Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > f1oresf1orab | Structured version Visualization version GIF version |
Description: Build a bijection by restricting the domain of a bijection. (Contributed by AV, 1-Aug-2022.) |
Ref | Expression |
---|---|
f1oresf1orab.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶) |
f1oresf1orab.2 | ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) |
f1oresf1orab.3 | ⊢ (𝜑 → 𝐷 ⊆ 𝐴) |
f1oresf1orab.4 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → (𝜒 ↔ 𝑥 ∈ 𝐷)) |
Ref | Expression |
---|---|
f1oresf1orab | ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1oresf1orab.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
2 | f1oresf1orab.2 | . . 3 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) | |
3 | f1oresf1orab.4 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → (𝜒 ↔ 𝑥 ∈ 𝐷)) | |
4 | 1, 2, 3 | f1oresrab 6960 | . 2 ⊢ (𝜑 → (𝐹 ↾ {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐷}):{𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐷}–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒}) |
5 | f1oresf1orab.3 | . . . . . 6 ⊢ (𝜑 → 𝐷 ⊆ 𝐴) | |
6 | dfss7 4169 | . . . . . 6 ⊢ (𝐷 ⊆ 𝐴 ↔ {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐷} = 𝐷) | |
7 | 5, 6 | sylib 221 | . . . . 5 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐷} = 𝐷) |
8 | 7 | eqcomd 2744 | . . . 4 ⊢ (𝜑 → 𝐷 = {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐷}) |
9 | 8 | reseq2d 5865 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐷) = (𝐹 ↾ {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐷})) |
10 | eqidd 2739 | . . 3 ⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ 𝜒} = {𝑦 ∈ 𝐵 ∣ 𝜒}) | |
11 | 9, 8, 10 | f1oeq123d 6673 | . 2 ⊢ (𝜑 → ((𝐹 ↾ 𝐷):𝐷–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒} ↔ (𝐹 ↾ {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐷}):{𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐷}–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒})) |
12 | 4, 11 | mpbird 260 | 1 ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ w3a 1089 = wceq 1543 ∈ wcel 2111 {crab 3066 ⊆ wss 3880 ↦ cmpt 5149 ↾ cres 5567 –1-1-onto→wf1o 6396 |
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 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5206 ax-nul 5213 ax-pr 5336 |
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 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3422 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-nul 4252 df-if 4454 df-sn 4556 df-pr 4558 df-op 4562 df-br 5068 df-opab 5130 df-mpt 5150 df-id 5469 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-fun 6399 df-fn 6400 df-f 6401 df-f1 6402 df-fo 6403 df-f1o 6404 |
This theorem is referenced by: (None) |
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