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Theorem dff3im 5340
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 5086 . 2 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
2 ffun 5076 . . . . . . . 8 (𝐹:𝐴𝐵 → Fun 𝐹)
32adantr 265 . . . . . . 7 ((𝐹:𝐴𝐵𝑥𝐴) → Fun 𝐹)
4 fdm 5078 . . . . . . . . 9 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
54eleq2d 2123 . . . . . . . 8 (𝐹:𝐴𝐵 → (𝑥 ∈ dom 𝐹𝑥𝐴))
65biimpar 285 . . . . . . 7 ((𝐹:𝐴𝐵𝑥𝐴) → 𝑥 ∈ dom 𝐹)
7 funfvop 5307 . . . . . . 7 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
83, 6, 7syl2anc 397 . . . . . 6 ((𝐹:𝐴𝐵𝑥𝐴) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
9 df-br 3793 . . . . . 6 (𝑥𝐹(𝐹𝑥) ↔ ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
108, 9sylibr 141 . . . . 5 ((𝐹:𝐴𝐵𝑥𝐴) → 𝑥𝐹(𝐹𝑥))
11 funfvex 5220 . . . . . . 7 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ V)
12 breq2 3796 . . . . . . . 8 (𝑦 = (𝐹𝑥) → (𝑥𝐹𝑦𝑥𝐹(𝐹𝑥)))
1312spcegv 2658 . . . . . . 7 ((𝐹𝑥) ∈ V → (𝑥𝐹(𝐹𝑥) → ∃𝑦 𝑥𝐹𝑦))
1411, 13syl 14 . . . . . 6 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝑥𝐹(𝐹𝑥) → ∃𝑦 𝑥𝐹𝑦))
153, 6, 14syl2anc 397 . . . . 5 ((𝐹:𝐴𝐵𝑥𝐴) → (𝑥𝐹(𝐹𝑥) → ∃𝑦 𝑥𝐹𝑦))
1610, 15mpd 13 . . . 4 ((𝐹:𝐴𝐵𝑥𝐴) → ∃𝑦 𝑥𝐹𝑦)
17 funmo 4945 . . . . . 6 (Fun 𝐹 → ∃*𝑦 𝑥𝐹𝑦)
182, 17syl 14 . . . . 5 (𝐹:𝐴𝐵 → ∃*𝑦 𝑥𝐹𝑦)
1918adantr 265 . . . 4 ((𝐹:𝐴𝐵𝑥𝐴) → ∃*𝑦 𝑥𝐹𝑦)
20 eu5 1963 . . . 4 (∃!𝑦 𝑥𝐹𝑦 ↔ (∃𝑦 𝑥𝐹𝑦 ∧ ∃*𝑦 𝑥𝐹𝑦))
2116, 19, 20sylanbrc 402 . . 3 ((𝐹:𝐴𝐵𝑥𝐴) → ∃!𝑦 𝑥𝐹𝑦)
2221ralrimiva 2409 . 2 (𝐹:𝐴𝐵 → ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦)
231, 22jca 294 1 (𝐹:𝐴𝐵 → (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wex 1397  wcel 1409  ∃!weu 1916  ∃*wmo 1917  wral 2323  Vcvv 2574  wss 2945  cop 3406   class class class wbr 3792   × cxp 4371  dom cdm 4373  Fun wfun 4924  wf 4926  cfv 4930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-fv 4938
This theorem is referenced by:  dff4im  5341
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