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Theorem fconst2g 7219
Description: A constant function expressed as a Cartesian product. (Contributed by NM, 27-Nov-2007.)
Assertion
Ref Expression
fconst2g (𝐵𝐶 → (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵})))

Proof of Theorem fconst2g
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fvconst 7177 . . . . . . 7 ((𝐹:𝐴⟶{𝐵} ∧ 𝑥𝐴) → (𝐹𝑥) = 𝐵)
21adantlr 713 . . . . . 6 (((𝐹:𝐴⟶{𝐵} ∧ 𝐵𝐶) ∧ 𝑥𝐴) → (𝐹𝑥) = 𝐵)
3 fvconst2g 7218 . . . . . . 7 ((𝐵𝐶𝑥𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵)
43adantll 712 . . . . . 6 (((𝐹:𝐴⟶{𝐵} ∧ 𝐵𝐶) ∧ 𝑥𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵)
52, 4eqtr4d 2768 . . . . 5 (((𝐹:𝐴⟶{𝐵} ∧ 𝐵𝐶) ∧ 𝑥𝐴) → (𝐹𝑥) = ((𝐴 × {𝐵})‘𝑥))
65ralrimiva 3135 . . . 4 ((𝐹:𝐴⟶{𝐵} ∧ 𝐵𝐶) → ∀𝑥𝐴 (𝐹𝑥) = ((𝐴 × {𝐵})‘𝑥))
7 ffn 6727 . . . . 5 (𝐹:𝐴⟶{𝐵} → 𝐹 Fn 𝐴)
8 fnconstg 6789 . . . . 5 (𝐵𝐶 → (𝐴 × {𝐵}) Fn 𝐴)
9 eqfnfv 7043 . . . . 5 ((𝐹 Fn 𝐴 ∧ (𝐴 × {𝐵}) Fn 𝐴) → (𝐹 = (𝐴 × {𝐵}) ↔ ∀𝑥𝐴 (𝐹𝑥) = ((𝐴 × {𝐵})‘𝑥)))
107, 8, 9syl2an 594 . . . 4 ((𝐹:𝐴⟶{𝐵} ∧ 𝐵𝐶) → (𝐹 = (𝐴 × {𝐵}) ↔ ∀𝑥𝐴 (𝐹𝑥) = ((𝐴 × {𝐵})‘𝑥)))
116, 10mpbird 256 . . 3 ((𝐹:𝐴⟶{𝐵} ∧ 𝐵𝐶) → 𝐹 = (𝐴 × {𝐵}))
1211expcom 412 . 2 (𝐵𝐶 → (𝐹:𝐴⟶{𝐵} → 𝐹 = (𝐴 × {𝐵})))
13 fconstg 6788 . . 3 (𝐵𝐶 → (𝐴 × {𝐵}):𝐴⟶{𝐵})
14 feq1 6708 . . 3 (𝐹 = (𝐴 × {𝐵}) → (𝐹:𝐴⟶{𝐵} ↔ (𝐴 × {𝐵}):𝐴⟶{𝐵}))
1513, 14syl5ibrcom 246 . 2 (𝐵𝐶 → (𝐹 = (𝐴 × {𝐵}) → 𝐹:𝐴⟶{𝐵}))
1612, 15impbid 211 1 (𝐵𝐶 → (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3050  {csn 4632   × cxp 5679   Fn wfn 6548  wf 6549  cfv 6553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5303  ax-nul 5310  ax-pr 5432
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4325  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5579  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-res 5693  df-ima 5694  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561
This theorem is referenced by:  fconst2  7221  fconst5  7222  snmapen  9073  repsdf2  14781  cnconst  23271  padct  32624  prv1n  35211  fconst7  44811  eufsnlem  48145  mofsn  48148  mofeu  48152
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