Theorem fconst2g 7132

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 7091	. . . . . . 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 7131	. . . . . . 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 2768	. . . . 5 ⊢ (((𝐹:𝐴⟶{𝐵} ∧ 𝐵 ∈ 𝐶) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = ((𝐴 × {𝐵})‘𝑥))
6	5	ralrimiva 3122	. . . 4 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝐵 ∈ 𝐶) → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = ((𝐴 × {𝐵})‘𝑥))
7		ffn 6647	. . . . 5 ⊢ (𝐹:𝐴⟶{𝐵} → 𝐹 Fn 𝐴)
8		fnconstg 6707	. . . . 5 ⊢ (𝐵 ∈ 𝐶 → (𝐴 × {𝐵}) Fn 𝐴)
9		eqfnfv 6959	. . . . 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 6706	. . 3 ⊢ (𝐵 ∈ 𝐶 → (𝐴 × {𝐵}):𝐴⟶{𝐵})
14		feq1 6625	. . 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 2110  ∀wral 3045  {csn 4574   × cxp 5612   Fn wfn 6472  ⟶wf 6473  ‘cfv 6477

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 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368

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 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-fv 6485

This theorem is referenced by:  fconst2  7134  fconst5  7135  snmapen  8955  repsdf2  14677  cnconst  23192  fconst7v  32593  padct  32691  prv1n  35443  fconst7  45280  eufsnlem  48851  mofsn  48854  mofeu  48858