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Theorem rnmptpr 41425
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 2832 . . . . 5 (𝑥 = 𝐴 → (𝑦 = 𝐶𝑦 = 𝐷))
5 rnmptpr.e . . . . . 6 (𝑥 = 𝐵𝐶 = 𝐸)
65eqeq2d 2832 . . . . 5 (𝑥 = 𝐵 → (𝑦 = 𝐶𝑦 = 𝐸))
74, 6rexprg 4627 . . . 4 ((𝐴𝑉𝐵𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶 ↔ (𝑦 = 𝐷𝑦 = 𝐸)))
81, 2, 7syl2anc 586 . . 3 (𝜑 → (∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶 ↔ (𝑦 = 𝐷𝑦 = 𝐸)))
9 rnmptpr.f . . . . 5 𝐹 = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐶)
109elrnmpt 5823 . . . 4 (𝑦 ∈ V → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶))
1110elv 3500 . . 3 (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶)
12 vex 3498 . . . 4 𝑦 ∈ V
1312elpr 4584 . . 3 (𝑦 ∈ {𝐷, 𝐸} ↔ (𝑦 = 𝐷𝑦 = 𝐸))
148, 11, 133bitr4g 316 . 2 (𝜑 → (𝑦 ∈ ran 𝐹𝑦 ∈ {𝐷, 𝐸}))
1514eqrdv 2819 1 (𝜑 → ran 𝐹 = {𝐷, 𝐸})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wo 843   = wceq 1533  wcel 2110  wrex 3139  Vcvv 3495  {cpr 4563  cmpt 5139  ran crn 5551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-br 5060  df-opab 5122  df-mpt 5140  df-cnv 5558  df-dm 5560  df-rn 5561
This theorem is referenced by:  sge0pr  42669
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