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Theorem f1mptrn 32552
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 3136 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
3 f1mptrn.2 . . 3 ((𝜑𝑦𝐶) → ∃!𝑥𝐴 𝑦 = 𝐵)
43ralrimiva 3136 . 2 (𝜑 → ∀𝑦𝐶 ∃!𝑥𝐴 𝑦 = 𝐵)
5 eqid 2726 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
65f1ompt 7117 . . 3 ((𝑥𝐴𝐵):𝐴1-1-onto𝐶 ↔ (∀𝑥𝐴 𝐵𝐶 ∧ ∀𝑦𝐶 ∃!𝑥𝐴 𝑦 = 𝐵))
7 dff1o2 6840 . . . 4 ((𝑥𝐴𝐵):𝐴1-1-onto𝐶 ↔ ((𝑥𝐴𝐵) Fn 𝐴 ∧ Fun (𝑥𝐴𝐵) ∧ ran (𝑥𝐴𝐵) = 𝐶))
87simp2bi 1143 . . 3 ((𝑥𝐴𝐵):𝐴1-1-onto𝐶 → Fun (𝑥𝐴𝐵))
96, 8sylbir 234 . 2 ((∀𝑥𝐴 𝐵𝐶 ∧ ∀𝑦𝐶 ∃!𝑥𝐴 𝑦 = 𝐵) → Fun (𝑥𝐴𝐵))
102, 4, 9syl2anc 582 1 (𝜑 → Fun (𝑥𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  wral 3051  ∃!wreu 3362  cmpt 5228  ccnv 5673  ran crn 5675  Fun wfun 6540   Fn wfn 6541  1-1-ontowf1o 6545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553
This theorem is referenced by:  esum2dlem  33938
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