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Theorem f1mptrn 29419
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 . . . 4 ((𝜑𝑥𝐴) → 𝐵𝐶)
21ralrimiva 2965 . . 3 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
3 f1mptrn.2 . . . 4 ((𝜑𝑦𝐶) → ∃!𝑥𝐴 𝑦 = 𝐵)
43ralrimiva 2965 . . 3 (𝜑 → ∀𝑦𝐶 ∃!𝑥𝐴 𝑦 = 𝐵)
52, 4jca 554 . 2 (𝜑 → (∀𝑥𝐴 𝐵𝐶 ∧ ∀𝑦𝐶 ∃!𝑥𝐴 𝑦 = 𝐵))
6 eqid 2621 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
76f1ompt 6380 . . 3 ((𝑥𝐴𝐵):𝐴1-1-onto𝐶 ↔ (∀𝑥𝐴 𝐵𝐶 ∧ ∀𝑦𝐶 ∃!𝑥𝐴 𝑦 = 𝐵))
8 dff1o2 6140 . . . 4 ((𝑥𝐴𝐵):𝐴1-1-onto𝐶 ↔ ((𝑥𝐴𝐵) Fn 𝐴 ∧ Fun (𝑥𝐴𝐵) ∧ ran (𝑥𝐴𝐵) = 𝐶))
98simp2bi 1076 . . 3 ((𝑥𝐴𝐵):𝐴1-1-onto𝐶 → Fun (𝑥𝐴𝐵))
107, 9sylbir 225 . 2 ((∀𝑥𝐴 𝐵𝐶 ∧ ∀𝑦𝐶 ∃!𝑥𝐴 𝑦 = 𝐵) → Fun (𝑥𝐴𝐵))
115, 10syl 17 1 (𝜑 → Fun (𝑥𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1482  wcel 1989  wral 2911  ∃!wreu 2913  cmpt 4727  ccnv 5111  ran crn 5113  Fun wfun 5880   Fn wfn 5881  1-1-ontowf1o 5885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894
This theorem is referenced by:  esum2dlem  30139
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