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Mirrors > Home > MPE Home > Th. List > f1oabexg | Structured version Visualization version 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 6612 | . . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴⟶𝐵) | |
3 | 2 | anim1i 618 | . . . 4 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝜑) → (𝑓:𝐴⟶𝐵 ∧ 𝜑)) |
4 | 3 | ss2abi 3954 | . . 3 ⊢ {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐵 ∧ 𝜑)} ⊆ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ 𝜑)} |
5 | eqid 2738 | . . . 4 ⊢ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ 𝜑)} = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ 𝜑)} | |
6 | 5 | fabexg 7658 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ 𝜑)} ∈ V) |
7 | ssexg 5188 | . . 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 590 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐵 ∧ 𝜑)} ∈ V) |
9 | 1, 8 | eqeltrid 2837 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝐹 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2113 {cab 2716 Vcvv 3397 ⊆ wss 3841 ⟶wf 6329 –1-1-onto→wf1o 6332 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2074 df-clab 2717 df-cleq 2730 df-clel 2811 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-br 5028 df-opab 5090 df-xp 5525 df-rel 5526 df-cnv 5527 df-dm 5529 df-rn 5530 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-f1o 6340 |
This theorem is referenced by: (None) |
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