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Theorem f1mptrn 30308
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 3179 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
3 f1mptrn.2 . . 3 ((𝜑𝑦𝐶) → ∃!𝑥𝐴 𝑦 = 𝐵)
43ralrimiva 3179 . 2 (𝜑 → ∀𝑦𝐶 ∃!𝑥𝐴 𝑦 = 𝐵)
5 eqid 2818 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
65f1ompt 6867 . . 3 ((𝑥𝐴𝐵):𝐴1-1-onto𝐶 ↔ (∀𝑥𝐴 𝐵𝐶 ∧ ∀𝑦𝐶 ∃!𝑥𝐴 𝑦 = 𝐵))
7 dff1o2 6613 . . . 4 ((𝑥𝐴𝐵):𝐴1-1-onto𝐶 ↔ ((𝑥𝐴𝐵) Fn 𝐴 ∧ Fun (𝑥𝐴𝐵) ∧ ran (𝑥𝐴𝐵) = 𝐶))
87simp2bi 1138 . . 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 1528  wcel 2105  wral 3135  ∃!wreu 3137  cmpt 5137  ccnv 5547  ran crn 5549  Fun wfun 6342   Fn wfn 6343  1-1-ontowf1o 6347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356
This theorem is referenced by:  esum2dlem  31250
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