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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 6837 | . . . 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 6852 | . . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐷 ⊆ 𝐴) → (𝐹 ↾ 𝐷):𝐷–1-1-onto→(𝐹 “ 𝐷)) | |
6 | 3, 4, 5 | syl2anc 582 | . 2 ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–1-1-onto→(𝐹 “ 𝐷)) |
7 | f1ofun 6840 | . . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Fun 𝐹) | |
8 | 1, 7 | syl 17 | . . . . 5 ⊢ (𝜑 → Fun 𝐹) |
9 | f1odm 6842 | . . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → dom 𝐹 = 𝐴) | |
10 | 1, 9 | syl 17 | . . . . . 6 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
11 | 4, 10 | sseqtrrd 4018 | . . . . 5 ⊢ (𝜑 → 𝐷 ⊆ dom 𝐹) |
12 | dfimafn 6960 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐷 ⊆ dom 𝐹) → (𝐹 “ 𝐷) = {𝑦 ∣ ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦}) | |
13 | 8, 11, 12 | syl2anc 582 | . . . 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 2794 | . . . . 5 ⊢ (𝜑 → {𝑦 ∣ ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜒)}) |
16 | df-rab 3419 | . . . . 5 ⊢ {𝑦 ∈ 𝐵 ∣ 𝜒} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜒)} | |
17 | 15, 16 | eqtr4di 2783 | . . . 4 ⊢ (𝜑 → {𝑦 ∣ ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦} = {𝑦 ∈ 𝐵 ∣ 𝜒}) |
18 | 13, 17 | eqtr2d 2766 | . . 3 ⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ 𝜒} = (𝐹 “ 𝐷)) |
19 | 18 | f1oeq3d 6835 | . 2 ⊢ (𝜑 → ((𝐹 ↾ 𝐷):𝐷–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒} ↔ (𝐹 ↾ 𝐷):𝐷–1-1-onto→(𝐹 “ 𝐷))) |
20 | 6, 19 | mpbird 256 | 1 ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 {cab 2702 ∃wrex 3059 {crab 3418 ⊆ wss 3944 dom cdm 5678 ↾ cres 5680 “ cima 5681 Fun wfun 6543 –1-1→wf1 6546 –1-1-onto→wf1o 6548 ‘cfv 6549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 |
This theorem is referenced by: f1oresf1o2 46809 |
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