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 7858. (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 30677 | . 2 ⊢ (𝜑 → (2nd ↾ 𝑈):𝑈–1-1→𝐴) |
7 | 1 | iunfo 10136 | . . 3 ⊢ (2nd ↾ 𝑈):𝑈–onto→∪ 𝑥 ∈ 𝑋 𝐶 |
8 | foeq3 6620 | . . . 4 ⊢ (∪ 𝑥 ∈ 𝑋 𝐶 = 𝐴 → ((2nd ↾ 𝑈):𝑈–onto→∪ 𝑥 ∈ 𝑋 𝐶 ↔ (2nd ↾ 𝑈):𝑈–onto→𝐴)) | |
9 | 8 | biimpa 480 | . . 3 ⊢ ((∪ 𝑥 ∈ 𝑋 𝐶 = 𝐴 ∧ (2nd ↾ 𝑈):𝑈–onto→∪ 𝑥 ∈ 𝑋 𝐶) → (2nd ↾ 𝑈):𝑈–onto→𝐴) |
10 | 5, 7, 9 | sylancl 589 | . 2 ⊢ (𝜑 → (2nd ↾ 𝑈):𝑈–onto→𝐴) |
11 | df-f1o 6376 | . 2 ⊢ ((2nd ↾ 𝑈):𝑈–1-1-onto→𝐴 ↔ ((2nd ↾ 𝑈):𝑈–1-1→𝐴 ∧ (2nd ↾ 𝑈):𝑈–onto→𝐴)) | |
12 | 6, 10, 11 | sylanbrc 586 | 1 ⊢ (𝜑 → (2nd ↾ 𝑈):𝑈–1-1-onto→𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 {csn 4531 ∪ ciun 4894 Disj wdisj 5008 × cxp 5538 ↾ cres 5542 –1-1→wf1 6366 –onto→wfo 6367 –1-1-onto→wf1o 6368 2nd c2nd 7749 |
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 ax-un 7512 |
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-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rmo 3062 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-uni 4810 df-iun 4896 df-disj 5009 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-2nd 7751 |
This theorem is referenced by: gsumpart 31006 |
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