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Mirrors > Home > MPE Home > Th. List > usgrexilem | Structured version Visualization version GIF version |
Description: Lemma for usgrexi 29326. (Contributed by AV, 12-Jan-2018.) (Revised by AV, 10-Nov-2021.) |
Ref | Expression |
---|---|
usgrexi.p | ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} |
Ref | Expression |
---|---|
usgrexilem | ⊢ (𝑉 ∈ 𝑊 → ( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1oi 6876 | . . . 4 ⊢ ( I ↾ 𝑃):𝑃–1-1-onto→𝑃 | |
2 | f1of1 6837 | . . . 4 ⊢ (( I ↾ 𝑃):𝑃–1-1-onto→𝑃 → ( I ↾ 𝑃):𝑃–1-1→𝑃) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ ( I ↾ 𝑃):𝑃–1-1→𝑃 |
4 | dmresi 6056 | . . . 4 ⊢ dom ( I ↾ 𝑃) = 𝑃 | |
5 | f1eq2 6789 | . . . 4 ⊢ (dom ( I ↾ 𝑃) = 𝑃 → (( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→𝑃 ↔ ( I ↾ 𝑃):𝑃–1-1→𝑃)) | |
6 | 4, 5 | ax-mp 5 | . . 3 ⊢ (( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→𝑃 ↔ ( I ↾ 𝑃):𝑃–1-1→𝑃) |
7 | 3, 6 | mpbir 230 | . 2 ⊢ ( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→𝑃 |
8 | usgrexi.p | . . . 4 ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} | |
9 | 8 | eqcomi 2734 | . . 3 ⊢ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} = 𝑃 |
10 | f1eq3 6790 | . . 3 ⊢ ({𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} = 𝑃 → (( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ↔ ( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→𝑃)) | |
11 | 9, 10 | mp1i 13 | . 2 ⊢ (𝑉 ∈ 𝑊 → (( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ↔ ( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→𝑃)) |
12 | 7, 11 | mpbiri 257 | 1 ⊢ (𝑉 ∈ 𝑊 → ( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 {crab 3418 𝒫 cpw 4604 I cid 5575 dom cdm 5678 ↾ cres 5680 –1-1→wf1 6546 –1-1-onto→wf1o 6548 ‘cfv 6549 2c2 12300 ♯chash 14325 |
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-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-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-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 |
This theorem is referenced by: usgrexi 29326 structtousgr 29330 |
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