Theorem fconst2g 7201

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 7160	. . . . . . 7 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
2	1	adantlr 727	. . . . . 6 ⊢ (((𝐹:𝐴⟶{𝐵} ∧ 𝐵 ∈ 𝐶) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
3		fvconst2g 7200	. . . . . . 7 ⊢ ((𝐵 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵)
4	3	adantll 726	. . . . . 6 ⊢ (((𝐹:𝐴⟶{𝐵} ∧ 𝐵 ∈ 𝐶) ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵)
5	2, 4	eqtr4d 2799	. . . . 5 ⊢ (((𝐹:𝐴⟶{𝐵} ∧ 𝐵 ∈ 𝐶) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = ((𝐴 × {𝐵})‘𝑥))
6	5	ralrimiva 3155	. . . 4 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝐵 ∈ 𝐶) → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = ((𝐴 × {𝐵})‘𝑥))
7		ffn 6705	. . . . 5 ⊢ (𝐹:𝐴⟶{𝐵} → 𝐹 Fn 𝐴)
8		fnconstg 6766	. . . . 5 ⊢ (𝐵 ∈ 𝐶 → (𝐴 × {𝐵}) Fn 𝐴)
9		eqfnfv 7025	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐴 × {𝐵}) Fn 𝐴) → (𝐹 = (𝐴 × {𝐵}) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = ((𝐴 × {𝐵})‘𝑥)))
10	7, 8, 9	syl2an 607	. . . 4 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝐵 ∈ 𝐶) → (𝐹 = (𝐴 × {𝐵}) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = ((𝐴 × {𝐵})‘𝑥)))
11	6, 10	mpbird 260	. . 3 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝐵 ∈ 𝐶) → 𝐹 = (𝐴 × {𝐵}))
12	11	expcom 418	. 2 ⊢ (𝐵 ∈ 𝐶 → (𝐹:𝐴⟶{𝐵} → 𝐹 = (𝐴 × {𝐵})))
13		fconstg 6765	. . 3 ⊢ (𝐵 ∈ 𝐶 → (𝐴 × {𝐵}):𝐴⟶{𝐵})
14		feq1 6683	. . 3 ⊢ (𝐹 = (𝐴 × {𝐵}) → (𝐹:𝐴⟶{𝐵} ↔ (𝐴 × {𝐵}):𝐴⟶{𝐵}))
15	13, 14	syl5ibrcom 250	. 2 ⊢ (𝐵 ∈ 𝐶 → (𝐹 = (𝐴 × {𝐵}) → 𝐹:𝐴⟶{𝐵}))
16	12, 15	impbid 215	1 ⊢ (𝐵 ∈ 𝐶 → (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵})))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1568   ∈ wcel 2141  ∀wral 3077  {csn 4588   × cxp 5659   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544

This theorem is referenced by:  fconst2  7203  fconst5  7204  snmapen  9034  repsdf2  14815  cnconst  23420  fconst7v  32931  padct  33029  prv1n  35889  fconst7  45949  eufsnlem  49586  mofsn  49589  mofeu  49593