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Theorem rnmptpr 44430
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 2737 . . . . 5 (𝑥 = 𝐴 → (𝑦 = 𝐶𝑦 = 𝐷))
5 rnmptpr.e . . . . . 6 (𝑥 = 𝐵𝐶 = 𝐸)
65eqeq2d 2737 . . . . 5 (𝑥 = 𝐵 → (𝑦 = 𝐶𝑦 = 𝐸))
74, 6rexprg 4695 . . . 4 ((𝐴𝑉𝐵𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶 ↔ (𝑦 = 𝐷𝑦 = 𝐸)))
81, 2, 7syl2anc 583 . . 3 (𝜑 → (∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶 ↔ (𝑦 = 𝐷𝑦 = 𝐸)))
9 rnmptpr.f . . . . 5 𝐹 = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐶)
109elrnmpt 5948 . . . 4 (𝑦 ∈ V → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶))
1110elv 3474 . . 3 (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶)
12 vex 3472 . . . 4 𝑦 ∈ V
1312elpr 4646 . . 3 (𝑦 ∈ {𝐷, 𝐸} ↔ (𝑦 = 𝐷𝑦 = 𝐸))
148, 11, 133bitr4g 314 . 2 (𝜑 → (𝑦 ∈ ran 𝐹𝑦 ∈ {𝐷, 𝐸}))
1514eqrdv 2724 1 (𝜑 → ran 𝐹 = {𝐷, 𝐸})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wo 844   = wceq 1533  wcel 2098  wrex 3064  Vcvv 3468  {cpr 4625  cmpt 5224  ran crn 5670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-mpt 5225  df-cnv 5677  df-dm 5679  df-rn 5680
This theorem is referenced by:  sge0pr  45664
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