Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6617 | . . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴⟶𝐵) | |
3 | 2 | anim1i 616 | . . . 4 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝜑) → (𝑓:𝐴⟶𝐵 ∧ 𝜑)) |
4 | 3 | ss2abi 4045 | . . 3 ⊢ {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐵 ∧ 𝜑)} ⊆ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ 𝜑)} |
5 | eqid 2823 | . . . 4 ⊢ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ 𝜑)} = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ 𝜑)} | |
6 | 5 | fabexg 7641 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ 𝜑)} ∈ V) |
7 | ssexg 5229 | . . 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 589 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐵 ∧ 𝜑)} ∈ V) |
9 | 1, 8 | eqeltrid 2919 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝐹 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {cab 2801 Vcvv 3496 ⊆ wss 3938 ⟶wf 6353 –1-1-onto→wf1o 6356 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-cnv 5565 df-dm 5567 df-rn 5568 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-f1o 6364 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |