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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 8081. (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 31851 | . 2 ⊢ (𝜑 → (2nd ↾ 𝑈):𝑈–1-1→𝐴) |
7 | 1 | iunfo 10529 | . . 3 ⊢ (2nd ↾ 𝑈):𝑈–onto→∪ 𝑥 ∈ 𝑋 𝐶 |
8 | foeq3 6799 | . . . 4 ⊢ (∪ 𝑥 ∈ 𝑋 𝐶 = 𝐴 → ((2nd ↾ 𝑈):𝑈–onto→∪ 𝑥 ∈ 𝑋 𝐶 ↔ (2nd ↾ 𝑈):𝑈–onto→𝐴)) | |
9 | 8 | biimpa 478 | . . 3 ⊢ ((∪ 𝑥 ∈ 𝑋 𝐶 = 𝐴 ∧ (2nd ↾ 𝑈):𝑈–onto→∪ 𝑥 ∈ 𝑋 𝐶) → (2nd ↾ 𝑈):𝑈–onto→𝐴) |
10 | 5, 7, 9 | sylancl 587 | . 2 ⊢ (𝜑 → (2nd ↾ 𝑈):𝑈–onto→𝐴) |
11 | df-f1o 6546 | . 2 ⊢ ((2nd ↾ 𝑈):𝑈–1-1-onto→𝐴 ↔ ((2nd ↾ 𝑈):𝑈–1-1→𝐴 ∧ (2nd ↾ 𝑈):𝑈–onto→𝐴)) | |
12 | 6, 10, 11 | sylanbrc 584 | 1 ⊢ (𝜑 → (2nd ↾ 𝑈):𝑈–1-1-onto→𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 {csn 4626 ∪ ciun 4995 Disj wdisj 5111 × cxp 5672 ↾ cres 5676 –1-1→wf1 6536 –onto→wfo 6537 –1-1-onto→wf1o 6538 2nd c2nd 7968 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5297 ax-nul 5304 ax-pr 5425 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4527 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-iun 4997 df-disj 5112 df-br 5147 df-opab 5209 df-mpt 5230 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-2nd 7970 |
This theorem is referenced by: gsumpart 32184 |
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