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Theorem dff3im 5682
Description: Property of a mapping. (Contributed by Jim Kingdon, 4-Jan-2019.)
Assertion
Ref Expression
dff3im (𝐹:𝐴𝐵 → (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦

Proof of Theorem dff3im
StepHypRef Expression
1 fssxp 5402 . 2 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
2 ffun 5387 . . . . . . . 8 (𝐹:𝐴𝐵 → Fun 𝐹)
32adantr 276 . . . . . . 7 ((𝐹:𝐴𝐵𝑥𝐴) → Fun 𝐹)
4 fdm 5390 . . . . . . . . 9 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
54eleq2d 2259 . . . . . . . 8 (𝐹:𝐴𝐵 → (𝑥 ∈ dom 𝐹𝑥𝐴))
65biimpar 297 . . . . . . 7 ((𝐹:𝐴𝐵𝑥𝐴) → 𝑥 ∈ dom 𝐹)
7 funfvop 5649 . . . . . . 7 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
83, 6, 7syl2anc 411 . . . . . 6 ((𝐹:𝐴𝐵𝑥𝐴) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
9 df-br 4019 . . . . . 6 (𝑥𝐹(𝐹𝑥) ↔ ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
108, 9sylibr 134 . . . . 5 ((𝐹:𝐴𝐵𝑥𝐴) → 𝑥𝐹(𝐹𝑥))
11 funfvex 5551 . . . . . . 7 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ V)
12 breq2 4022 . . . . . . . 8 (𝑦 = (𝐹𝑥) → (𝑥𝐹𝑦𝑥𝐹(𝐹𝑥)))
1312spcegv 2840 . . . . . . 7 ((𝐹𝑥) ∈ V → (𝑥𝐹(𝐹𝑥) → ∃𝑦 𝑥𝐹𝑦))
1411, 13syl 14 . . . . . 6 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝑥𝐹(𝐹𝑥) → ∃𝑦 𝑥𝐹𝑦))
153, 6, 14syl2anc 411 . . . . 5 ((𝐹:𝐴𝐵𝑥𝐴) → (𝑥𝐹(𝐹𝑥) → ∃𝑦 𝑥𝐹𝑦))
1610, 15mpd 13 . . . 4 ((𝐹:𝐴𝐵𝑥𝐴) → ∃𝑦 𝑥𝐹𝑦)
17 funmo 5250 . . . . . 6 (Fun 𝐹 → ∃*𝑦 𝑥𝐹𝑦)
182, 17syl 14 . . . . 5 (𝐹:𝐴𝐵 → ∃*𝑦 𝑥𝐹𝑦)
1918adantr 276 . . . 4 ((𝐹:𝐴𝐵𝑥𝐴) → ∃*𝑦 𝑥𝐹𝑦)
20 eu5 2085 . . . 4 (∃!𝑦 𝑥𝐹𝑦 ↔ (∃𝑦 𝑥𝐹𝑦 ∧ ∃*𝑦 𝑥𝐹𝑦))
2116, 19, 20sylanbrc 417 . . 3 ((𝐹:𝐴𝐵𝑥𝐴) → ∃!𝑦 𝑥𝐹𝑦)
2221ralrimiva 2563 . 2 (𝐹:𝐴𝐵 → ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦)
231, 22jca 306 1 (𝐹:𝐴𝐵 → (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wex 1503  ∃!weu 2038  ∃*wmo 2039  wcel 2160  wral 2468  Vcvv 2752  wss 3144  cop 3610   class class class wbr 4018   × cxp 4642  dom cdm 4644  Fun wfun 5229  wf 5231  cfv 5235
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243
This theorem is referenced by:  dff4im  5683
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