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Theorem elrnmpoid 42274
 Description: Membership in the range of an operation class abstraction. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypothesis
Ref Expression
elrnmpoid.1 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
Assertion
Ref Expression
elrnmpoid ((𝑥𝐴𝑦𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝐶𝑉) → (𝑥𝐹𝑦) ∈ ran 𝐹)
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem elrnmpoid
StepHypRef Expression
1 elrnmpoid.1 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
21fnmpo 7778 . . 3 (∀𝑥𝐴𝑦𝐵 𝐶𝑉𝐹 Fn (𝐴 × 𝐵))
323ad2ant3 1133 . 2 ((𝑥𝐴𝑦𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝐶𝑉) → 𝐹 Fn (𝐴 × 𝐵))
4 simp1 1134 . 2 ((𝑥𝐴𝑦𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝐶𝑉) → 𝑥𝐴)
5 simp2 1135 . 2 ((𝑥𝐴𝑦𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝐶𝑉) → 𝑦𝐵)
6 fnovrn 7326 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝑥𝐴𝑦𝐵) → (𝑥𝐹𝑦) ∈ ran 𝐹)
73, 4, 5, 6syl3anc 1369 1 ((𝑥𝐴𝑦𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝐶𝑉) → (𝑥𝐹𝑦) ∈ ran 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112  ∀wral 3071   × cxp 5527  ran crn 5530   Fn wfn 6336  (class class class)co 7157   ∈ cmpo 7159 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5174  ax-nul 5181  ax-pr 5303  ax-un 7466 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3700  df-csb 3809  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-nul 4229  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4803  df-iun 4889  df-br 5038  df-opab 5100  df-mpt 5118  df-id 5435  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-fv 6349  df-ov 7160  df-oprab 7161  df-mpo 7162  df-1st 7700  df-2nd 7701 This theorem is referenced by:  smflimlem1  43816  smflimlem2  43817
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