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Theorem ofc1g 6143
Description: Left operation by a constant. (Contributed by Mario Carneiro, 24-Jul-2014.)
Hypotheses
Ref Expression
ofc1.1 (𝜑𝐴𝑉)
ofc1.2 (𝜑𝐵𝑊)
ofc1.3 (𝜑𝐹 Fn 𝐴)
ofc1.4 ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)
ofc1g.ex ((𝜑𝑋𝐴) → (𝐵𝑅𝐶) ∈ 𝑈)
Assertion
Ref Expression
ofc1g ((𝜑𝑋𝐴) → (((𝐴 × {𝐵}) ∘𝑓 𝑅𝐹)‘𝑋) = (𝐵𝑅𝐶))

Proof of Theorem ofc1g
StepHypRef Expression
1 ofc1.2 . . 3 (𝜑𝐵𝑊)
2 fnconstg 5443 . . 3 (𝐵𝑊 → (𝐴 × {𝐵}) Fn 𝐴)
31, 2syl 14 . 2 (𝜑 → (𝐴 × {𝐵}) Fn 𝐴)
4 ofc1.3 . 2 (𝜑𝐹 Fn 𝐴)
5 ofc1.1 . 2 (𝜑𝐴𝑉)
6 inidm 3368 . 2 (𝐴𝐴) = 𝐴
7 fvconst2g 5764 . . 3 ((𝐵𝑊𝑋𝐴) → ((𝐴 × {𝐵})‘𝑋) = 𝐵)
81, 7sylan 283 . 2 ((𝜑𝑋𝐴) → ((𝐴 × {𝐵})‘𝑋) = 𝐵)
9 ofc1.4 . 2 ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)
10 ofc1g.ex . 2 ((𝜑𝑋𝐴) → (𝐵𝑅𝐶) ∈ 𝑈)
113, 4, 5, 5, 6, 8, 9, 10ofvalg 6132 1 ((𝜑𝑋𝐴) → (((𝐴 × {𝐵}) ∘𝑓 𝑅𝐹)‘𝑋) = (𝐵𝑅𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2164  {csn 3618   × cxp 4653   Fn wfn 5241  cfv 5246  (class class class)co 5910  𝑓 cof 6120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-setind 4565
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4322  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-iota 5207  df-fun 5248  df-fn 5249  df-f 5250  df-f1 5251  df-fo 5252  df-f1o 5253  df-fv 5254  df-ov 5913  df-oprab 5914  df-mpo 5915  df-of 6122
This theorem is referenced by:  ofnegsub  8971
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