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 6665 | . . . 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 6680 | . . 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 6668 | . . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Fun 𝐹) | |
8 | 1, 7 | syl 17 | . . . . 5 ⊢ (𝜑 → Fun 𝐹) |
9 | f1odm 6670 | . . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → dom 𝐹 = 𝐴) | |
10 | 1, 9 | syl 17 | . . . . . 6 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
11 | 4, 10 | sseqtrrd 3947 | . . . . 5 ⊢ (𝜑 → 𝐷 ⊆ dom 𝐹) |
12 | dfimafn 6780 | . . . . 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 2807 | . . . . 5 ⊢ (𝜑 → {𝑦 ∣ ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜒)}) |
16 | df-rab 3070 | . . . . 5 ⊢ {𝑦 ∈ 𝐵 ∣ 𝜒} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜒)} | |
17 | 15, 16 | eqtr4di 2796 | . . . 4 ⊢ (𝜑 → {𝑦 ∣ ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦} = {𝑦 ∈ 𝐵 ∣ 𝜒}) |
18 | 13, 17 | eqtr2d 2778 | . . 3 ⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ 𝜒} = (𝐹 “ 𝐷)) |
19 | 18 | f1oeq3d 6663 | . 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 1543 ∈ wcel 2110 {cab 2714 ∃wrex 3062 {crab 3065 ⊆ wss 3871 dom cdm 5556 ↾ cres 5558 “ cima 5559 Fun wfun 6379 –1-1→wf1 6382 –1-1-onto→wf1o 6384 ‘cfv 6385 |
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 2708 ax-sep 5197 ax-nul 5204 ax-pr 5327 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3415 df-dif 3874 df-un 3876 df-in 3878 df-ss 3888 df-nul 4243 df-if 4445 df-sn 4547 df-pr 4549 df-op 4553 df-uni 4825 df-br 5059 df-opab 5121 df-id 5460 df-xp 5562 df-rel 5563 df-cnv 5564 df-co 5565 df-dm 5566 df-rn 5567 df-res 5568 df-ima 5569 df-iota 6343 df-fun 6387 df-fn 6388 df-f 6389 df-f1 6390 df-fo 6391 df-f1o 6392 df-fv 6393 |
This theorem is referenced by: f1oresf1o2 44463 |
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