Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
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 6607 | . . . 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 6622 | . . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐷 ⊆ 𝐴) → (𝐹 ↾ 𝐷):𝐷–1-1-onto→(𝐹 “ 𝐷)) | |
6 | 3, 4, 5 | syl2anc 586 | . 2 ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–1-1-onto→(𝐹 “ 𝐷)) |
7 | f1ofun 6610 | . . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Fun 𝐹) | |
8 | 1, 7 | syl 17 | . . . . 5 ⊢ (𝜑 → Fun 𝐹) |
9 | f1odm 6612 | . . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → dom 𝐹 = 𝐴) | |
10 | 1, 9 | syl 17 | . . . . . 6 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
11 | 4, 10 | sseqtrrd 4006 | . . . . 5 ⊢ (𝜑 → 𝐷 ⊆ dom 𝐹) |
12 | dfimafn 6721 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐷 ⊆ dom 𝐹) → (𝐹 “ 𝐷) = {𝑦 ∣ ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦}) | |
13 | 8, 11, 12 | syl2anc 586 | . . . 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 2883 | . . . . 5 ⊢ (𝜑 → {𝑦 ∣ ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜒)}) |
16 | df-rab 3145 | . . . . 5 ⊢ {𝑦 ∈ 𝐵 ∣ 𝜒} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜒)} | |
17 | 15, 16 | syl6eqr 2872 | . . . 4 ⊢ (𝜑 → {𝑦 ∣ ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦} = {𝑦 ∈ 𝐵 ∣ 𝜒}) |
18 | 13, 17 | eqtr2d 2855 | . . 3 ⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ 𝜒} = (𝐹 “ 𝐷)) |
19 | 18 | f1oeq3d 6605 | . 2 ⊢ (𝜑 → ((𝐹 ↾ 𝐷):𝐷–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒} ↔ (𝐹 ↾ 𝐷):𝐷–1-1-onto→(𝐹 “ 𝐷))) |
20 | 6, 19 | mpbird 259 | 1 ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1531 ∈ wcel 2108 {cab 2797 ∃wrex 3137 {crab 3140 ⊆ wss 3934 dom cdm 5548 ↾ cres 5550 “ cima 5551 Fun wfun 6342 –1-1→wf1 6345 –1-1-onto→wf1o 6347 ‘cfv 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 |
This theorem is referenced by: f1oresf1o2 43481 |
Copyright terms: Public domain | W3C validator |