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	⊢ (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		vex 3418	. . . . . 6 ⊢ 𝑦 ∈ V
2		rnmptpr.f	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐶)
3	2	elrnmpt 5606	. . . . . 6 ⊢ (𝑦 ∈ V → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶))
4	1, 3	ax-mp 5	. . . . 5 ⊢ (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶)
5	4	a1i 11	. . . 4 ⊢ (𝜑 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶))
6		rnmptpr.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
7		rnmptpr.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
8		rnmptpr.d	. . . . . . 7 ⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷)
9	8	eqeq2d 2836	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑦 = 𝐶 ↔ 𝑦 = 𝐷))
10		rnmptpr.e	. . . . . . 7 ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸)
11	10	eqeq2d 2836	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑦 = 𝐶 ↔ 𝑦 = 𝐸))
12	9, 11	rexprg 4455	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶 ↔ (𝑦 = 𝐷 ∨ 𝑦 = 𝐸)))
13	6, 7, 12	syl2anc 581	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶 ↔ (𝑦 = 𝐷 ∨ 𝑦 = 𝐸)))
14	1	elpr 4421	. . . . . 6 ⊢ (𝑦 ∈ {𝐷, 𝐸} ↔ (𝑦 = 𝐷 ∨ 𝑦 = 𝐸))
15	14	bicomi 216	. . . . 5 ⊢ ((𝑦 = 𝐷 ∨ 𝑦 = 𝐸) ↔ 𝑦 ∈ {𝐷, 𝐸})
16	15	a1i 11	. . . 4 ⊢ (𝜑 → ((𝑦 = 𝐷 ∨ 𝑦 = 𝐸) ↔ 𝑦 ∈ {𝐷, 𝐸}))
17	5, 13, 16	3bitrd 297	. . 3 ⊢ (𝜑 → (𝑦 ∈ ran 𝐹 ↔ 𝑦 ∈ {𝐷, 𝐸}))
18	17	alrimiv 2028	. 2 ⊢ (𝜑 → ∀𝑦(𝑦 ∈ ran 𝐹 ↔ 𝑦 ∈ {𝐷, 𝐸}))
19		dfcleq 2820	. 2 ⊢ (ran 𝐹 = {𝐷, 𝐸} ↔ ∀𝑦(𝑦 ∈ ran 𝐹 ↔ 𝑦 ∈ {𝐷, 𝐸}))
20	18, 19	sylibr 226	1 ⊢ (𝜑 → ran 𝐹 = {𝐷, 𝐸})

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