Theorem dff4im 5848

Description: Property of a mapping. (Contributed by Jim Kingdon, 4-Jan-2019.)

Assertion

Ref	Expression
dff4im	⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝐹𝑦))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦

Proof of Theorem dff4im

Step	Hyp	Ref	Expression
1		dff3im 5847	. 2 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦))
2		df-br 4129	. . . . . . . 8 ⊢ (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
3		ssel 3242	. . . . . . . . 9 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
4		opelxp2 4807	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑦 ∈ 𝐵)
5	3, 4	syl6 33	. . . . . . . 8 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → 𝑦 ∈ 𝐵))
6	2, 5	biimtrid 152	. . . . . . 7 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (𝑥𝐹𝑦 → 𝑦 ∈ 𝐵))
7	6	pm4.71rd 398	. . . . . 6 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (𝑥𝐹𝑦 ↔ (𝑦 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
8	7	eubidv 2094	. . . . 5 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (∃!𝑦 𝑥𝐹𝑦 ↔ ∃!𝑦(𝑦 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
9		df-reu 2535	. . . . 5 ⊢ (∃!𝑦 ∈ 𝐵 𝑥𝐹𝑦 ↔ ∃!𝑦(𝑦 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
10	8, 9	bitr4di 198	. . . 4 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (∃!𝑦 𝑥𝐹𝑦 ↔ ∃!𝑦 ∈ 𝐵 𝑥𝐹𝑦))
11	10	ralbidv 2550	. . 3 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦 ↔ ∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝐹𝑦))
12	11	pm5.32i 458	. 2 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦) ↔ (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝐹𝑦))
13	1, 12	sylib 122	1 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝐹𝑦))

Syntax hints:   → wi 4   ∧ wa 104  ∃!weu 2086   ∈ wcel 2209  ∀wral 2528  ∃!wreu 2530   ⊆ wss 3220  ⟨cop 3711   class class class wbr 4128   × cxp 4770  ⟶wf 5371

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-fv 5383

This theorem is referenced by: (None)