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Theorem f1mptrn 30295
Description: Express injection for a mapping operation. (Contributed by Thierry Arnoux, 3-May-2020.)
Hypotheses
Ref Expression
f1mptrn.1 ((𝜑𝑥𝐴) → 𝐵𝐶)
f1mptrn.2 ((𝜑𝑦𝐶) → ∃!𝑥𝐴 𝑦 = 𝐵)
Assertion
Ref Expression
f1mptrn (𝜑 → Fun (𝑥𝐴𝐵))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵   𝑥,𝐶,𝑦   𝜑,𝑥,𝑦
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem f1mptrn
StepHypRef Expression
1 f1mptrn.1 . . 3 ((𝜑𝑥𝐴) → 𝐵𝐶)
21ralrimiva 3187 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
3 f1mptrn.2 . . 3 ((𝜑𝑦𝐶) → ∃!𝑥𝐴 𝑦 = 𝐵)
43ralrimiva 3187 . 2 (𝜑 → ∀𝑦𝐶 ∃!𝑥𝐴 𝑦 = 𝐵)
5 eqid 2826 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
65f1ompt 6871 . . 3 ((𝑥𝐴𝐵):𝐴1-1-onto𝐶 ↔ (∀𝑥𝐴 𝐵𝐶 ∧ ∀𝑦𝐶 ∃!𝑥𝐴 𝑦 = 𝐵))
7 dff1o2 6617 . . . 4 ((𝑥𝐴𝐵):𝐴1-1-onto𝐶 ↔ ((𝑥𝐴𝐵) Fn 𝐴 ∧ Fun (𝑥𝐴𝐵) ∧ ran (𝑥𝐴𝐵) = 𝐶))
87simp2bi 1140 . . 3 ((𝑥𝐴𝐵):𝐴1-1-onto𝐶 → Fun (𝑥𝐴𝐵))
96, 8sylbir 236 . 2 ((∀𝑥𝐴 𝐵𝐶 ∧ ∀𝑦𝐶 ∃!𝑥𝐴 𝑦 = 𝐵) → Fun (𝑥𝐴𝐵))
102, 4, 9syl2anc 584 1 (𝜑 → Fun (𝑥𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  wral 3143  ∃!wreu 3145  cmpt 5143  ccnv 5553  ran crn 5555  Fun wfun 6346   Fn wfn 6347  1-1-ontowf1o 6351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360
This theorem is referenced by:  esum2dlem  31237
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