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Theorem fovcl 5548
 Description: Closure law for an operation. (Contributed by NM, 19-Apr-2007.)
Hypothesis
Ref Expression
fovcl.1 𝐹:(𝑅 × 𝑆)⟶𝐶
Assertion
Ref Expression
fovcl ((A 𝑅 B 𝑆) → (A𝐹B) 𝐶)

Proof of Theorem fovcl
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fovcl.1 . . 3 𝐹:(𝑅 × 𝑆)⟶𝐶
2 ffnov 5547 . . . 4 (𝐹:(𝑅 × 𝑆)⟶𝐶 ↔ (𝐹 Fn (𝑅 × 𝑆) x 𝑅 y 𝑆 (x𝐹y) 𝐶))
32simprbi 260 . . 3 (𝐹:(𝑅 × 𝑆)⟶𝐶x 𝑅 y 𝑆 (x𝐹y) 𝐶)
41, 3ax-mp 7 . 2 x 𝑅 y 𝑆 (x𝐹y) 𝐶
5 oveq1 5462 . . . 4 (x = A → (x𝐹y) = (A𝐹y))
65eleq1d 2103 . . 3 (x = A → ((x𝐹y) 𝐶 ↔ (A𝐹y) 𝐶))
7 oveq2 5463 . . . 4 (y = B → (A𝐹y) = (A𝐹B))
87eleq1d 2103 . . 3 (y = B → ((A𝐹y) 𝐶 ↔ (A𝐹B) 𝐶))
96, 8rspc2v 2656 . 2 ((A 𝑅 B 𝑆) → (x 𝑅 y 𝑆 (x𝐹y) 𝐶 → (A𝐹B) 𝐶))
104, 9mpi 15 1 ((A 𝑅 B 𝑆) → (A𝐹B) 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390  ∀wral 2300   × cxp 4286   Fn wfn 4840  ⟶wf 4841  (class class class)co 5455 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fv 4853  df-ov 5458 This theorem is referenced by:  ixxssxr  8539  fzof  8771  elfzoelz  8774  fzoval  8775
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