Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rnmptpr Structured version   Visualization version   GIF version

Theorem rnmptpr 40168
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 3418 . . . . . 6 𝑦 ∈ V
2 rnmptpr.f . . . . . . 7 𝐹 = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐶)
32elrnmpt 5606 . . . . . 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 2836 . . . . . 6 (𝑥 = 𝐴 → (𝑦 = 𝐶𝑦 = 𝐷))
10 rnmptpr.e . . . . . . 7 (𝑥 = 𝐵𝐶 = 𝐸)
1110eqeq2d 2836 . . . . . 6 (𝑥 = 𝐵 → (𝑦 = 𝐶𝑦 = 𝐸))
129, 11rexprg 4455 . . . . 5 ((𝐴𝑉𝐵𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶 ↔ (𝑦 = 𝐷𝑦 = 𝐸)))
136, 7, 12syl2anc 581 . . . 4 (𝜑 → (∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶 ↔ (𝑦 = 𝐷𝑦 = 𝐸)))
141elpr 4421 . . . . . 6 (𝑦 ∈ {𝐷, 𝐸} ↔ (𝑦 = 𝐷𝑦 = 𝐸))
1514bicomi 216 . . . . 5 ((𝑦 = 𝐷𝑦 = 𝐸) ↔ 𝑦 ∈ {𝐷, 𝐸})
1615a1i 11 . . . 4 (𝜑 → ((𝑦 = 𝐷𝑦 = 𝐸) ↔ 𝑦 ∈ {𝐷, 𝐸}))
175, 13, 163bitrd 297 . . 3 (𝜑 → (𝑦 ∈ ran 𝐹𝑦 ∈ {𝐷, 𝐸}))
1817alrimiv 2028 . 2 (𝜑 → ∀𝑦(𝑦 ∈ ran 𝐹𝑦 ∈ {𝐷, 𝐸}))
19 dfcleq 2820 . 2 (ran 𝐹 = {𝐷, 𝐸} ↔ ∀𝑦(𝑦 ∈ ran 𝐹𝑦 ∈ {𝐷, 𝐸}))
2018, 19sylibr 226 1 (𝜑 → ran 𝐹 = {𝐷, 𝐸})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wo 880  wal 1656   = wceq 1658  wcel 2166  wrex 3119  Vcvv 3415  {cpr 4400  cmpt 4953  ran crn 5344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-rex 3124  df-rab 3127  df-v 3417  df-sbc 3664  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4875  df-opab 4937  df-mpt 4954  df-cnv 5351  df-dm 5353  df-rn 5354
This theorem is referenced by:  sge0pr  41403
  Copyright terms: Public domain W3C validator