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Theorem rnmptpr 39877
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 3354 . . . . . 6 𝑦 ∈ V
2 rnmptpr.f . . . . . . 7 𝐹 = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐶)
32elrnmpt 5509 . . . . . 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 2781 . . . . . 6 (𝑥 = 𝐴 → (𝑦 = 𝐶𝑦 = 𝐷))
10 rnmptpr.e . . . . . . 7 (𝑥 = 𝐵𝐶 = 𝐸)
1110eqeq2d 2781 . . . . . 6 (𝑥 = 𝐵 → (𝑦 = 𝐶𝑦 = 𝐸))
129, 11rexprg 4373 . . . . 5 ((𝐴𝑉𝐵𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶 ↔ (𝑦 = 𝐷𝑦 = 𝐸)))
136, 7, 12syl2anc 573 . . . 4 (𝜑 → (∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶 ↔ (𝑦 = 𝐷𝑦 = 𝐸)))
141elpr 4339 . . . . . 6 (𝑦 ∈ {𝐷, 𝐸} ↔ (𝑦 = 𝐷𝑦 = 𝐸))
1514bicomi 214 . . . . 5 ((𝑦 = 𝐷𝑦 = 𝐸) ↔ 𝑦 ∈ {𝐷, 𝐸})
1615a1i 11 . . . 4 (𝜑 → ((𝑦 = 𝐷𝑦 = 𝐸) ↔ 𝑦 ∈ {𝐷, 𝐸}))
175, 13, 163bitrd 294 . . 3 (𝜑 → (𝑦 ∈ ran 𝐹𝑦 ∈ {𝐷, 𝐸}))
1817alrimiv 2007 . 2 (𝜑 → ∀𝑦(𝑦 ∈ ran 𝐹𝑦 ∈ {𝐷, 𝐸}))
19 dfcleq 2765 . 2 (ran 𝐹 = {𝐷, 𝐸} ↔ ∀𝑦(𝑦 ∈ ran 𝐹𝑦 ∈ {𝐷, 𝐸}))
2018, 19sylibr 224 1 (𝜑 → ran 𝐹 = {𝐷, 𝐸})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 836  wal 1629   = wceq 1631  wcel 2145  wrex 3062  Vcvv 3351  {cpr 4319  cmpt 4864  ran crn 5251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-mpt 4865  df-cnv 5258  df-dm 5260  df-rn 5261
This theorem is referenced by:  sge0pr  41125
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