ↾ 𝑈):𝑈–onto→𝐴)) | |
12 | 6, 10, 11 | sylanbrc 583 | 1 ⊢ (𝜑 → (2nd ↾ 𝑈):𝑈–1-1-onto→𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 {csn 4628 ∪ ciun 4997 Disj wdisj 5113 × cxp 5674 ↾ cres 5678 –1-1→wf1 6540 –onto→wfo 6541 –1-1-onto→wf1o 6542 2nd c2nd 7973 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-disj 5114 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-2nd 7975 |
This theorem is referenced by: gsumpart 32202 |
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