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Theorem fconst2g 6611
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 6573 . . . . . . 7 ((𝐹:𝐴⟶{𝐵} ∧ 𝑥𝐴) → (𝐹𝑥) = 𝐵)
21adantlr 694 . . . . . 6 (((𝐹:𝐴⟶{𝐵} ∧ 𝐵𝐶) ∧ 𝑥𝐴) → (𝐹𝑥) = 𝐵)
3 fvconst2g 6610 . . . . . . 7 ((𝐵𝐶𝑥𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵)
43adantll 693 . . . . . 6 (((𝐹:𝐴⟶{𝐵} ∧ 𝐵𝐶) ∧ 𝑥𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵)
52, 4eqtr4d 2808 . . . . 5 (((𝐹:𝐴⟶{𝐵} ∧ 𝐵𝐶) ∧ 𝑥𝐴) → (𝐹𝑥) = ((𝐴 × {𝐵})‘𝑥))
65ralrimiva 3115 . . . 4 ((𝐹:𝐴⟶{𝐵} ∧ 𝐵𝐶) → ∀𝑥𝐴 (𝐹𝑥) = ((𝐴 × {𝐵})‘𝑥))
7 ffn 6183 . . . . 5 (𝐹:𝐴⟶{𝐵} → 𝐹 Fn 𝐴)
8 fnconstg 6231 . . . . 5 (𝐵𝐶 → (𝐴 × {𝐵}) Fn 𝐴)
9 eqfnfv 6453 . . . . 5 ((𝐹 Fn 𝐴 ∧ (𝐴 × {𝐵}) Fn 𝐴) → (𝐹 = (𝐴 × {𝐵}) ↔ ∀𝑥𝐴 (𝐹𝑥) = ((𝐴 × {𝐵})‘𝑥)))
107, 8, 9syl2an 583 . . . 4 ((𝐹:𝐴⟶{𝐵} ∧ 𝐵𝐶) → (𝐹 = (𝐴 × {𝐵}) ↔ ∀𝑥𝐴 (𝐹𝑥) = ((𝐴 × {𝐵})‘𝑥)))
116, 10mpbird 247 . . 3 ((𝐹:𝐴⟶{𝐵} ∧ 𝐵𝐶) → 𝐹 = (𝐴 × {𝐵}))
1211expcom 398 . 2 (𝐵𝐶 → (𝐹:𝐴⟶{𝐵} → 𝐹 = (𝐴 × {𝐵})))
13 fconstg 6230 . . 3 (𝐵𝐶 → (𝐴 × {𝐵}):𝐴⟶{𝐵})
14 feq1 6164 . . 3 (𝐹 = (𝐴 × {𝐵}) → (𝐹:𝐴⟶{𝐵} ↔ (𝐴 × {𝐵}):𝐴⟶{𝐵}))
1513, 14syl5ibrcom 237 . 2 (𝐵𝐶 → (𝐹 = (𝐴 × {𝐵}) → 𝐹:𝐴⟶{𝐵}))
1612, 15impbid 202 1 (𝐵𝐶 → (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  {csn 4316   × cxp 5247   Fn wfn 6024  wf 6025  cfv 6029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-fv 6037
This theorem is referenced by:  fconst2  6613  fconst5  6614  snmapen  8190  repsdf2  13730  cnconst  21305  padct  29833  fconst7  39999
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