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Theorem rnmptpr 45182
Description: Range of a function defined on an unordered pair. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
rnmptpr.a (𝜑𝐴𝑉)
rnmptpr.b (𝜑𝐵𝑊)
rnmptpr.f 𝐹 = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐶)
rnmptpr.d (𝑥 = 𝐴𝐶 = 𝐷)
rnmptpr.e (𝑥 = 𝐵𝐶 = 𝐸)
Assertion
Ref Expression
rnmptpr (𝜑 → ran 𝐹 = {𝐷, 𝐸})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐷   𝑥,𝐸
Allowed substitution hints:   𝜑(𝑥)   𝐶(𝑥)   𝐹(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem rnmptpr
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 rnmptpr.a . . . 4 (𝜑𝐴𝑉)
2 rnmptpr.b . . . 4 (𝜑𝐵𝑊)
3 rnmptpr.d . . . . . 6 (𝑥 = 𝐴𝐶 = 𝐷)
43eqeq2d 2748 . . . . 5 (𝑥 = 𝐴 → (𝑦 = 𝐶𝑦 = 𝐷))
5 rnmptpr.e . . . . . 6 (𝑥 = 𝐵𝐶 = 𝐸)
65eqeq2d 2748 . . . . 5 (𝑥 = 𝐵 → (𝑦 = 𝐶𝑦 = 𝐸))
74, 6rexprg 4697 . . . 4 ((𝐴𝑉𝐵𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶 ↔ (𝑦 = 𝐷𝑦 = 𝐸)))
81, 2, 7syl2anc 584 . . 3 (𝜑 → (∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶 ↔ (𝑦 = 𝐷𝑦 = 𝐸)))
9 rnmptpr.f . . . . 5 𝐹 = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐶)
109elrnmpt 5969 . . . 4 (𝑦 ∈ V → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶))
1110elv 3485 . . 3 (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶)
12 vex 3484 . . . 4 𝑦 ∈ V
1312elpr 4650 . . 3 (𝑦 ∈ {𝐷, 𝐸} ↔ (𝑦 = 𝐷𝑦 = 𝐸))
148, 11, 133bitr4g 314 . 2 (𝜑 → (𝑦 ∈ ran 𝐹𝑦 ∈ {𝐷, 𝐸}))
1514eqrdv 2735 1 (𝜑 → ran 𝐹 = {𝐷, 𝐸})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wo 848   = wceq 1540  wcel 2108  wrex 3070  Vcvv 3480  {cpr 4628  cmpt 5225  ran crn 5686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-mpt 5226  df-cnv 5693  df-dm 5695  df-rn 5696
This theorem is referenced by:  sge0pr  46409
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