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Theorem rnmptc 39870
 Description: Range of a constant function in map to notation. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
rnmptc.f 𝐹 = (𝑥𝐴𝐵)
rnmptc.b ((𝜑𝑥𝐴) → 𝐵𝐶)
rnmptc.a (𝜑𝐴 ≠ ∅)
Assertion
Ref Expression
rnmptc (𝜑 → ran 𝐹 = {𝐵})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝜑,𝑥
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem rnmptc
StepHypRef Expression
1 rnmptc.f . . 3 𝐹 = (𝑥𝐴𝐵)
2 fconstmpt 5320 . . 3 (𝐴 × {𝐵}) = (𝑥𝐴𝐵)
31, 2eqtr4i 2785 . 2 𝐹 = (𝐴 × {𝐵})
4 rnmptc.b . . . . 5 ((𝜑𝑥𝐴) → 𝐵𝐶)
54, 1fmptd 6549 . . . 4 (𝜑𝐹:𝐴𝐶)
6 ffn 6206 . . . 4 (𝐹:𝐴𝐶𝐹 Fn 𝐴)
75, 6syl 17 . . 3 (𝜑𝐹 Fn 𝐴)
8 rnmptc.a . . 3 (𝜑𝐴 ≠ ∅)
9 fconst5 6636 . . 3 ((𝐹 Fn 𝐴𝐴 ≠ ∅) → (𝐹 = (𝐴 × {𝐵}) ↔ ran 𝐹 = {𝐵}))
107, 8, 9syl2anc 696 . 2 (𝜑 → (𝐹 = (𝐴 × {𝐵}) ↔ ran 𝐹 = {𝐵}))
113, 10mpbii 223 1 (𝜑 → ran 𝐹 = {𝐵})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1632   ∈ wcel 2139   ≠ wne 2932  ∅c0 4058  {csn 4321   ↦ cmpt 4881   × cxp 5264  ran crn 5267   Fn wfn 6044  ⟶wf 6045 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fo 6055  df-fv 6057 This theorem is referenced by:  limsup0  40447  limsuppnfdlem  40454  limsup10ex  40526  liminf10ex  40527  fourierdlem60  40904  fourierdlem61  40905  sge0z  41113
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