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Theorem f1mptrn 32645
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 3146 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
3 f1mptrn.2 . . 3 ((𝜑𝑦𝐶) → ∃!𝑥𝐴 𝑦 = 𝐵)
43ralrimiva 3146 . 2 (𝜑 → ∀𝑦𝐶 ∃!𝑥𝐴 𝑦 = 𝐵)
5 eqid 2737 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
65f1ompt 7131 . . 3 ((𝑥𝐴𝐵):𝐴1-1-onto𝐶 ↔ (∀𝑥𝐴 𝐵𝐶 ∧ ∀𝑦𝐶 ∃!𝑥𝐴 𝑦 = 𝐵))
7 dff1o2 6853 . . . 4 ((𝑥𝐴𝐵):𝐴1-1-onto𝐶 ↔ ((𝑥𝐴𝐵) Fn 𝐴 ∧ Fun (𝑥𝐴𝐵) ∧ ran (𝑥𝐴𝐵) = 𝐶))
87simp2bi 1147 . . 3 ((𝑥𝐴𝐵):𝐴1-1-onto𝐶 → Fun (𝑥𝐴𝐵))
96, 8sylbir 235 . 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 395   = wceq 1540  wcel 2108  wral 3061  ∃!wreu 3378  cmpt 5225  ccnv 5684  ran crn 5686  Fun wfun 6555   Fn wfn 6556  1-1-ontowf1o 6560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568
This theorem is referenced by:  esum2dlem  34093
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