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Mirrors > Home > ILE Home > Th. List > f1oabexg | GIF version |
Description: The class of all 1-1-onto functions mapping one set to another is a set. (Contributed by Paul Chapman, 25-Feb-2008.) |
Ref | Expression |
---|---|
f1oabexg.1 | ⊢ 𝐹 = {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐵 ∧ 𝜑)} |
Ref | Expression |
---|---|
f1oabexg | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝐹 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1oabexg.1 | . 2 ⊢ 𝐹 = {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐵 ∧ 𝜑)} | |
2 | f1of 5426 | . . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴⟶𝐵) | |
3 | 2 | anim1i 338 | . . . 4 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝜑) → (𝑓:𝐴⟶𝐵 ∧ 𝜑)) |
4 | 3 | ss2abi 3209 | . . 3 ⊢ {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐵 ∧ 𝜑)} ⊆ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ 𝜑)} |
5 | eqid 2164 | . . . 4 ⊢ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ 𝜑)} = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ 𝜑)} | |
6 | 5 | fabexg 5369 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ 𝜑)} ∈ V) |
7 | ssexg 4115 | . . 3 ⊢ (({𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐵 ∧ 𝜑)} ⊆ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ 𝜑)} ∧ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ 𝜑)} ∈ V) → {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐵 ∧ 𝜑)} ∈ V) | |
8 | 4, 6, 7 | sylancr 411 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐵 ∧ 𝜑)} ∈ V) |
9 | 1, 8 | eqeltrid 2251 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝐹 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1342 ∈ wcel 2135 {cab 2150 Vcvv 2721 ⊆ wss 3111 ⟶wf 5178 –1-1-onto→wf1o 5181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-xp 4604 df-rel 4605 df-cnv 4606 df-dm 4608 df-rn 4609 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-f1o 5189 |
This theorem is referenced by: (None) |
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