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Theorem fovrnd 5881
Description: An operation's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
Hypotheses
Ref Expression
fovrnd.1 (𝜑𝐹:(𝑅 × 𝑆)⟶𝐶)
fovrnd.2 (𝜑𝐴𝑅)
fovrnd.3 (𝜑𝐵𝑆)
Assertion
Ref Expression
fovrnd (𝜑 → (𝐴𝐹𝐵) ∈ 𝐶)

Proof of Theorem fovrnd
StepHypRef Expression
1 fovrnd.1 . 2 (𝜑𝐹:(𝑅 × 𝑆)⟶𝐶)
2 fovrnd.2 . 2 (𝜑𝐴𝑅)
3 fovrnd.3 . 2 (𝜑𝐵𝑆)
4 fovrn 5879 . 2 ((𝐹:(𝑅 × 𝑆)⟶𝐶𝐴𝑅𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝐶)
51, 2, 3, 4syl3anc 1199 1 (𝜑 → (𝐴𝐹𝐵) ∈ 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1463   × cxp 4505  wf 5087  (class class class)co 5740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-fv 5099  df-ov 5743
This theorem is referenced by:  eroveu  6486  isxmet2d  12423  ismet2  12429  comet  12574  bdmetval  12575  txmetcnp  12593  limccnp2lem  12720  limccnp2cntop  12721
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