Theorem rnmptpr 46334

Description: Range of a function defined on an unordered pair. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
rnmptpr.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
rnmptpr.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
rnmptpr.f	⊢ 𝐹 = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐶)
rnmptpr.d	⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷)
rnmptpr.e	⊢ (𝑥 = 𝐵 → 𝐶 = 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.

Step	Hyp	Ref	Expression
1		rnmptpr.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		rnmptpr.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
3		rnmptpr.d	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷)
4	3	eqeq2d 2774	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦 = 𝐶 ↔ 𝑦 = 𝐷))
5		rnmptpr.e	. . . . . 6 ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸)
6	5	eqeq2d 2774	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑦 = 𝐶 ↔ 𝑦 = 𝐸))
7	4, 6	rexprg 4658	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶 ↔ (𝑦 = 𝐷 ∨ 𝑦 = 𝐸)))
8	1, 2, 7	syl2anc 596	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶 ↔ (𝑦 = 𝐷 ∨ 𝑦 = 𝐸)))
9		rnmptpr.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐶)
10	9	elrnmpt 5945	. . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶))
11	10	elv 3459	. . 3 ⊢ (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶)
12		vex 3458	. . . 4 ⊢ 𝑦 ∈ V
13	12	elpr 4609	. . 3 ⊢ (𝑦 ∈ {𝐷, 𝐸} ↔ (𝑦 = 𝐷 ∨ 𝑦 = 𝐸))
14	8, 11, 13	3bitr4g 317	. 2 ⊢ (𝜑 → (𝑦 ∈ ran 𝐹 ↔ 𝑦 ∈ {𝐷, 𝐸}))
15	14	eqrdv 2761	1 ⊢ (𝜑 → ran 𝐹 = {𝐷, 𝐸})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∨ wo 861   = wceq 1569   ∈ wcel 2146  ∃wrex 3090  Vcvv 3454  {cpr 4590   ↦ cmpt 5189  ran crn 5656

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-mpt 5190  df-cnv 5663  df-dm 5665  df-rn 5666

This theorem is used by:  sge0pr  47546