Theorem f1oabexg 5335

Description: The class of all 1-1-onto functions mapping one set to another is a set. (Contributed by Paul Chapman, 25-Feb-2008.)

Hypothesis

Ref	Expression
f1oabexg.1	⊢ 𝐹 = {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐵 ∧ 𝜑)}

Assertion

Ref	Expression
f1oabexg	⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝐹 ∈ V)

Distinct variable groups:   𝐴,𝑓   𝐵,𝑓

Allowed substitution hints:   𝜑(𝑓)   𝐶(𝑓)   𝐷(𝑓)   𝐹(𝑓)

Proof of Theorem f1oabexg

Step	Hyp	Ref	Expression
1		f1oabexg.1	. 2 ⊢ 𝐹 = {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐵 ∧ 𝜑)}
2		f1of 5323	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴⟶𝐵)
3	2	anim1i 336	. . . 4 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝜑) → (𝑓:𝐴⟶𝐵 ∧ 𝜑))
4	3	ss2abi 3135	. . 3 ⊢ {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐵 ∧ 𝜑)} ⊆ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ 𝜑)}
5		eqid 2115	. . . 4 ⊢ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ 𝜑)} = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ 𝜑)}
6	5	fabexg 5268	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ 𝜑)} ∈ V)
7		ssexg 4027	. . 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 408	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐵 ∧ 𝜑)} ∈ V)
9	1, 8	syl5eqel 2201	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝐹 ∈ V)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1314   ∈ wcel 1463  {cab 2101  Vcvv 2657   ⊆ wss 3037  ⟶wf 5077  –1-1-onto→wf1o 5080

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-opab 3950  df-xp 4505  df-rel 4506  df-cnv 4507  df-dm 4509  df-rn 4510  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-f1o 5088

This theorem is referenced by: (None)