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Theorem rnmptpr 38353
 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 vex 3176 . . . . . 6 𝑦 ∈ V
2 rnmptpr.f . . . . . . 7 𝐹 = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐶)
32elrnmpt 5293 . . . . . 6 (𝑦 ∈ V → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶))
41, 3ax-mp 5 . . . . 5 (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶)
54a1i 11 . . . 4 (𝜑 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶))
6 rnmptpr.a . . . . 5 (𝜑𝐴𝑉)
7 rnmptpr.b . . . . 5 (𝜑𝐵𝑊)
8 rnmptpr.d . . . . . . 7 (𝑥 = 𝐴𝐶 = 𝐷)
98eqeq2d 2620 . . . . . 6 (𝑥 = 𝐴 → (𝑦 = 𝐶𝑦 = 𝐷))
10 rnmptpr.e . . . . . . 7 (𝑥 = 𝐵𝐶 = 𝐸)
1110eqeq2d 2620 . . . . . 6 (𝑥 = 𝐵 → (𝑦 = 𝐶𝑦 = 𝐸))
129, 11rexprg 4182 . . . . 5 ((𝐴𝑉𝐵𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶 ↔ (𝑦 = 𝐷𝑦 = 𝐸)))
136, 7, 12syl2anc 691 . . . 4 (𝜑 → (∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶 ↔ (𝑦 = 𝐷𝑦 = 𝐸)))
141elpr 4146 . . . . . 6 (𝑦 ∈ {𝐷, 𝐸} ↔ (𝑦 = 𝐷𝑦 = 𝐸))
1514bicomi 213 . . . . 5 ((𝑦 = 𝐷𝑦 = 𝐸) ↔ 𝑦 ∈ {𝐷, 𝐸})
1615a1i 11 . . . 4 (𝜑 → ((𝑦 = 𝐷𝑦 = 𝐸) ↔ 𝑦 ∈ {𝐷, 𝐸}))
175, 13, 163bitrd 293 . . 3 (𝜑 → (𝑦 ∈ ran 𝐹𝑦 ∈ {𝐷, 𝐸}))
1817alrimiv 1842 . 2 (𝜑 → ∀𝑦(𝑦 ∈ ran 𝐹𝑦 ∈ {𝐷, 𝐸}))
19 dfcleq 2604 . 2 (ran 𝐹 = {𝐷, 𝐸} ↔ ∀𝑦(𝑦 ∈ ran 𝐹𝑦 ∈ {𝐷, 𝐸}))
2018, 19sylibr 223 1 (𝜑 → ran 𝐹 = {𝐷, 𝐸})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173  {cpr 4127   ↦ cmpt 4643  ran crn 5039 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-mpt 4645  df-cnv 5046  df-dm 5048  df-rn 5049 This theorem is referenced by:  sge0pr  39287
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