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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 7794. (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 30494 | . 2 ⊢ (𝜑 → (2nd ↾ 𝑈):𝑈–1-1→𝐴) |
7 | 1 | iunfo 9984 | . . 3 ⊢ (2nd ↾ 𝑈):𝑈–onto→∪ 𝑥 ∈ 𝑋 𝐶 |
8 | foeq3 6567 | . . . 4 ⊢ (∪ 𝑥 ∈ 𝑋 𝐶 = 𝐴 → ((2nd ↾ 𝑈):𝑈–onto→∪ 𝑥 ∈ 𝑋 𝐶 ↔ (2nd ↾ 𝑈):𝑈–onto→𝐴)) | |
9 | 8 | biimpa 481 | . . 3 ⊢ ((∪ 𝑥 ∈ 𝑋 𝐶 = 𝐴 ∧ (2nd ↾ 𝑈):𝑈–onto→∪ 𝑥 ∈ 𝑋 𝐶) → (2nd ↾ 𝑈):𝑈–onto→𝐴) |
10 | 5, 7, 9 | sylancl 590 | . 2 ⊢ (𝜑 → (2nd ↾ 𝑈):𝑈–onto→𝐴) |
11 | df-f1o 6335 | . 2 ⊢ ((2nd ↾ 𝑈):𝑈–1-1-onto→𝐴 ↔ ((2nd ↾ 𝑈):𝑈–1-1→𝐴 ∧ (2nd ↾ 𝑈):𝑈–onto→𝐴)) | |
12 | 6, 10, 11 | sylanbrc 587 | 1 ⊢ (𝜑 → (2nd ↾ 𝑈):𝑈–1-1-onto→𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2112 {csn 4515 ∪ ciun 4876 Disj wdisj 4990 × cxp 5515 ↾ cres 5519 –1-1→wf1 6325 –onto→wfo 6326 –1-1-onto→wf1o 6327 2nd c2nd 7685 |
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 ax-un 7452 |
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-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-ral 3073 df-rex 3074 df-rmo 3076 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-uni 4792 df-iun 4878 df-disj 4991 df-br 5026 df-opab 5088 df-mpt 5106 df-id 5423 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-2nd 7687 |
This theorem is referenced by: gsumpart 30826 |
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