Theorem fconst2g 7146

Description: A constant function expressed as a Cartesian product. (Contributed by NM, 27-Nov-2007.)

Assertion

Ref	Expression
fconst2g	⊢ (𝐵 ∈ 𝐶 → (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵})))

Proof of Theorem fconst2g

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvconst 7105	. . . . . . 7 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
2	1	adantlr 715	. . . . . 6 ⊢ (((𝐹:𝐴⟶{𝐵} ∧ 𝐵 ∈ 𝐶) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
3		fvconst2g 7145	. . . . . . 7 ⊢ ((𝐵 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵)
4	3	adantll 714	. . . . . 6 ⊢ (((𝐹:𝐴⟶{𝐵} ∧ 𝐵 ∈ 𝐶) ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵)
5	2, 4	eqtr4d 2771	. . . . 5 ⊢ (((𝐹:𝐴⟶{𝐵} ∧ 𝐵 ∈ 𝐶) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = ((𝐴 × {𝐵})‘𝑥))
6	5	ralrimiva 3126	. . . 4 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝐵 ∈ 𝐶) → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = ((𝐴 × {𝐵})‘𝑥))
7		ffn 6659	. . . . 5 ⊢ (𝐹:𝐴⟶{𝐵} → 𝐹 Fn 𝐴)
8		fnconstg 6719	. . . . 5 ⊢ (𝐵 ∈ 𝐶 → (𝐴 × {𝐵}) Fn 𝐴)
9		eqfnfv 6973	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐴 × {𝐵}) Fn 𝐴) → (𝐹 = (𝐴 × {𝐵}) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = ((𝐴 × {𝐵})‘𝑥)))
10	7, 8, 9	syl2an 596	. . . 4 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝐵 ∈ 𝐶) → (𝐹 = (𝐴 × {𝐵}) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = ((𝐴 × {𝐵})‘𝑥)))
11	6, 10	mpbird 257	. . 3 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝐵 ∈ 𝐶) → 𝐹 = (𝐴 × {𝐵}))
12	11	expcom 413	. 2 ⊢ (𝐵 ∈ 𝐶 → (𝐹:𝐴⟶{𝐵} → 𝐹 = (𝐴 × {𝐵})))
13		fconstg 6718	. . 3 ⊢ (𝐵 ∈ 𝐶 → (𝐴 × {𝐵}):𝐴⟶{𝐵})
14		feq1 6637	. . 3 ⊢ (𝐹 = (𝐴 × {𝐵}) → (𝐹:𝐴⟶{𝐵} ↔ (𝐴 × {𝐵}):𝐴⟶{𝐵}))
15	13, 14	syl5ibrcom 247	. 2 ⊢ (𝐵 ∈ 𝐶 → (𝐹 = (𝐴 × {𝐵}) → 𝐹:𝐴⟶{𝐵}))
16	12, 15	impbid 212	1 ⊢ (𝐵 ∈ 𝐶 → (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵})))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3049  {csn 4577   × cxp 5619   Fn wfn 6484  ⟶wf 6485  ‘cfv 6489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-fv 6497

This theorem is referenced by:  fconst2  7148  fconst5  7149  snmapen  8970  repsdf2  14695  cnconst  23209  fconst7v  32614  padct  32712  prv1n  35486  fconst7  45375  eufsnlem  48955  mofsn  48958  mofeu  48962