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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > f1oresf1o | Structured version Visualization version GIF version |
Description: Build a bijection by restricting the domain of a bijection. (Contributed by AV, 31-Jul-2022.) |
Ref | Expression |
---|---|
f1oresf1o.1 | ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) |
f1oresf1o.2 | ⊢ (𝜑 → 𝐷 ⊆ 𝐴) |
f1oresf1o.3 | ⊢ (𝜑 → (∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦 ↔ (𝑦 ∈ 𝐵 ∧ 𝜒))) |
Ref | Expression |
---|---|
f1oresf1o | ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1oresf1o.1 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) | |
2 | f1of1 6589 | . . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–1-1→𝐵) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹:𝐴–1-1→𝐵) |
4 | f1oresf1o.2 | . . 3 ⊢ (𝜑 → 𝐷 ⊆ 𝐴) | |
5 | f1ores 6604 | . . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐷 ⊆ 𝐴) → (𝐹 ↾ 𝐷):𝐷–1-1-onto→(𝐹 “ 𝐷)) | |
6 | 3, 4, 5 | syl2anc 587 | . 2 ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–1-1-onto→(𝐹 “ 𝐷)) |
7 | f1ofun 6592 | . . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Fun 𝐹) | |
8 | 1, 7 | syl 17 | . . . . 5 ⊢ (𝜑 → Fun 𝐹) |
9 | f1odm 6594 | . . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → dom 𝐹 = 𝐴) | |
10 | 1, 9 | syl 17 | . . . . . 6 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
11 | 4, 10 | sseqtrrd 3956 | . . . . 5 ⊢ (𝜑 → 𝐷 ⊆ dom 𝐹) |
12 | dfimafn 6703 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐷 ⊆ dom 𝐹) → (𝐹 “ 𝐷) = {𝑦 ∣ ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦}) | |
13 | 8, 11, 12 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (𝐹 “ 𝐷) = {𝑦 ∣ ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦}) |
14 | f1oresf1o.3 | . . . . . 6 ⊢ (𝜑 → (∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦 ↔ (𝑦 ∈ 𝐵 ∧ 𝜒))) | |
15 | 14 | abbidv 2862 | . . . . 5 ⊢ (𝜑 → {𝑦 ∣ ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜒)}) |
16 | df-rab 3115 | . . . . 5 ⊢ {𝑦 ∈ 𝐵 ∣ 𝜒} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜒)} | |
17 | 15, 16 | eqtr4di 2851 | . . . 4 ⊢ (𝜑 → {𝑦 ∣ ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦} = {𝑦 ∈ 𝐵 ∣ 𝜒}) |
18 | 13, 17 | eqtr2d 2834 | . . 3 ⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ 𝜒} = (𝐹 “ 𝐷)) |
19 | 18 | f1oeq3d 6587 | . 2 ⊢ (𝜑 → ((𝐹 ↾ 𝐷):𝐷–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒} ↔ (𝐹 ↾ 𝐷):𝐷–1-1-onto→(𝐹 “ 𝐷))) |
20 | 6, 19 | 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 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {cab 2776 ∃wrex 3107 {crab 3110 ⊆ wss 3881 dom cdm 5519 ↾ cres 5521 “ cima 5522 Fun wfun 6318 –1-1→wf1 6321 –1-1-onto→wf1o 6323 ‘cfv 6324 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 |
This theorem is referenced by: f1oresf1o2 43847 |
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